Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

1.   Introduction

In its original formulation (see ^[54]) the Signorini problem in linear elastostatics consists in finding the equilibrium configuration of an elastic body $\Omega$ resting on a frictionless rigid support $E\!\subset\! \partial \Omega$ in its natural configuration and subject to body forces and surface forces acting on $\partial \Omega\!\setminus\! E$; precisely, if $\mathbf u:\Omega\to\mathbb R^3$ denotes the displacement field of the body, $\mathbb C$ represents the classical linear elasticity tensor and $\mathbb E$ denotes the linear strain tensor, we assume that

is the corresponding strain energy density (see ^[27]) and that the body is subject to a load system of forces ${\mathbf f}:\Omega\rightarrow {{\mathbb R}}^3$ and ${\mathbf g}:\partial\Omega\!\setminus\! E\rightarrow {{\mathbb R}}^3$ such that

is the load potential, where $\mathcal H^2$ is the two-dimensional Hausdorff measure. Assuming that $\mathcal H^2(E)\! > \! 0$, the variational formulation of the Signorini problem consists in finding a minimizer of the functional

among all ${\mathbf{u}}$ in the Sobolev space $H^1(\Omega; \mathbb R^3)$ such that ${\mathbf{u}}\cdot\mathbf n\ge 0$ $\, \mathcal H^2$-a.e. on $E$, where $\mathbf n$ is the inward unit vector normal to $\partial \Omega$. A classical result (see ^[22]) states that a solution of (1.2) exists if the following condition is verified: Every infinitesimal rigid displacement ${\mathbf{v}}$ fulfills $\mathcal L({\mathbf{v}})\!\le\! 0\,$ if $\, {\mathbf{v}}\cdot{\mathbf{n}}\!\ge\! 0$ $\, \mathcal H^2$-a.e. on $E$ and $\mathcal L({\mathbf{v}}) \! = \! 0$ if and only if ${\mathbf{v}}\cdot{\mathbf{n}} \equiv 0$ $\, \mathcal H^2$-a.e. on $E$. Moreover if $E$ is planar, that is $E\subset \partial \Omega\cap\{x_3 = 0\}$, and if $\mathcal L(\mathbf e_3) < 0,$ $\mathbf f\in C^{0, \alpha}(\overline \Omega; {{\mathbb R}}^3)$ and $\mathbf g\in L^2(\partial \Omega\setminus E; {{\mathbb R}}^3)$ then a minimizer of (1.2) exists if and only if the above condition holds (see [22, Theorem XXXII] and ^[10]): In particular if $\Omega$ is the cylinder

with $a\ge 0, \, R > 0, \, H > 0,$ and $\mathbf f\! = \!-\mathbf e_3, \ \mathbf g\! = \!0,$ then a minimum is attained if and only if $aH < 2R$ that is $0\le \vartheta\! : = \! \arctan a < \arctan {2R}/{H}$ where $\vartheta$ is the inclination of the cylinder with respect to the $x_3$-axis.

More recent formulations of constrained problems in the calculus of variations use the notion of capacity (see Section 2 for details) leading to consider more general geometries since any set of null capacity has null $\mathcal H^2$ measure (see [57, Theorem 4]) but there exist sets of null $\mathcal H^2$ measure and strictly positive capacity (see [1, Theorem 5.4.1]). Indeed, a proper generalization of the latter case is to assume that the set $E\subset \{x_3\ge 0\}$ has positive capacity and accordingly modify the obstacle condition by requiring $x_3+u_3({{\bf x}})\ge 0$ on $E$ up to a set of null capacity (shortly, q.e. on $E$): The existence of minimizers for this general setting was proved by [12, Theorem 4.5]. Although the original obstacle formulation given in ^[54] may look different from the generalized notion exploited in this work, it can be shown (see Remark 2.3) that if the set $E\!\subset\!\partial\Omega$ is regular in an appropriate sense (see Remark 2.3) then the two frameworks coincide.

Like in the analysis of many problems in linear elastostatics, it is quite natural to ask whether there exists a sequence of functionals in finite elasticity whose minimizing sequences converge to a minimizer of (1.2), possibly under suitable compatibility conditions on the functional $\mathcal L$: Such an approach provides a variational justification of the linearized theory and could help finding other reasonable models rigorously deduced.

In this paper we show sharp conditions on $\mathcal L$ entailing that a wide class of energy functionals in finite elasticity fulfill this variational property in the context of obstacle problems; in addition we also show examples of loads leading to the failure of this convergence. In this perspective, denoting by $\mathbf y:\Omega\to\mathbb R^3$ the deformation field and by $h > 0$ an adimensional parameter, we introduce a family of energy functionals defined by

where $\mathcal L$ is defined as in (1.1) and $\mathcal W:\Omega\times\mathbb R^{3\times 3}\to[0, +\infty]$ is the strain energy density. For every ${{\bf x}}\in \Omega$, the function $\mathcal W({{\bf x}}, \cdot)$ is assumed to be frame indifferent and attaining its minimum value $0$ at rigid deformations only. We also assume that $\mathcal W$ is $C^2$-regular in a neighborhood of rigid deformations and satisfies a natural coercivity condition, see (2.22).

According to a standard approach in the deduction of linearized theories in continuum mechanics, if $\mathbf y_h$ is a minimizing sequence of $\mathcal F_h$ (see (2.40)) in a class $\mathcal A_h$ of deformations satisfying a suitable obstacle constraint, we aim to investigate whether $\mathcal F({\mathbf{y}}_h)$ converges, as $h$ goes to 0, to the minimum of $\mathcal E$ (with $\mathbb C = D^2\mathcal W({\mathbf{x}}, \mathbf I)$) among displacements fulfilling $u_3({\mathbf{x}})+x_3\ge0$ q.e. on $E$. Since here the aim is the deduction of the Signorini problem in linear elasticity, it is natural to assume that the unilateral constraint in nonlinear approximating problems takes the form $\, x_3+h^{-1}({\mathbf y}_{h, 3}-x_3)\ge 0$ on $E$ that is

We define the functionals $\mathcal G_h$ coupling the energies $\mathcal F_h$ with the unilateral constraint due to rigid obstacle:

where $E$, the portion of the elastic body sensitive to the obstacle, has an horizontal projection with non negligible capacity. Moreover we have to assume that

for every deformation $\mathbf y = \mathbf y({\mathbf{x}})$ fulfilling (1.4) and such that

On the contrary if ${\mathbf{y}}$ satisfied (1.4) and (1.7) but $\mathcal L(\mathbf y-\mathbf x) > 0$ then

Under our assumptions on $\mathcal W$, Eq (1.7) holds true if and only if $\mathbf y$ is rigid deformation, i.e., $\mathbf y({{\bf x}}) = \mathbf R {{\bf x}}+\mathbf c$ for some $\mathbf R\!\in\! SO(3)$ and $\mathbf c\in \mathbb R^3$, while (1.4) is fulfilled by these ${\mathbf{y}}$ if and only if $(\mathbf R{\mathbf{x}})_3+c_3\ge (1-h)x_3$ on $E$, a condition which is satisfied for every $h\! > \!0$ if and only if

Thus, due to (1.6) we have to assume this geometrical compatibility between load and obstacle

Nevertheless though (1.9) entails the compatibility assumptions of Theorem 4.5 in ^[12] and though compatibility assumptions of Theorem 4.5 in ^[12] entail the existence of minimizers of the Signorini functional in linear elasticity, these compatibility assumptions alone do not warrant the equiboundedness from below of approximating functionals $\mathcal G_h$ (see Example 3.6 below). In the main result of this paper (Theorem 2.4) we show that if $\mathcal L$ satisfies the necessary condition (1.9) together with $\mathcal L(\mathbf e_3) < 0$ and

then, under some capacitary assumptions on $E$ (see (2.13)), we have

where

Along this paper $E_{ess}$ denotes the essential part of $E$ with respect to the capacity (see (2.9)), a closed canonical subset of $\overline E$ such that $E\setminus E_{ess}$ has null capacity.

Under the hypotheses detailed previously we will show (see Lemma 3.8) that either $\mathcal S_{\mathcal L, E}\equiv \{\mathbf I\}$ or $\mathcal S_{\mathcal L, E}\, = \, \left\{ \mathbf R\in SO(3): \mathbf R\mathbf e_3 = \mathbf e_3 \right\}.$ If $\mathcal S_{\mathcal L, E}\!\equiv\! \{\mathbf I\}$ then clearly ${\mathcal G}\!\equiv\! \mathcal E$, hence in this case the minimum of Signorini problem in linearized elasticity is the limit of the $\inf \mathcal G_h$ but, quite surprisingly, the second alternative is much more subtle and indeed we are able to exhibit examples such that

namely a gap between $\lim_{h\to 0}(\inf\mathcal G_h)$ and $\min\mathcal E$ may appear (see Section 5). However the coincidence of minimizers of $\mathcal G$ and $\mathcal E$ may hold true even if $\mathcal S_{\mathcal L, E}$ is not reduced to the identity matrix: In particular, if $\Omega$ is contained in the upper half-space, $E$ is either $\overline\Omega$ or $\partial\Omega$, the load

satisfies condition (1.9) and $\mathcal L(\mathbf e_3) < 0$, then $\mathcal L({\mathbf{v}}) = \mathcal L(\mathbf R{\mathbf{v}})$ for every $\mathbf R\in \mathcal S_{\mathcal L, E}$ hence $\min \mathcal G\! = \!\min \mathcal E,$ say the energy of minimizing sequences for $\mathcal G_h$ converges to the minimum energy of $\mathcal E$. On the other hand, it is always possible to rotate the external forces in such a way that $\mathcal G_{\mathbf R}$ and $\mathcal E_{\mathbf R}$ (the functionals obtained replacing the load functional $\mathcal L$ with $\mathcal L_{\mathbf R}$ defined by $\mathcal L_{\mathbf R}({\mathbf{v}}): = \mathcal L (\mathbf R{\mathbf{v}})$) have the same minimum as shown in Theorem 5.5.

For several contributions facing issues strictly connected with the context of the present paper we refer to ^{[3,4,5,6,8,9,11,13,14,15,16,17,18,19,21,23,25,26,27,28,29,32,33,34,35,36,37,38,39,40,41,42,43,46,47,48,49,50,51,52,53,55,56]}.

2.   Notation and main results

We set $a^+\!\!: = \max\{a, 0\}, \ a^-\!\!: = \max\{ -a, 0\}$ for every $a\in {{\mathbb R}}$; notations ${\mathbf{x}} = (x_1, x_2, x_3)$ and ${\mathbf{y}} = (y_1, y_2, y_3)$ represent generic points in ${{\mathbb R}}^3$; $\mathbf e_j$, $j = 1, 2, 3$ denote the unitary basis vectors of ${{\mathbb R}}^3$, $\mathbb R^{3\times 3}$ is the set of $3\times 3$ real matrices, endowed with the Euclidean norm $|\mathbf F| = \sqrt{ {{\rm{Tr}}}(\mathbf F^T\mathbf F)}$. $\mathbb R^{3\times 3}_{\rm sym}$ (resp. $\mathbb R^{3\times 3}_{\rm skew})$ denotes the subset of symmetric (resp. skew-symmetric) matrices. For every $\mathbf F\in \mathbb R^{3\times 3}$ we define ${\rm sym\, }\mathbf F: = \frac{1}{2}(\mathbf F+\mathbf F^T),$ $SO(3)$ will denote the special orthogonal group and for every $\mathbf R\!\in\! SO(3)$ there exist $\vartheta\in [0, 2\pi]$ and $\mathbf a \in \mathbb R^3, \ |\mathbf a| = 1$ such that the following Euler-Rodrigues representation formula holds

For every compact set $K\subset\mathbb R^N$ we define the capacity of $K$ by setting (see [1, Definition 2.2.1])

If $G\subset {{\mathbb R}}^N$ is open we define (see [1, Definition 2.2.2])

and, since (see [1, Proposition 2.2.3])

we may extend the above definitions to an arbitrary set by setting (see [1, Definition 2.2.4])

A straightforward consequence of (2.3) and (2.5) is that

On the other hand, for every $E\subset {{\mathbb R}}^N$ the Bessel capacity is defined as (see [1, Definition 2.3.3])

where $g_1\!\in\! L^1({{\mathbb R}}^3)$ is the first-order Bessel kernel in ${{\mathbb R}}^N$ defined as the inverse Fourier transform of $(1+|\boldsymbol\xi|^2)^{-1/2}$, say

Notice that since $f\ge 0$ a.e. we have that $f*g_1$ is everywhere defined if we allow it to take the value $+\infty$ (see [1, Definition 2.3.1]) and that $f*g_1$ is l.s.c. by Proposition 2.3.2 of ^[1]. Thus inequality $(f*g_1)({{\bf x}})\ge 1$ for every ${{\bf x}}\in E$ appearing in formula (2.7) has a precise meaning.

In addition it is possible to show that there exist two constants $\alpha, \beta > 0$ such that

(see [1, Definition 2.2.6 and Proposition 2.3.13]).

A property is said to hold quasi-everywhere (q.e. for short) if it holds true outside a set of zero capacity. It is convenient to introduce (see ^[12]) a canonical representative of the set $E$, called the essential part of $E$ and denoted by $E_{ess}$, which nevertheless coincides with $E$ itself whenever it is a smooth closed manifold or the closure of an open subset of $\mathbb R^N$.

For every set $E\subset{{\mathbb R}}^3$ we define the essential part $E_{ess}$ of $E$ (with respect to the capacity) by

It has been shown in ^[12] that

In the following ${\mathop{{{\rm{co}}}}\nolimits} A$, ${\mathop{{{\rm{aff}}}}\nolimits} A$, ${\mathop{{{\rm{ri}}}}\nolimits} A$, ${\rm r}\partial A$ and ${\mathop{{{\rm{proj}}}}\nolimits} A$ denote respectively, the closed convex hull of the set $A\!\subset\! {{\mathbb R}}^3$ (say, the intersection of all convex sets containing $A$), the affine hull of the set $A$ (say, the smallest affine space containing $A$), the relative interior of $A$ (say, the interior part of $A$ with respect to the affine hull of $A$), the relative boundary of $A$ (say, the boundary of $A$ with respect to the affine hull of $A$: ${\mathop{{{\rm{ri}}}}\nolimits}\partial A = \overline A\setminus{\mathop{{{\rm{ri}}}}\nolimits} A$) and the projection of $A$ onto the horizontal plane $\{x_3 = 0\}$.

Throughout the paper we will assume that

Notice that ${\mathop{{{\rm{cap}}}}\nolimits} E\! > \!0$ does not imply (2.13) whereas the converse is true: Indeed, if ${\mathop{{{\rm{cap}}}}\nolimits} E\! = \!0$ then by (2.12) we get $E_{ess}\! = \!\emptyset$ so ${\mathop{{{\rm{proj}}}}\nolimits} ({\mathop{{{\rm{co}}}}\nolimits} E_{ess})\! = \emptyset$ and ${\mathop{{{\rm{cap}}}}\nolimits} ({\mathop{{{\rm{proj}}}}\nolimits} ({\mathop{{{\rm{co}}}}\nolimits} E_{ess}))\! = \! 0$, a contradiction to (2.13).

In the following $\Omega$ will denote the reference configuration of an elastic body and it is always assumed to be a nonempty, bounded, connected, Lipschitz open set in $\mathbb R^3$. We need to show that any function in the Sobolev space $H^{1}(\Omega; {{\mathbb R}}^3)$ actually has a precise representative defined quasi-everywhere on the whole $\overline\Omega$ with respect to the capacity. Indeed, if ${\mathbf{uu}}\!\in\! H^{1}(\Omega; {{\mathbb R}}^3)$ and ${\mathbf{v}}\!\in\! H^{1}({{\mathbb R}}^3;{{\mathbb R}}^3)$ is a Sobolev extension of ${\mathbf{uu}}$, it is well known (see [1, Proposition 6.1.3]) that the limit

exists for q.e. ${{\bf x}}\!\in\!{{\mathbb R}}^3.$ The function ${{\mathbf{v}}}^*$ is called the precise representative of ${\mathbf{v}}$ and is a quasicontinuous function in ${{\mathbb R}}^3$, that is to say, for every ${\varepsilon} > 0$ there exists an open set $V\subset {{\mathbb R}}^3$ such that ${\mathop{{{\rm{cap}}}}\nolimits} V < {\varepsilon}$ and ${{\mathbf{v}}}^*$ is continuous in ${{\mathbb R}}^3\!\setminus\! V$. We claim that if ${\mathbf{v}}_1, \ {\mathbf{v}}_2$ are two distinct Sobolev extensions of ${\mathbf{uu}}$ then

The claim is trivial for q.e. ${\mathbf{x}}\in \Omega$, thus we are left to show (2.15) for q.e. ${\mathbf{x}}\in \partial \Omega$.

Ler $R > 0$ such that $\overline \Omega\subset B_R(0)$ and let $\Omega_R: = B_R(0)\setminus \overline \Omega$. Since $\Omega$ has Lipschitz boundary it is well known (see ^[2]) that

and for $\mathcal H^2$ a.e. ${\mathbf{x}}\in \partial \Omega$,

where we have denoted again with ${\mathbf{uu}}$ the trace of ${\mathbf{uu}}$ on $\partial \Omega$. Hence

so, by taking account $\partial \Omega\subset\partial \Omega_R$ and by recalling that $\partial \Omega$ is Lipschitz, we may apply Theorem 2.1 of ^[20] to ${\mathbf{v}}_1-{\mathbf{v}}_2\in H^1(\Omega_R; {{\mathbb R}}^3)$ and we get

Since ${\mathbf{v}}_1 = {\mathbf{v}}_2 = {\mathbf{uu}}$ a.e. in $B_r({\mathbf{x}})\!\setminus\! \Omega_R$ the claim follows easily by (2.14). Therefore if ${\mathbf{uu}}\in H^{1}(\Omega; {{\mathbb R}}^3)$ we may define its precise representative for quasi-every ${\mathbf{x}}$ on $\overline \Omega$ by

where ${\mathbf{v}}$ is any Sobolev extension of ${\mathbf{uu}}$.

The function ${{\mathbf{uu}}}^*$ is pointwise quasi-everywhere defined by (2.18) and is quasicontinuous on $\overline \Omega$ i.e., for every ${\varepsilon} > 0$ there exists a relatively open set $V\subset \overline \Omega$ such that ${\mathop{{{\rm{cap}}}}\nolimits} V < {\varepsilon}$ and ${{\mathbf{uu}}}^*$ is continuous in $\overline \Omega\setminus V$.

2.1.   The elastic energy density

Let ${\mathcal L}^3$ and ${\mathcal B}^3$ denote respectively the $\sigma\mbox{-algebras}$ of Lebesgue measurable and Borel measurable subsets of ${{\mathbb R}}^3$ and let $\mathcal W : \Omega \times \mathbb R^{3 \times 3} \to [0, +\infty ]$ be ${\mathcal L}^3\! \times\! {\mathcal B}^{9}$-measurable satisfying the following assumptions, see also ^[3,45]:

and as far as it concerns the regularity of $\mathcal W$, we assume that there exist an open neighborhood $\mathcal U$ of $SO(3)$ in ${{\mathbb R}}^{3\times3}$, an increasing function $\omega:\mathbb R_+\to\mathbb R$ satisfying $\lim_{t\to0^+}\omega(t) = 0$ and a constant $K > 0$ such that for a.e. ${{\bf x}}\in \Omega$

Moreover we assume that there exists $C > 0$ such that for a.e. ${{\bf x}}\in\Omega$

where $d(\, \cdot\, ,SO(3))$ denotes the Euclidean distance function from the set of rotations.

The frame indifference assumption (2.19) implies that there exists a function $\mathcal V$ such that

By (2.19)–(2.21), for a.e. ${{\bf x}}\in\Omega$, we have $\mathcal W({{\bf x}}, \mathbf R) = D\mathcal W({{\bf x}}, \mathbf R) = \mathbf 0$ for any $\mathbf R\in SO(3).$ By (2.23), for a.e. ${{\bf x}}\in \Omega$, given $\mathbf B\in\mathbb R^{3\times 3}$ and $h > 0$, we have

and (2.20), (2.21) together imply

Hence, by (2.22) and polar decomposition ^[27], we obtain, for a.e. ${{\bf x}}\!\in\! \Omega$ and every $\mathbf B\!\in\!\mathbb R^{3\times3}$, eventually as $h\to 0_+$ (since $\det(\mathbf I+h\mathbf B)\! > \!0$ for small $h$)

Moreover, as noticed also in ^[44], by expressing the remainder of Taylor's expansion in terms of the ${{\bf x}}$-independent modulus of continuity $\omega$ of $D^2 {\mathcal W}({{\bf x}}, \cdot)$ on the set $\mathcal U$ from (2.21), we have

for any small enough $h$ (such that $h\mathbf B\in\mathcal U$). Similarly, $\mathcal V({{\bf x}}, \cdot)$ is $C^2$ in a neighborhood of the origin in $\mathbb R^{3\times 3}$, with an ${{\bf x}}$-independent modulus of continuity $\eta:\mathbb R_+\to \mathbb R$, which is increasing and such that $\lim_{t\to0^+}\eta(t) = 0$, and we have

for any small enough $h$.

We mention a general class of energy densities ${\mathcal W}$ (the so called Yeoh materials) fulfilling the assumptions above (2.19)–(2.22) and for which the main result of the present paper (see Theorem 2.4 below) applies.

Example 2.1. For simplicity, we consider the homogeneous case and assume that a standard isochoric-volumetric decomposition of elastic energy density by setting

where ${\mathcal W}_{ \rm{iso}}$ is an energy density of Yeoh type which is defined by choosing

with coefficients $c_k > 0$ and $\mathcal{W}_{ \rm{vol}}(\mathbf{F}) = g(\det\mathbf{F})$ for some convex $g\in C^2({{\mathbb R}}_{+})$ such that

It is easy to check that with this choice the energy density satisfies all assumptions from (2.19) to (2.21) while inequality (2.22) has been proven in ^[43].

It is worth noticing that when material constants are suitably chosen then also Ogden-type energies may fulfil assumptions (2.19)–(2.22) and we refer to ^[43] for all details.

2.2.   External forces

We introduce a body force field $\mathbf f\in L^{6/5}(\Omega, {{\mathbb R}}^3)$ and a surface force field $\mathbf g\in L^{4/3}(\partial\Omega, {{\mathbb R}}^3)$. From now on, $\mathbf f$ and $\mathbf g$ will always be understood to satisfy these summability assumptions. The load functional is the following linear functional

We note that since $\Omega$ is a bounded Lipschitz domain, the Sobolev embedding $H^{1}(\Omega, \mathbb R^3)\hookrightarrow L^{6}(\Omega, \mathbb R^3)$ and the Sobolev trace embedding $H^{1}(\Omega, \mathbb R^3)\hookrightarrow L^{4}(\partial\Omega, \mathbb R^3)$ imply that $\mathcal L$ is a bounded functional over $H^{1}(\Omega, \mathbb R^3)$ and throughout the paper we denote its norm with $\|\mathcal L\|_*$.

For every $\mathbf R\!\in\! SO(3)$ we set

and, as we have observed in the Introduction, we must assume the following geometrical compatibility between load and obstacle

together with

the summation convention over repeated index $\alpha = 1, 2$ being understood all along this paper. It can be shown that condition (2.31) is equivalent to (see Remark 3.4 below)

where we have set

and from now on we will use (2.33) in place of (2.31). On the other hand Remark 4.5 below will show that also condition (2.32) is in fact unavoidable.

2.3.   Energy functionals

If $E\!\subset\!\overline \Omega\!\subset\! \{\mathbf x\!: x_3\!\ge\! 0\}$, the classical Signorini problem in linear elasticity can be described as the minimization of the functional $\mathcal E: H^1(\Omega, \mathbb R^3)\to\mathbb R\cup\{+\infty\}$ defined by

where $\mathbb E({\mathbf{u}}): = \! {\mathop{{{\rm{sym}}}}\nolimits} \nabla{\mathbf{u}}$, ${\mathcal Q} ({{\bf x}}, \mathbf F)\! = \tfrac12\, \mathbf F^T\mathbb C\, \mathbf F$ with $\mathbb C = D^2\mathcal W({{\bf x}}, \mathbf I)$ and $\mathcal A$ denotes the set of admissible displacements, defined by

The meaning of such constraint is that, if the portion $E$ of the elastic body is contained in $\{x_3\!\ge\! 0\}$ in the reference configuration, then the deformed configuration of $E$, namely $\{\mathbf y({\mathbf{x}})\!: = {\mathbf{x}}+{\mathbf{u}}({\mathbf{x}}), \, {\mathbf{x}}\!\in\! E\}$, is constrained to remain in $\{y_3\ge 0\}$.

For every $\mathbf y\in H^1(\Omega, \mathbb R^3)$ we introduce the set

Thus, due to the rigidity inequality of ^[24], there exists a constant $C = C(\Omega) > 0$ such that for every ${\mathbf{y}}\in H^1(\Omega, \mathbb R^{3})$ and every $\mathbf R\in \mathcal M(\mathbf y)$

where $d\big(\mathbf F, SO(3)\big): = \min\{|\mathbf F-\mathbf R|:\mathbf R\in SO(3)\}$.

We introduce the set of admissible deformations $\mathcal A_h$ as

and the rescaled finite elasticity functionals $\mathcal G_h: H^1(\Omega, {{\mathbb R}}^3)\to {{\mathbb R}}\cup\{+\infty\}$ by setting

It is readily seen that, for every $\mathbf R\in SO(3)$ and for every $\mathbf c\in {{\mathbb R}}^3$ such that

the map ${\mathbf{y}} ({\mathbf{x}}): = \mathbf R{\mathbf{x}}+\mathbf c$ belongs to $\mathcal A_h$ for every $h > 0$. In the sequel we use the short notations $\mathcal G_j: = \mathcal G_{h_j}$ and $\mathcal A_j: = \mathcal A_{h_j}$ whenever $\{h_j\}_{j\in \mathbb N}$ is a sequence of strictly positive real numbers such that $h_j\to 0^+$ as $j\to +\infty$.

We say that $(\mathbf y_j)_{j\in\mathbb N}\!\subset\! H^1(\Omega, \mathbb R^3)$ is a minimizing sequence of the sequence of functionals $\mathcal G_j$ if

The main focus of the paper is to investigate whether minimizers of (2.34) can be approximated by minimizing sequences of the sequence of functionals $\mathcal G_j$, as defined by (1.5) and (2.40).

To this end we introduce the functionals $\mathcal I, \ \widetilde{\mathcal G}, \ \mathcal G :H^1(\Omega, \mathbb R^3)\to\mathbb R\cup\{+\infty\}$ defined by

and

where

Remark 2.2. It is worth noticing that $\mathcal G\le \mathcal E$, since $\mathbf I\in\mathcal S_{\mathcal L, E}$ and it is straightforward checking that $\mathcal I, \ \widetilde{\mathcal G}, \ \mathcal G$ are all continuous with respect to the strong convergence in $H^1(\Omega; \mathbb R^3)$.

Before stating the main result in Theorem 2.4, we show the next Remark with some insight on technicalities implied by precise obstacle formulation in the Sobolev space $H^1(\Omega)$.

Remark 2.3. If $w\in H^1(\Omega)$ then $w^-\!\in\! H^1(\Omega)$ too. Moreover, both $(w^-)^*$ and $(w^*)^-$ are quasicontinuous in $\overline \Omega$ and $(w^-)^* = (w^*)^- = w^-$ a.e. in $\Omega$. Then, by ^[30], $(w^-)^* = (w^*)^-$ q.e. in $\overline \Omega$. Therefore the condition $(w^-)^* = 0$ q.e. in $E_{ess}$ is equivalent to $w^*\ge 0$ q.e. in $E_{ess}$.

In particular we claim that

is equivalent to

Indeed if $w\ge 0$ a.e. in $\Omega$ then $(w^-)^* = 0$ a.e. in $\Omega$ and hence $(w^-)^* = 0$ q.e. in $\Omega$.

If $w\ge 0\ $ $\mathcal H^2\,$a.e. on $\partial \Omega$ and $v$ is a Sobolev extension of $w^-$ then

and by taking (2.16) into account we get

By recalling that $\Omega$ is a Lipschitz set, it is easily checked that $\partial \Omega$ is Ahlfors $2$-regular, that is there are constants $c_1, c_2 > 0$ such that

for every $0\! < \! r \! < \! \hbox{diam}(\partial\Omega)$ and for every ${\mathbf{x}}\in\partial \Omega$. Therefore if we choose $R\! > \! 0$ such that $\overline \Omega\!\subset\! B_R(0)$ we may apply Proposition 6.1.3. of ^[1] and Theorem 2.1 of ^[20] both in $H^1(\Omega)$ and in $H^1(B_R(0)\setminus\overline \Omega)$ and we get

that is (2.46) implies (2.45).

Conversely if (2.45) holds then $(w^-)^* = 0$ a.e. in $\Omega$ and $\mathcal H^2$ a.e. on $\partial \Omega$. Therefore $w\ge 0$ a.e. in $\Omega$ and by recalling again that the negative part of the trace of $w$ and the trace of its negative part coincide $\mathcal H^2$ a.e. on $\partial \Omega$ we get $w\ge 0\,$ $\mathcal H^2$ a.e. on $\partial \Omega$ thus proving (2.46) and the claim.

Similarly, again by Theorem 2.1 of ^[20], if $E_{ess}\subset \partial \Omega$ is Ahlfors $2$-regular then the condition $w\ge 0\ $ q.e. on $E$ is equivalent to $w\ge 0\ \ \mathcal H^2$ a.e. on $E,$ so the classical framework of ^[12,31,54] is equivalent to ours in this case as it was claimed in the Introduction.

2.4.   The convergence result

The convergence result is stated in the next theorem, referring to (2.36) and (2.40).

Theorem 2.4. Assume (2.13), (2.19)–(2.22), (2.32)–(2.33) and $\mathcal L(\mathbf e_3) < 0$. Let $h_j\to 0^+$ as $j\to+\infty$ and let $(\overline{\mathbf y}_j)_{j\in\mathbb N}\!\subset\! H^1(\Omega, \mathbb R^3)$ be a minimizing sequence of $\mathcal G_j$. If $\mathbf R_j\!\in\!\mathcal M(\overline{\mathbf y}_j)$ for every $j\in\mathbb N$, then there are $\overline{\mathbf c}_j\!\in\! \mathbb R^3$ such that the sequence

is weakly compact in $H^1(\Omega, \mathbb R^3)$. Therefore up to subsequences, $\overline{{\mathbf{uu}}}_j{\rightharpoonup} \overline{\mathbf{u}}$ in $H^1(\Omega, \mathbb R^3)$ and also

Remark 2.5. Since $\widetilde{\mathcal G}\le \mathcal G$ then equality $\min\widetilde{\mathcal G} = \min\mathcal G$ is equivalent to ${\mathop{{{\rm{argmin}}}}\nolimits} \mathcal G\subset {\mathop{{{\rm{argmin}}}}\nolimits} \widetilde{\mathcal G}$ with possible strict inclusion (see Remark 5.6), thus in general $\overline{\mathbf{u}}$ may not belong to ${\mathop{{{\rm{argmin}}}}\nolimits} \mathcal G$.

Remark 2.6. Conditions (2.32) and (2.33) are compatible. Indeed set

and $\mathbf f\! = {\bf 0}, \ \mathbf g\! = \!-\mathbf e_3 {\mathbf 1}_{E}.$ It is readily seen that $\mathcal L(\mathbf e_3) < 0 = \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2)$ and

thus both (2.32) and (2.33) are fulfilled.

On the other hand condition (2.32) does not entail (2.33). Indeed if $\Omega$ and $E$ are as in (2.50), $\mathbf f\! = -\mathbf e_3\ \mathbf g\! = 0$ then is not since if $\overline{\mathbf R} = \mathbf e_1\otimes \mathbf e_1-\mathbf e_2\otimes \mathbf e_2-\mathbf e_3\otimes \mathbf e_3$ a direct calculation yields

Eventually (2.33) does not imply (2.32), see Remark 4.5.

Example 2.7. Here we show an example where the all the assumptions in Theorem 2.4 concerning the geometry of the material body $\Omega$ and its portion $E$ sensitive to the constraint and their compatibility with the loads are fulfilled. Set

Then $\mathcal L({\mathbf{u}}) = \int_\Omega \mathbf f\cdot {\mathbf{u}}\, d{{\bf x}}$, condition (2.33) is satisfied and $\mathcal S_{\mathcal L, E} = \{\mathbf R\in SO(3): \mathbf R\mathbf e_3 = \mathbf e_3\}$.

Indeed it is readily seen that $\mathcal L(\mathbf e_3) = p| \Omega| < 0 = \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2)$; moreover if $\mathbf R\in SO(3)$ and we denote its entries as $R_{ij}\ i, j = 1, 2, 3$ then, taking into account $p < 0$, we get

and $\Phi(\mathbf R, E, \mathcal L) = 0$ if and only if $R_{33} = 1$ that is $\mathbf R\mathbf e_3 = \mathbf e_3$ as claimed.

Both conditions (2.13) and (2.32) are trivially fulfilled.

We emphasize that the above claims still hold true if the assumption on $E$ in (2.52) is weakened by allowing any $E\subset\Omega$ such that $E$ fulfills ${\mathop{{{\rm{co}}}}\nolimits} E_{ess} = \overline\Omega.$

3.   Properties of admissible loads

This section makes explicit the properties of admissible loads by exploiting the conditions stated by (2.32) and (2.33).

Lemma 3.1. Assume that (2.32) holds. Then

Proof. By the Euler-Rodrigues formula (2.32) entails

for every $\vartheta \in (0, 2\pi)$ and by letting $\vartheta\to 0^+$ we get $\mathcal L ((\mathbf a\wedge {{\bf x}})_{\alpha}\mathbf e_\alpha)\le 0$ for every $\mathbf a\in \mathbb R^3$ hence $\mathcal L ((\mathbf a\wedge {{\bf x}})_{\alpha}\mathbf e_\alpha) = 0$ for every $\mathbf a\in \mathbb R^3$. The second inequality in (3.1) follows by the previous one. □

Remark 3.2. It is worth noticing that, by inserting $\mathbf a = \mathbf e_1$ or $\mathbf a = \mathbf e_2$, the condition (3.1) entails $\mathcal L(x_3\mathbf e_2) = 0$ and $\mathcal L(x_3\mathbf e_1) = 0$ respectively.

Lemma 3.3. Assume (2.13) and (2.31). Then

(1) $\, \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0$ and $\mathcal L(\mathbf e_3) \le 0;$

(2) $\, \mathcal L(\mathbf e_3\wedge {\mathbf{x}}) = 0;$

(3) $\, \mathcal L\big(\mathbf e_3\wedge (\mathbf e_3\wedge{\mathbf{x}})\big) \le 0;$

(4) there exists ${\mathbf{x}}_{\mathcal L} \in {\mathop{{{\rm{ri}}}}\nolimits} {\mathop{{{\rm{co}}}}\nolimits} E_{ess}$ such that $\ \mathcal L\big(\, \mathbf a \wedge ({\mathbf{x}}-{\mathbf{x}}_{\mathcal L})\big) = 0$ $\ \forall\mathbf a\in{{\mathbb R}}^3$.

Proof. By choosing $\mathbf R = \mathbf I$ in (2.31) we get $\mathcal L(\mathbf c)\le 0$ for every $\mathbf c\in \mathcal C_{\mathbf I} = \{\mathbf c\!\in\! {{\mathbb R}}^3\!: c_3\ge 0\}$. Since $c_1\, \mathbf e_1+c_2\, \mathbf e_2\in C_{\mathbf I}$ for every $c_1, c_2\in {{\mathbb R}}$, we get $\mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0$. Moreover $c_3\, \mathbf e_3\in C_{\mathbf I}$ for $c_3\ge 0$ entails $\mathcal L(\mathbf e_3) \le 0$. Thus (1) is proved and (2.31) entails

that is by the Euler-Rodrigues formula

If $\mathbf a\! = \!\mathbf e_3$ then $\mathbf R\mathbf e_3\! = \!\mathbf e_3$ and (3.4) reads

By $\varphi(0) = \varphi(2\pi) = 0$ and $\varphi(\vartheta)\le 0$ we get $0\ge \varphi'(0) = \mathcal L(\mathbf e_3\wedge {\mathbf{x}}) = \varphi'(2\pi)\ge 0$. Thus (2) is proved. By (2) and (3.5) we get $\mathcal L\big(\mathbf e_3 \wedge (\mathbf e_3 \wedge {\mathbf{x}})\big)\le 0$, say (3). In order to show (4), first we notice that $\mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0$ entail for every ${\boldsymbol{\xi}}\in{{\mathbb R}}^3$

moreover, $\mathcal L(\mathbf e_3\wedge {\mathbf{x}}) = 0$ entails

Let us assume first that $\mathcal L(\mathbf e_3) < 0$. In this case we can set

hence, by (3.6)–(3.8),

say

Since $\varphi_{\mathbf a}(0) = \varphi_{\mathbf a}(2\pi) = 0$ and $\varphi_{\mathbf a}(\vartheta)\le 0$ for every $\mathbf a\in{{\mathbb R}}^3, |\mathbf a| = 1$ and for every $\vartheta \in [0, 2\pi],$ (3.4) entails

Hence

so, by (3.6), (3.8) and (3.9),

By taking account of $\mathcal L(\mathbf e_3) < 0$, we find

hence, by linearity and by homogeneity,

By subtracting $(\mathbf a \wedge \widetilde{\mathbf{x}})_3$ on each term of inequality (3.13) we get

for every $\mathbf a\in{{\mathbb R}}^3$ and for every $\widetilde {\mathbf{x}}\in \{(\widetilde x_1, \widetilde x_2, z): z\in \mathbb R\}$.

We claim that at least one of the above inequalities is strict for every $\mathbf a\in \mathbb R^3$ such that $a_1^{\, 2}+a_2^{\, 2}\neq 0$. Indeed, if by contradiction there was $\overline {\mathbf a} = (\overline a_1, \overline a_2, \overline a_3)$ with $\overline a_1^{\, 2}+\overline a_2^{\, 2}\neq 0$ such that

then

Since the plane $\{\overline a_1x_2-\overline a_2x_1 = 0\}$ is orthogonal to $\{x_3 = 0\}$ we obtain

hence

which contradicts (2.13).

Without loss of generality we can proceed by assuming that the first inequality is strict, say

for every $\mathbf a\in \mathbb R^3$ such that $a_1^{\, 2}+a_2^{\, 2}\neq 0$ and for every $\widetilde{\mathbf{x}}\in \{(\widetilde x_1, \widetilde x_2, z), \, z\in{{\mathbb R}}\}$. Hence, by setting $\, T: = {\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess}-\widetilde {\mathbf{x}})$, we get

for every $\mathbf a\in \mathbb R^3$ such that $a_1^{\, 2}+a_2^{\, 2}\neq 0$. For every $(a_1, a_2)\in \mathbb R^2$ such that $a_1^{\, 2}+a_2^{\, 2}\neq 0$ we set now

then $\big\{\, \Gamma(a_1, a_2):\ a_1^{\, 2}+a_2^{\, 2} = 1 \, \big\}$ is the set of half-planes supporting $T$. Since $T$ is closed and convex, we get

By (3.14), we get

Hence we deduce the existence of $(\widetilde a_1, \widetilde a_2) :\ \widetilde a_1^{\, 2}+\widetilde a_2^{\, 2} = 1$ such that

so $(0, 0)\, \in \, {\mathop{{{\rm{ri}}}}\nolimits} T$ that is $(\widetilde x_1, \widetilde x_2, 0)\in {\mathop{{{\rm{ri}}}}\nolimits}{\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess})$.

We are left to show that there exists $\widetilde x_3$ such that ${\mathbf{x}}_{\mathcal L}: = (\widetilde x_1, \widetilde x_2, \widetilde x_3) \!\in\! {\mathop{{{\rm{ri}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess}$. To this aim it is readily seen that by taking account of ${\mathop{{{\rm{cap}}}}\nolimits}({\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess})) > 0$, we get ${\mathop{{{\rm{aff}}}}\nolimits}({\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess}))) = \{x_3 = 0\}$ so there exists $\overline r > 0$ such that

Let now

and assume that $(x_1, x_2, z)\in \, ({\mathop{{{\rm{co}}}}\nolimits} E_{ess})\!\setminus ({\mathop{{{\rm{ri}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess})$ for every $z\in J$. Then

for every $z\in J$ and for every $r > 0$, therefore by recalling that ${\mathop{{{\rm{aff}}}}\nolimits}{\mathop{{{\rm{proj}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess}\, = \, \{x_3 = 0\}$

for every $r > 0$. This is a contradiction since

for some $r > 0$ thus (4) is proved in this case since by construction $\mathcal L(\mathbf a\wedge ({\mathbf{x}}-{\mathbf{x}}_{\mathcal L}) = 0$ for every $\mathbf a \in \mathbb R^3$. Eventually, if $\mathcal L(\mathbf e_3) = 0$, (2.33) reduces to

say $\sin\vartheta\, \mathcal L(\mathbf a \wedge {\mathbf{x}})+(1-\cos\vartheta) \mathcal L\big(\mathbf a\wedge(\mathbf a \wedge {\mathbf{x}})\big)\le 0$ for all $\mathbf a\in {{\mathbb R}}^3$, thus, by repeating the analysis made on (3.5), we get

and, since ${\mathop{{{\rm{ri}}}}\nolimits} {\mathop{{{\rm{proj}}}}\nolimits} {\mathop{{{\rm{co}}}}\nolimits} E_{ess} \neq\emptyset$ due to (2.13), by exploiting identity (3.6) with $\boldsymbol \xi = \widetilde {\mathbf{x}}$ we obtain, for whatever choice of $\widetilde{\mathbf{x}}\in{\mathop{{{\rm{ri}}}}\nolimits} {\mathop{{{\rm{proj}}}}\nolimits} {\mathop{{{\rm{co}}}}\nolimits} E_{ess}$

that is (4) is proven also in this case. □

Remark 3.4. Conditions (2.31) and (2.33) are equivalent as claimed in Subsection 2.2.

Indeed, as it has been pointed out in the proof of Lemma 3.3, condition (2.31) implies that $\mathcal L(\mathbf e_3)\le 0, \ \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0$ and

Conversely if the latter condition holds and $\mathcal L(\mathbf e_3)\le 0, \ \ \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0$, then

for every $\mathbf c\in {{\mathbb R}}^3$ such that $c_3\ge -\min_{{{\bf x}}\in E_{ess}}((\mathbf R{{\bf x}})_3-x_3)\}$.

Remark 3.5. We emphasize that conditions (1) and (4) in Lemma 3.3 together with (2.13) and $\mathcal L(\mathbf e_3) < 0$ coincide with conditions (4.9)–(4.11) of Theorem 4.5 of ^[12], which provides the solution to Signorini problem in linear elasticity.

The whole set of conditions (1)–(4) appearing in the claim of Lemma 3.3 together with condition (2.13) on the set $E$ is not equivalent to admissibility of the loads as expressed by (2.33): This phenomenon is made explicit by subsequent Example 3.6.

Example 3.6. Let $\Omega = \{{\mathbf{x}}:\, x_1^2+x_2^2 < 1, \, 0\! < \!x_3\! < \!H\}$, $E\! = \!E_{ess}\! = \!\{\, (x_1, x_2, 0)\!:\, x_1^2+x_2^2\!\le\! 1\}$ and $\mathcal L({\mathbf{v}}) = \!\int_\Omega p\, v_3\, d{\mathbf{x}}$ with $p < 0$, say $\mathbf f = p\, \mathbf e_3, \ \mathbf g = \mathbf 0$.

Then $E$ fulfills (2.13), since ${\mathop{{{\rm{cap}}}}\nolimits} E > 0$ and ${\mathop{{{\rm{proj}}}}\nolimits} (\, {\mathop{{{\rm{co}}}}\nolimits} E_{ess}) = E_{ess}\, \subset\, \overline{\Omega}\, \cap \{x_3 = 0\}$; moreover all claims (1)–(4) of Lemma 3.3 hold true: Indeed

eventually, by choosing ${\mathbf{x}}_{\mathcal L} = (0, 0, 0)\in {\mathop{{{\rm{ri}}}}\nolimits}\ ({\mathop{{{\rm{co}}}}\nolimits} E_{ess}) = E$ and by taking the symmetry of $\Omega$ into account, we get

Nevertheless, condition (2.33) is violated, since we can consider the $\pi$ radians rotation around axis $\mathbf e_1$ which keeps $E$ above the horizontal plane (obstacle boundary) but capsizes the body below the horizontal plane, namely $\widetilde{\mathbf R}\in SO(3)$ defined by $\widetilde{\mathbf R}{\mathbf{x}} = x_1\mathbf e_1-x_2\mathbf e_2-x_3\mathbf e_3$. Thus

The assumptions in Lemma 3.3 and in Theorem 2.4 cannot be weakened by assuming only ${\mathop{{{\rm{cap}}}}\nolimits} E > 0$ in place of (2.13) as it is shown in the next example, thus proving that Theorem 2.4 is a sharp result with respect to the sets $E$ subject to the constraint that are admissible.

Example 3.7. Choose $\mathbf f = -\mathbf e_3, \ \mathbf g = \mathbf 0$ and

It is readily seen that condition (2.32) is fulfilled since $\mathcal L((\mathbf R{{\bf x}}-{{\bf x}})_{\alpha}\mathbf e_\alpha) = 0$ moreover, since $\mathcal L(\mathbf e_3) < 0 = \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2),$

for every $\mathbf R\in SO(3),$ then also condition (2.33) is satisfied. Nevertheless it can be easily checked that ${\mathop{{{\rm{cap}}}}\nolimits} E > 0$ but ${\mathop{{{\rm{cap}}}}\nolimits} \big({\mathop{{{\rm{proj}}}}\nolimits} (\, {\mathop{{{\rm{co}}}}\nolimits} E_{ess}\, )\big) = 0$, thus $E$ does not fulfil (2.13).

If there was $\overline{\mathbf{x}}\in {\mathop{{{\rm{ri}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess}$ such that $\mathcal L(\mathbf a\wedge({\mathbf{x}}-\overline {\mathbf{x}})) = 0$ for every $\mathbf a\in \mathbb R^3$ then we get

then, since $\mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0$, we could apply (3.6) and find

hence $\overline x_1 = 0, \ \overline x_2 = \frac 2 {3\pi}$, thus

a contradiction. Thus claim (4) of Lemma 3.3 cannot hold true in this case.

Moreover if we set $\mathbf w_k({{\bf x}}): = -k\, (\, x_2\, \mathbf e_3-x_3\, \mathbf e_2\, ),$ then $\mathbb E(\mathbf w_k) = {\bf 0}, \ w_{k, 3}+x_3\equiv x_3\ge 0$ on $E$ whence $\mathbf w_k\in \mathcal A$ and it is readily seen that

hence

that is $\inf_{\mathcal A}\mathcal G = -\infty$ so the convergence of the energies claimed in Theorem 2.4 fails to be true in this case thus showing sharpness of condition (2.13).

Lemma 3.8. Assume that (2.13), (2.33) hold and that $\mathcal L(\mathbf e_3) < 0$. Then

Proof. First, we prove the inclusion

Indeed if $\mathbf R$ belongs to $\mathcal S_{\mathcal L, E}$ and $\mathbf a$ is a rotation axis of $\mathbf R$ with $|\mathbf a| = 1$, then

for every $\vartheta\in[0, 2\pi]$. By arguing now as in the proof of (4) of Lemma 3.3, we get

and, since $\mathcal L(\mathbf e_3) < 0,$ we get

that is the function

is constant on $E_{ess}$ hence it is constant in ${\mathop{{{\rm{co}}}}\nolimits}(E_{ess})$ thus ${\mathop{{{\rm{cap}}}}\nolimits} {\mathop{{{\rm{proj}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess} > 0$ entails $a_1 = a_2 = 0$ that is $\mathbf R\mathbf e_3 = \mathbf e_3$ as claimed.

We notice that $\mathbf I\in \mathcal S_{\mathcal L, E}$ and, the other hand, if $\mathcal S_{\mathcal L, E}\not\equiv\{\, \mathbf I\, \}$ then there is

such that $\widetilde {\mathbf R}\not = \mathbf I$ and

for every ${\mathbf{x}}\in{{\mathbb R}}^3$ and for some suitable $\widetilde\vartheta\in(0, 2\pi)$. By taking (2) of Lemma 3.3 into account, we get

thus $\ \mathcal L \big(\mathbf e_3\wedge (\mathbf e_3\wedge {\mathbf{x}})\big) = 0$.

Therefore for any other $\mathbf R\in SO(3), \ \mathbf R\neq\mathbf I$ such that $\mathbf R\mathbf e_3 = \mathbf e_3$ there is $\vartheta\in(0, 2\pi)$ such that

thus, by taking again (2) of Lemma 3.3 into account, we get

that is $\mathbf R$ belongs to $\mathcal S_{\mathcal L, E}$ thus concluding the proof of the lemma. □

Remark 3.9. It is possible to show that both alternatives in Lemma 3.8 can actually occur. Indeed in Example 2.7 we have exhibited an example in which $\mathcal S_{\mathcal L, E} = \{\mathbf R\in SO(3): \mathbf R\mathbf e_3 = \mathbf e_3\}$ and we show here that also the other alternative may occur. Indeed set

where $\partial_{l} \Omega$ is the lateral boundary of $\Omega$ and $\mathbf n$ the unit outward vector normal to $\partial_{l} \Omega$. If $\mathbf R\in SO(3)$ and we denote its entries as $R_{ij}\ i, j = 1, 2, 3,$ then

that is condition (2.32) is satisfied. Moreover since

and

and equality holds if and only if $R_{11} = R_{22} = R_{33} = 1$ then condition (2.33) is satisfied and $\mathcal S_{\mathcal L, E}\equiv \{\mathbf I\}$ as claimed.

Lemma 3.10. Assume (2.13), (2.33) and $\mathcal L(\mathbf e_3) < 0$. Let $\mathbf R_j\in SO(3)$ be a sequence of rotations such that $\, \mathbf R_j\mathbf e_3\neq\mathbf e_3$, $\, \mathbf R_j\mathbf e_3\to \mathbf e_3$ as $j\to +\infty$. Then

Proof. $\Phi (\mathbf R_j, E, \mathcal L)\le 0$, by (2.33). Hence the $\limsup$ in (3.24) cannot be strictly positive.

We assume by contradiction

By Euler-Rodrigues formula there are sequences $\mathbf a_j\in {{\mathbb R}}^3$ and $\vartheta_j\in [0, 2\pi]$, such that $|\mathbf a_j| = 1$ and

thus a direct computation yields

By taking account of $\mathbf R_j\mathbf e_3\!\neq\!\mathbf e_3$ and $\mathbf R_j\mathbf e_3\to \mathbf e_3$ as $j\!\to\! +\infty$, we get $\mathbf a_j\neq \mathbf e_3$, $\vartheta_j\!\in\! (0, 2\pi)$ and therefore, up to subsequences, we may assume: that $\mathbf a_j\to \mathbf a, \ \vartheta_j\to \vartheta\in [0, 2\pi],$ that either $\vartheta\in \{0, 2\pi\}$ or $a_3 = 1$ and that $\mu_j\, {a_{i, j}}\to\, \alpha_i, \ i = 1, 2$ with $\alpha_1^2+\alpha_2^2 = 1$, where we have set

For every ${\mathbf{x}}\in\Omega$ and ${\mathbf{v}}\in \mathbb R^2, |{\mathbf{v}}| = 1,$ set

where $(\widetilde x_1, \widetilde x_2, \widetilde x_3) = {\mathbf{x}}_{\mathcal L}\in {\mathop{{{\rm{ri}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess}$ is chosen as in the proof of (4) in Lemma 3.3. Hence, by taking account of (1) and (4) of Lemma 3.3, we have

that is

By (2) of Lemma 3.3 we know $\mathcal L(\mathbf e_3\wedge {\mathbf{x}}) = 0$, then (3.8) entails

and by (3) of Lemma 3.3 we have

By taking (3.29) into account and by recalling that either $\vartheta\in \{0, 2\pi\}$ or $a_3 = 1$, we get

that is

Let now $\eta\in C([0, 2\pi])$ such that $\eta(\vartheta) = \left (2(1-\cos\vartheta)\right)^{-\frac{1}{2}}\sin (\vartheta)$ for every $\vartheta\in (0, 2\pi)$.

By recalling that either $a_3 = 1$ or $\vartheta \in \{0, 2\pi\}$ we get

so, by taking (3.30)–(3.32) into account we obtain

since $(\widetilde x_1, \widetilde x_2, \widetilde x_3)\in {\mathop{{{\rm{ri}}}}\nolimits}{\mathop{{{\rm{co}}}}\nolimits} E_{ess}$. Therefore the function

attains its minimum on ${\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess})$ at $(\widetilde x_1, \widetilde x_2)\in {\mathop{{{\rm{ri}}}}\nolimits} ({\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess}))$ hence, by taking into account of ${\mathop{{{\rm{aff}}}}\nolimits} ({\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess})) = \{x_3 = 0\},$ we get

a contradiction. Thus

and the proof is achieved. □

Remark 3.11. If ${\mathop{{{\rm{cap}}}}\nolimits} ({\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess})) = 0$ then the claim of Lemma 3.10 may be false even if ${\mathop{{{\rm{cap}}}}\nolimits} E > 0$. For instance, set for every $j\in \mathbb N\setminus \{0\}$

let

and $\mathbf f: = -\mathbf e_3, \ \mathbf g = \mathbf 0.$ It is straightforward checking that $\mathbf R_j\mathbf e_3\neq \mathbf e_3,$ $\mathbf R_j \to \mathbf I$, moreover since $\mathcal L(\mathbf e_3) < 0 = \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2),$

and

conditions (2.32) and (2.33) are satisfied. Nevertheless

and the claim of Lemma 3.10 cannot be true in this case.

4.   Proof of the variational convergence result

This section contains the proof of our main result. We start by showing that sequences of deformations with equibounded energy correspond (up to suitably tuned rotations and translations of the horizontal components) to displacements that are equibounded in $H^1$.

Lemma 4.1. (compactness) Assume that $E$, $\mathcal L$ and $\mathcal W$ fulfil (2.13), (2.19)–(2.22), (2.33) and $\mathcal L(\mathbf e_3) < 0$. If $\, 0 < h_j\to 0^+$ as $j\to +\infty$ then for every ${\mathbf{y}}_j\in H^1(\Omega; \mathbb R^3)$ with $\mathcal G_j({\mathbf{y}}_j)\le M < +\infty$ there are $\mathbf R_j\in SO(3), \ \mathbf c_j\in \mathcal C_{\mathbf R_j}$ such that

and the sequence

is bounded in $H^1(\Omega; \mathbb R^3)$.

Proof. Referring to (2.36), we can choose ${\mathbf R}_j\in \mathcal M({\mathbf{y}}_j)$ in such a way that, up to subsequence and without relabelling, ${\mathbf R}_j\to\mathbf R$. Then we define $\mathbf c_j = (c_{j, 1}, c_{j, 2}, c_{j, 3})$ by

By the rigidity inequality (^[24]) there exists a constant $C = C(\Omega) > 0$ such that

Thus, by (2.33), $\mathcal L(\mathbf e_1)\! = \!\mathcal L(\mathbf e_2)\! = \!0$ and the definition of $c_{j, 3}$, we get

and Poincaré inequality entails, for every $\varepsilon\! > \!0$,

Estimates (4.5) and (4.6) together with Young inequality provide

By choosing $\varepsilon = C/ (C_P+1)$, we get

Thus, if we show that $\ h_j^{-1} \| ({\mathbf{y}}_j-\mathbf R_j{\mathbf{x}}-\mathbf c_j)_3 \|_{L^2(\Omega)}\,$ is uniformly bounded then, due to estimate (4.8), $\|h_j^{-1}(\nabla{\mathbf{y}}_j-\mathbf R_j)\|_{L^2(\Omega)}$ is equibounded too. So $\, h_j^{-1} \, ({\mathbf{y}}_j-\mathbf R_j{\mathbf{x}}-\mathbf c_j)\,$ is uniformly bounded in $H^{1}(\Omega; {{\mathbb R}}^3)$ and we set

To this aim we assume by contradiction that, up to subsequences,

and set $w_j \ : = t_j^{-1} h_j^{-1} \, ({\mathbf{y}}_j-\mathbf R_j{\mathbf{x}}-\mathbf c_j)_3.$ Then

and by choosing $\varepsilon = 2C/C_P$ in (4.12) we get

while, by choosing $\varepsilon = C/C_P$, (4.12) yields

Thus

Normalization $\|w_j\|_{L^2} = 1$ entails, for every $\varepsilon > 0$,

and choosing $\varepsilon = C\, t_j^2$ therein we get, by (4.13),

thus $\int_\Omega |\nabla w_j|^2\, \, d{\mathbf{x}}\to 0$ so by (4.11) $w_j\to w$ in $H^1(\Omega; {{\mathbb R}}^3)$ with $\nabla w = 0$ a.e. in $\Omega$ that is $w$ is a constant function since $\Omega$ is a connected open set.

Combining estimates (4.15)–(4.17), we get

hence

and

Moreover, by (4.13), we get

Hence, due to $\mathcal L (w_j\, \mathbf e_3)\to \mathcal L (w\, \mathbf e_3) = w\, \mathcal L(\mathbf e_3)$, we have $w\, \mathcal L (\mathbf e_3)\!\ge\! 0,$ thus, by taking into account of $\mathcal L (\mathbf e_3) < 0,$ we get $w\le 0$ and eventually, by $\|w_j\|_{L^2} = 1$, we obtain $w\ < \ 0$. Then, by (2.33), (4.4) and (4.9), we get

Hence, due to $\mathbf R_j\to \mathbf R,$ we have $\Phi(\mathbf R, E, \mathcal L) = 0$ thus $\mathbf R\in \mathcal S_{\mathcal L, E}$ and (4.1) is proven.

We notice that either $\mathbf R_j\mathbf e_3\not = \mathbf e_3$ for $j$ large enough or $\mathbf R_j\mathbf e_3 = \mathbf e_3$ for infinitely many $j$. In the first case, by taking account of $\mathcal L(\mathbf e_3) < 0,$ Lemma 3.10 entails

By (4.21) we get $\Phi(\mathbf R_j, E, \mathcal L)\ge - (M+M_1) h_j$ hence

and for large enough $j$ (4.23) yields

Therefore, for every ${\mathbf{x}}\in\Omega$

hence, by taking into account of $c_{j, 3}\! = \!-\min\, \left\{\, (\mathbf R_j{\mathbf{x}})_3-{\mathbf{x}}_3\, :\, {\mathbf{x}}\in E_{ess}\, \right\}$, we get for q.e. ${\mathbf{x}}\!\in\! E$

a contradiction since ${\mathbf{y}}_{j, 3}^*\ge (1-h_j)x_3$ for q.e. ${\mathbf{x}}\in E$, that is $(h_j\, t_j)^{-1}\big({\mathbf{y}}_{j, 3}^*-{\mathbf{x}}_3\big)\ge -\, x_3/t_j\longrightarrow 0$ as $j\to +\infty$ for q.e. ${\mathbf{x}}\in E_{ess}$. Therefore in this case the sequence $t_j$ is bounded so $h_j^{-1}\left({\mathbf{y}}_j-\mathbf R_j{\mathbf{x}}-\mathbf c_j\right)$ is equibounded in $H^1(\Omega; {{\mathbb R}}^3)$ and in particular $h_j^{-1}\left({\mathbf{y}}_j-\mathbf R_j{\mathbf{x}}-\mathbf c_j\right)_{\alpha}\mathbf e_{\alpha}$ is equibounded in $H^1(\Omega; {{\mathbb R}}^3)$.

In the second case we may assume that $\mathbf R_j\mathbf e_3 = \mathbf e_3$ for every $j$ so $c_{j, 3} = 0$ for every $j$. By arguing as in the previous case we may assume that

and by setting $w_j \ : = t_j^{-1}h_j^{-1} \, (y_{j, 3}- x_3)$ we get $w_j\to w < 0$ as before which is again a contradiction, so $t_j$ is a bounded sequence. Eventually we are left to show that, in the first case, $h_j^{-1}(y_{j, 3}-x_3)$ is equibounded in $H^1(\Omega; {{\mathbb R}}^3)$. To this aim let $C > 0$ such that

for every $j\in \mathbb N$ and assume that for every $n\in \mathbb N$ there exists $j_n$ such that

Then for every $n > C$ we have $\mathbf R_{j_n}\mathbf e_3\not = \mathbf e_3$ otherwise $c_{j_n, 3} = 0$ and

a contradiction. By taking account of (4.23) and (4.25) there exists $\widetilde C > 0$ such that

for every $n > C$, hence

thus showing that

which contradicts (4.27) and proves that $h_j^{-1}(y_{j, 3}-x_3)$ is equibounded in $H^1(\Omega; {{\mathbb R}}^3)$. □

Remark 4.2. If ${\mathop{{{\rm{cap}}}}\nolimits} ({\mathop{{{\rm{proj}}}}\nolimits}({\mathop{{{\rm{co}}}}\nolimits} E_{ess}))\! = \!0$, the claim of Lemma 4.1 may fail even if ${\mathop{{{\rm{cap}}}}\nolimits} E > 0.$ Indeed, choose $\Omega, E, \mathbf f, \mathbf g, \mathbf R_j$ as in Remark 3.11 and set $h_j = j^{-1}.$ Thus both (2.32) and (2.33) are satisfied but (2.13) is not. It is readily seen that ${\mathbf{y}}_j({\mathbf{x}}): = \mathbf R_j{\mathbf{x}}$ belongs to $\mathcal A_j$ since $y_{j, 3} = (1-j^{-1})x_3$ on $E$ and that $\mathcal G_j({\mathbf{y}}_j) = \pi|\, \Omega|/2$ for every $j$, but

is not equibounded in $H^1(\Omega; {{\mathbb R}}^3)$ as $j\!\to\!+\infty$: thus claim of Lemma 4.1 fails in this case.

Lemma 4.3. Assume that $E$, $\mathcal L$ and $\mathcal W$ fulfil conditions (2.13), (2.19)–(2.22), (2.33) and $\mathcal L(\mathbf e_3) < 0.$ Choose ${\mathbf{y}}_j, \ \mathbf R_j$ as in Lemma 4.1 and set

Then there exist $b_1, \ b_2, \ b_3\in \mathbb R$ such that by setting

we have, up to subsequences, ${\mathbf{z}}_j {\rightharpoonup} {\mathbf{z}}$ in $w^*-W^{1, \infty}(\Omega; \mathbb R^3)$.

Proof. We may assume that $\mathbf R_j\mathbf e_3\neq \mathbf e_3$ for infinitely many $j$ otherwise ${\mathbf{z}}_j\equiv 0$ for $j$ large enough and thesis is obvious. Therefore by Euler-Rodrigues formula, there are sequences $\mathbf a_j\in {{\mathbb R}}^3$ and $\vartheta_j\in (0, 2\pi)$, s.t. $|\mathbf a_j| = 1, \ \mathbf a_j\neq \mathbf e_3$ and

By recalling (3.27) and (4.1) we have, up to subsequences, $\mathbf R_j\mathbf e_3\to \mathbf e_3$. Then, up to subsequences, we may assume: that $\mathbf a_j\to \mathbf a, \ \vartheta_j\to \vartheta\in [0, 2\pi],$ that either $\vartheta\in \{0, 2\pi\}$ or $a_3 = 1$ and that $\mu_j\, {a_{i, j}}\to\, \alpha_i, \ i = 1, 2$ with $\alpha_1^2+\alpha_2^2 = 1$, where we set $\mu_j: = (a_{1, j}^{\, 2}+a_{2, j}^{\, 2})^{-\frac{1}{2}}$. By recallling (4.24) we may assume that, up to subsequences,

for some $\beta\in \mathbb R$. Moreover by exploiting (3.27), (3.26), we get

where $\eta\in C([0, 2\pi])$ is such that $\eta(\vartheta) = \left (2(1-\cos\vartheta)\right)^{-\frac{1}{2}}\sin\vartheta$ for every $\vartheta\in (0, 2\pi)$.

By arguing as in (3.32) and by taking (4.24) into account we get, up to subsequences,

for some $\lambda\ge 0$. On the other hand $\nabla{\mathbf{z}}_j = h_j^{-1}(\mathbf R_j\mathbf e_3-\mathbf e_3)$ and (4.24) entail $\|\nabla{\mathbf{z}}_j\|_{\infty}\le C$ for some $C > 0$ so ${\mathbf{z}}_j{\rightharpoonup} {\mathbf{z}}$ in $w^*-W^{1, \infty}(\Omega:\mathbb R^3)$ whenever $b_1 = \lambda(-\alpha_{2}\eta(\vartheta)+\alpha_1\sin\frac{\vartheta}{2}), \ b_2 = \lambda(\alpha_{1}\eta(\vartheta)+\alpha_2\sin\frac{\vartheta}{2}), \ b_3 = -\beta.$ □

Lemma 4.4. (Lower bound) Assume that $E$, $\mathcal L$, $\mathcal W$ fulfil the conditions (2.13)–(2.22), (2.32), (2.33) and $\mathcal L(\mathbf e_3) < 0$. If $h_j\to 0^+$ as $j\to +\infty$ then, for every sequence of deformations ${\mathbf{y}}_j\in H^1(\Omega; \mathbb R^3)$ such that $\mathcal G_j({\mathbf{y}}_j)\le M < +\infty$ and for every $\mathbf R_j\in \mathcal M({\mathbf{y}}_j)$ there exist $\mathbf c_j\in \mathcal C_{\mathbf R_j}$ such that by setting

there is ${\mathbf{u}}\in\mathcal A$ such that up to subsequences ${\mathbf{u}}_j{\rightharpoonup} {\mathbf{u}}$ weakly in $H^1(\Omega; {{\mathbb R}}^3)$ and

Proof. Due to Lemma 4.1, the sequence defined in (4.34) is equibounded in $H^1(\Omega; \mathbb R^3)$ hence there exists ${\mathbf{u}}\in H^1(\Omega; \mathbb R^3)$ such that up to subsequences ${\mathbf{u}}_j{\rightharpoonup}{\mathbf{u}}$ in $H^1(\Omega; \mathbb R^3)$. By recalling Lemma A1 of ^[12] we get, again up to subsequences, ${\mathbf{u}}_j^*({\mathbf{x}})\to {\mathbf{uu}}^*({\mathbf{x}})$ for q.e. ${\mathbf{x}}\in E$ hence by taking account of

for q.e. ${\mathbf{x}}\in E$ we get $u_{3}^*\ge -x_3$ for q.e. ${\mathbf{x}}\in E$ that is ${\mathbf{uu}}\in \mathcal A$.

By taking account of $\mathcal G_j({\mathbf{y}}_j)\le M$ and by arguing as in Lemma 4.1 the sequence $h_j^{-1}({\mathbf{y}}_j-\mathbf R_j-\mathbf c_j)$ is bounded in $H^1(\Omega; \mathbb R^3)$ hence (4.4) entails

Therefore, by recalling (4.2), (4.3) and that, up to subsequences, $\mathbf R_j\to \mathbf R$ we get $\mathbf R\in \mathcal S_{\mathcal L, E}$. By defining ${\mathbf{z}}_j$ as in Lemma 4.3 and by setting

a straightforward calculation shows that

If now

we immediately notice that, by Tchebycheff inequality, $| \Omega\setminus B_j|\to 0$ as $j\to +\infty$ and that for large enough $j$

Moreover by Lemma 4.3 there exists $C > 0$ such that

hence by defining ${\mathbf{z}}$ as in (4.30) and by taking account of $\mathbf R_j\to\mathbf R\in \mathcal S_{\mathcal L, E}$, we get

By taking account of $\mathcal L(\mathbf e_\alpha) = 0$ for $\alpha = 1, 2$, (2.32) entails

thus, since $\eta$ is increasing, by (2.21)–(2.23), (2.25), (4.37)–(4.40) we get for large $j$

Since $h_{j}\nabla{\mathbf{u}}_{j}^T\nabla{\mathbf{u}}_j\to 0$ a.e. in $\Omega$ and $|{\mathbf 1}_{B_j}h_{j}\nabla{\mathbf{u}}_{j}^T\nabla{\mathbf{u}}_j| \le 1,$ by taking account of $| \Omega\setminus B_j|\to 0$ as $j\to +\infty$ we get ${\mathbf 1}_{B_j}h_{j}\nabla{\mathbf{u}}_{j}^T\nabla{\mathbf{u}}_j{\rightharpoonup} 0$ weakly in $L^2(\Omega, \mathbb R^{3\times3})$. By taking account of ${\mathbf 1}_{B_j}\nabla{\mathbf{u}}_j{\rightharpoonup} \nabla{\mathbf{u}}$ weakly in $L^2(\Omega, \mathbb R^{3\times3})$ and (4.40), we then obtain

weakly in $L^2(\Omega, \mathbb R^{3\times 3})$. Since $\mathbf R_j\to\mathbf R\in \mathcal S_{\mathcal L, E}$, then

and by (2.21) and (4.42), the weak $L^2(\Omega, \mathbb R^{3\times3})$ lower semicontinuity of the map $\mathbf B\mapsto\int_\Omega \mathcal Q({{\bf x}}, \mathbf B)d{{\bf x}}$ entails

which, by recalling (4.30), ends the proof. □

Remark 4.5. If condition (2.32) is not satisfied then the thesis of Lemma 4.4 may fail. Indeed let $\mathbf f: = -\mathbf e_3+6(x_3-\frac{1}{2})\mathbf e_1, \ \mathbf g = \mathbf 0$ and

It is straightforward checking that $\mathcal L(\mathbf e_3) < 0 = \mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2)$ and

for every $\mathbf R\in SO(3)$. On the other hand if $R_{13} > 0$ we have

so (2.33) is satisfied while (2.32) is not. Choose now $h_j: = j^{-1},$

and set ${\mathbf{y}}_j: = \mathbf R_j{{\bf x}}+j^{-1}\sqrt{2 -j^{-2}}\mathbf e_3$. It is readily seen that

hence ${\mathbf{y}}_j\in \mathcal A_j$ and by taking (4.2) into account we get $({\mathbf{y}}_j-\mathbf R_j{{\bf x}}- \mathbf c_j)_{\alpha} \equiv 0, \ \alpha = 1, 2$. Therefore bearing in mind that $\mathbf R_j^T\to \mathbf I$ we have

and by Lemma 3.8 we get $\mathbf R{\mathbf{u}} = {\mathbf{u}}$ for every $\mathbf R\in \mathcal S_{\mathcal L, E},$ hence

On the other hand by taking account of

and

it is straightforward checking that

thus proving that the claim of Lemma 4.4 fails in this case.

Lemma 4.6. (Upper bound) Assume (2.13), (2.19)–(2.22), (2.32), (2.33)$, \mathcal L(\mathbf e_3)\! < \!0$ and let $0 < h_j\to 0^+$ as $j\to +\infty$. For every ${\mathbf{u}}\in C^1(\overline\Omega, \mathbb R^3)$ there exists $\widetilde{{\mathbf{y}}}_j\in C^1(\overline\Omega, \mathbb R^3)$ such that

Proof. We assume without loss of generality that ${\mathbf{u}}\in \mathcal A$ and let

It is readily seen that $\widetilde{\mathbf{u}}\in \mathcal A$, that $\mathbb E(\widetilde{\mathbf{u}}) = \mathbb E({\mathbf{u}})+\frac{1}{2}b_{\alpha}^*(\mathbf e_{\alpha}\otimes\mathbf e_3+\mathbf e_{3}\otimes\mathbf e_\alpha)$ hence

Moreover, by (3.1) of Lemma 3.1 and Remark 3.2 we obtain

Therefore by choosing

we get

By setting $\widetilde{\mathbf y}_j: = \widetilde{\mathbf R}({{\bf x}}+h_j\widetilde{\mathbf{u}})$, taking account $\mathcal S_{\mathcal L, E}\subset \{\mathbf R: \mathbf R\mathbf e_3 = \mathbf e_3\}$ and Lemma 3.8, we get $\mathcal L(\widetilde{\mathbf R}{{\bf x}}-{{\bf x}}) = 0$ and for q.e. ${\mathbf{x}}\in E$

Therefore $\widetilde{{\mathbf{y}}}_j\in \mathcal A_j$ and by (2.24) we get

which proves the lemma. □

We are now in a position to prove our main theorem.

Proof of Theorem 2.4. If $(\overline{\mathbf y}_j)_{j\in\mathbb N}\subset H^1(\Omega, \mathbb R^3)$ is a minimizing sequence for $\mathcal G_j$ then $\mathcal G_j(\overline{\mathbf y}_j)\le \mathcal G_j(\mathbf x) = 0,$ moreover if $\mathbf R_j$ belong $\mathcal A(\overline{\mathbf y}_j)$ and $\overline{\mathbf c}_j$ is defined by (4.2) and (4.3), then Lemma 4.1 entails that the sequence

is bounded in $H^1(\Omega; \mathbb R^3)$. Therefore up to subsequences ${\overline{\mathbf{u}}}_j\to \overline{\mathbf{u}}$ weakly in $H^1(\Omega; {{\mathbb R}}^3),$ so, by Lemma 4.4, we have $\overline{\mathbf{u}}\in \mathcal A$ and

On the other hand, by Lemma 4.6, for every ${\mathbf{u}}\in C^1(\overline\Omega, \mathbb R^3)\cap\mathcal A$ there exists a sequence $\mathbf y_j\in C^1(\overline\Omega, \mathbb R^3)$ such that

Since

by passing to the limit as $j\to +\infty$, we get

Now fix a generic ${\mathbf{u}}\!\in\! \mathcal A$ and denote again by ${\mathbf{u}}$ a Sobolev extension of ${\mathbf{u}}$ to the whole $\mathbb R^3$. We claim that there exists ${\mathbf{u}}_j\in C^1(\overline\Omega, \mathbb R^3)\cap\mathcal A$ such that ${\mathbf{u}}_j\to {\mathbf{u}}$ in $H^1(\Omega; \mathbb R^3)$: Indeed, since $\widetilde u_3({\mathbf{x}})+x_3\ge 0$ for q.e. ${\mathbf{x}}\in E$, by Lemma A.4 it is enough to choose $u_{3, j}: = \eta_j-x_3$ where $\eta_j\in C^1(\mathbb R^3), \ \eta_j\ge 0\ \text{q.e. in}\ E, \ \eta_j \to u_3+x_3$ in $H^1(\mathbb R^3)$ (here $u_3+x_3$ denotes also an extension to the whole $H^1(\mathbb R^3)$) and $u_{\alpha, j}: = u_\alpha*\rho_j, \ \alpha = 1, 2$ where $\rho_j$ is a sequence of smooth mollifiers. By (4.50) we have

whence by Remark 2.2,

that is $\overline{\mathbf{u}}\in{\mathop{{{\rm{argmin}}}}\nolimits} \widetilde{\mathcal G}$.

We show that ${\mathcal G}_j(\mathbf y_j)\to \widetilde{\mathcal G}(\overline{\mathbf{u}})$: By Lemma A.4 in the Appendix, for every ${\varepsilon} > 0$ there is $\overline{\mathbf{u}}_{\varepsilon}\in C^1(\overline \Omega; \mathbb R^3))\cap\mathcal A$ such that

and by Lemma 4.6 there exists $\mathbf y_{j, {\varepsilon}}\in C^1(\overline \Omega; \mathbb R^3)$ such that by taking account of (4.49) we have

for every ${\varepsilon} > 0$. Since by Lemma 4.4,

we get ${\mathcal G}_j(\mathbf y_j)\to \widetilde{\mathcal G}(\overline{\mathbf{u}})$ as claimed.

We are only left to show that $\min\widetilde{\mathcal G} = \min\mathcal G.$ To this aim we show first that for every ${\mathbf{u}}\in \mathcal A$ there exists ${\mathbf{u}}_*\in \mathcal A$ such that $\mathcal G({\mathbf{u}}_*) = \widetilde{\mathcal G}({\mathbf{u}})$. Indeed if $\widetilde{\mathbf{u}}$ is defined as in (4.46) then by (4.47) and (4.48) we get

as claimed. By recalling that $\widetilde{\mathcal G}(\overline{\mathbf{u}}) = \min \widetilde{\mathcal G}\le \inf \mathcal G$ let us assume that inequality is strict. Then by (4.51) there exists ${\overline{\mathbf{u}}}_*\in \mathcal A$ such that $\mathcal G({\overline{\mathbf{u}}}_*) = \widetilde{\mathcal G}(\overline{\mathbf{u}}) < \inf \mathcal G$, a contradiction. Thus again by (4.51) $\mathcal G({\overline{\mathbf{u}}}_*) = \widetilde{\mathcal G}(\overline{\mathbf{u}}) = \min \mathcal G$. □

5.   The gap with Signorini problem

In this section we will exhibit a choice of energy density $\mathcal W$, open set $\Omega$, dead loads $\mathbf f, \mathbf g$ and set $E\!\subset\! \overline \Omega$ fulfilling all the assumptions of Theorem 2.4 but such that the minimum of the limit functional $\mathcal G$ is strictly less than the minimum of the Signorini functional (see ^[45] for a counterexample exhibiting an analogous gap for unconstrained pure traction problem).

We shall consider the energy density already defined in (2.26) by setting

where ${\mathcal W}_{ \rm{iso}}$ is the energy density of Yeoh type defined in (2.27) with $2c_1 = \mu > 0$ and $\mathcal W_{ \rm{vol}}(\mathbf F) = g(\det\mathbf F)$ where $g:\mathbb R_+\to\mathbb R$ is the convex $C^2$ function (satisfying (2.28) with $r = 2$) defined by

By recalling Example 2.1 it is readily seen that $\mathcal W$ satisfies (2.14)–(2.17) and by taking into account of

and ${{\rm{Tr}}} (\mathbf B^T \mathbf B) \! = \! |\mathbf B|^2$ for every $\mathbf B\in \mathbb R^{3\times3}$, we obtain as $h\to 0$

for every $\mathbf B\in \mathbb R^{3\times3}_{\mathrm{sym}}$. Moreover by recalling (2.27) (with $2c_1 = \mu$)

so

Let

and $\varphi\in C^2(\overline E)$ such that

(for instance $\phi(r): = 1-6r^2+9r^4-4r^6$ fulfills (5.4)). It is readily seen that condition (2.13) is fulfilled and that $E_{ess} = \overline E$. We define

Condition (2.32) is satisfied since

Moreover $\mathcal L(\mathbf e_1) = \mathcal L(\mathbf e_2) = 0, \ \mathcal L(\mathbf e_3) < 0$ and

so (2.33) is fulfilled too. By taking account of $\mathbf R_*\mathbf e_3 = \mathbf e_3$, we get

whence $\mathbf R_*\in \mathcal S_{\mathcal L, E}$ and Lemma 3.8 entails $\mathcal S_{\mathcal L, E} = \{\mathbf R: \mathbf R\mathbf e_3 = \mathbf e_3\}$. Since $E$ fulfills (2.13), $\mathcal W$ satisfies (2.14)–(2.17) and $\mathcal L$ satisfies (2.27) together with $\mathcal L(\mathbf e_3) < 0$ then, by taking into account of (5.2), Theorem 2.4 entails

We set

and, for every $\mathbf R\in \mathcal S_{\mathcal L, E},$

We aim to show

so that, once (5.10) is proved, we deduce

In order to show inequality (5.10) we need some properties of minimizers of $\mathcal E$ which have been essentially proven in ^[45]. In the following $\overline{\mathbb E}(\cdot)$ will denote the upper-left $2\times2$ submatrix of $\mathbb E(\cdot)$ and $\overline{\mathbf R}\in SO(2)$ the upper-left $2\times2$ submatrix of any $\mathbf R\in \mathcal S_{\mathcal L, E}$.

Lemma 5.1. Let ${\mathbf{u}}\in \mathcal A$ and let

where

Then ${\mathbf{v}}\in \mathcal A$ and

where $\overline{\mathbf{v}}: = v_\alpha\mathbf e_\alpha$, and

In particular if $\mathbf R = \mathbf I$, then (5.13) reduces to $\mathcal E({\mathbf{u}})\ge \mathcal J(\overline{\mathbf{v}})$ having set $\mathcal J: = \mathcal J_{\overline{\mathbf I}}$.

Proof. Since $u_{3}^*\ge 0$ q.e. on $E = \partial \Omega\cap \{x_3 = 0\}$ then by Remark 2.3 we get $u_3\ge 0\ \ \mathcal H^2$- q.e. in $E$ that is $v_3(0)\ge 0$ hence, again by Remark 2.3, ${\mathbf{v}}\in \mathcal A$. Moreover, by using the notation $u_{3, 3}: = \partial_3u_3$, Jensen inequality entails

thus proving the lemma. □

We need now the following characterization of minimizers of $\mathcal J$ which has been given in ^[45].

Lemma 5.2. There exists $\overline{\Phi}\in H^2(E)$ such that

where we have used the notation $\Phi_{, \alpha\beta}: = \partial^2_{\alpha\beta}\Phi$.

A straightforward application of Lemma 5.1 (with $\mathbf R = \mathbf I$) and Lemma 5.2 yields the following precise calculation of the energy level of ${\mathbf{u}}\in {\mathop{{{\rm{argmin}}}}\nolimits}_{\mathcal A}\mathcal E$.

Lemma 5.3. There holds

Proof. It is readily seen that any displacement of the kind $(\nabla\Phi(x_1, x_2), v_3(x_3))\in \mathcal A$ if and only if $\Phi\in H^2(E), \ v_3\in H^1(0, 1)$ and $v_3(0)\ge 0$. Therefore, by Lemmas 5.1 and 5.2, we get

The opposite inequality follows by choosing ${\mathbf{v}}: = (\nabla\Phi, 0)$ with $\Phi\in H^2(B)$ and by taking into account of

for every choice of $\Phi\in H^2(E)$. □ Let now $\Phi\in H^2(E)$. Then ${\mathbf{v}}: = \Phi_{, 2}\mathbf e_1 -\Phi_{, 1}\mathbf e_2\in \mathcal A$ and a direct computation shows that

Therefore inequality (5.10) is an immediate consequence of the next proposition.

Proposition 5.4. There holds

Proof. The proof is the same of formula (5.14) of ^[45]. □

The previous explicit example shows that a gap phenomenon may actually develop. Nevertheless one can prove that whenever $\mathbf f, \ \mathbf g$ satisfy (2.33) then they can be suitable rotated in order to avoid the gap. In order to state such result, we introduce suitable notation: Set

say $\mathcal L_{\mathbf R}$ is the load functional associated to the external forces $\mathbf R^T\mathbf f, \mathbf R^T\mathbf g$ and let $\mathcal E_{\mathbf R}$ be the functional defined by replacing $\mathcal L$ with $\mathcal L_{\mathbf R}$ in the definition of $\mathcal E$.

Theorem 5.5. Assume (2.13), (2.33), $\mathcal L(\mathbf e_3) < 0$ and $\mathbf R\in \mathcal S_{\mathcal L, E}$. Then the functional $\mathcal L_{\mathbf R}$ fulfills (2.33) and $\mathcal S_{\mathcal L, E}\equiv\mathcal S_{\mathcal L_{\mathbf R}, E}$. Moreover, if ${\mathbf{u}}$ minimizes $\mathcal G$ over $H^1 \Omega, {{\mathbb R}}^3),$ $\mathbf R\in\mathcal S_{\mathcal L, E}$ attains the maximum in definition (1.12) of $\mathcal G({\mathbf{u}})$ then ${\mathbf{u}}\in {\mathop{{{\rm{argmin}}}}\nolimits} {\mathcal E}_{\mathbf R}$ and

Proof. By Lemma 3.8 we have either $\mathcal S_{\mathcal L, E}\equiv \{\mathbf I\}$ or $\mathcal S_{\mathcal L, E} = \{ \mathbf R: \mathbf R\mathbf e_3 = \mathbf e_3\}$: In the first case there is nothing to prove, in the second one by (2.33) we get

Therefore for any other $\mathbf S\in SO(3)$ by taking account of $\mathbf R\mathbf e_3 = \mathbf e_3$ and (5.22) we have

that is $\mathcal L_{\mathbf R}$ satisfies (2.33). By Remark 3.5 conditions (4.9)–(4.11) of Theorem 4.5 in ^[12] are fulfilled hence $\mathcal E_{\mathbf R}$ achieves a finite minimum. Moreover since $\mathbf R\mathbf e_3 = \mathbf e_3$ implies $\mathbf R^2\mathbf e_3 = \mathbf e_3$, (5.23) together with Lemma 3.8 entails

whence $\mathbf R\in \mathcal S_{\mathcal L_{\mathbf R}, E}$ whenever $\mathbf R\in \mathcal S_{\mathcal L, E}$. Then since $\mathcal L_{\mathbf R}(\mathbf e_3) = \mathcal L(\mathbf R\mathbf e_3) = \mathcal L(\mathbf e_3) < 0$ we get, again by Lemma 3.8, $\mathcal S_{\mathcal L_{\mathbf R}, E}\not\equiv \{\mathbf I\}$ hence $\mathcal S_{\mathcal L_{\mathbf R}, E} = \{ \mathbf R: \mathbf R\mathbf e_3 = \mathbf e_3\} = \mathcal S_{\mathcal L, E}$ as claimed.

We conclude by checking that if ${\mathbf{u}}$ minimizes $\mathcal G$ then it is also a minimizer of $\mathcal E_{\mathbf R}$ over $H^1(\Omega, {{\mathbb R}}^3)$. If ${\mathbf{u}}\in H^1(\Omega, \mathbb R^3)$ minimizes $\mathcal G$ and $\mathbf R$ attains the maximum then

Thus since $\mathcal G\le \mathcal E_{\mathbf R}$ then (5.21) is proven. □

Remark 5.6. By choosing $\mathcal W$ as in (5.1), $\Omega, E$ as in (5.3) we provide an example where the inclusion ${\mathop{{{\rm{argmin}}}}\nolimits} \mathcal G \subset {\mathop{{{\rm{argmin}}}}\nolimits} \widetilde{\mathcal G}$ is strict. Indeed Lemma 5.1 shows that for every $\mathbf R\in \mathcal S_{\mathcal L, E}$ there exists $\mathbf w\in{\mathop{{{\rm{argmin}}}}\nolimits}\mathcal E_{\mathbf R}$ such that $\mathbf w({{\bf x}}) = w_{\alpha}(x_1, x_2)\mathbf e_{\alpha}$. By Theorem 5.5 there exists ${\mathbf{u}}\in {\mathop{{{\rm{argmin}}}}\nolimits} \mathcal G\subset {\mathop{{{\rm{argmin}}}}\nolimits} \widetilde{\mathcal G}$ such that ${\mathbf{u}}({{\bf x}}) = u_{\alpha}(x_1, x_2)\mathbf e_{\alpha}$. Then we can set ${\mathbf{u}}_*({{\bf x}}): = {\mathbf{u}}({{\bf x}})+x_3\mathbf e_1$. By taking account of Lemma 3.1 and Remark 3.2, we get $\mathcal L(\mathbf R{\mathbf{u}}) = \mathcal L(\mathbf R{\mathbf{u}}_*)$ for every $\mathbf R\in \mathcal S_{\mathcal L, E}$ hence $\widetilde {\mathcal G}({\mathbf{u}}) = \widetilde {\mathcal G}({\mathbf{u}}_*)$ and ${\mathbf{u}}_*\in {\mathop{{{\rm{argmin}}}}\nolimits} \widetilde{\mathcal G}$. Moreover, again by taking account of Lemma 3.1 and Remark 3.2, we have

thus ${\mathbf{u}}_*\not\in {\mathop{{{\rm{argmin}}}}\nolimits} \mathcal G$ and the inclusion is strict in this case.

6.   Conclusions

We have showed a rigorous variational linearization for a classical obstacle problem in nonlinear elasticity, namely an elastic body subject to pure traction load, supported on a unilateral rigid plane. Under suitable geometric admissibility conditions on the loads we obtain coincidence of minima with the classical Signorini problem in linear elasticity. On the other hand, we have shown the existence of loads violating such admissibility condition and entailing a gap between the minimum of the two problems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Francesco Maddalena and Franco Tomarelli are members of the Unione Matematica Italiana and the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors are partially funded by INdAM–GNAMPA 2023 Project Codice CUP–E53C22001930001, "Modelli variazionali ed evolutivi per problemi di adesione e di contatto", Italian M.U.R. PRIN grant number 2022J4FYNJ "Variational Methods for Stationary and Evolution Problems with Singularities and Interfaces", INdAM–GNAMPA 2023 Project Codice CUP E53C22001930001, "Modelli variazionali ed evolutivi per problemi di adesione e di contatto".

Research Project of National Relevance "Evolution problems involving interacting scales" granted by the Italian Ministry of Education, University and Research (MIUR Prin 2022, project code 2022M9BKBC, Grant No. CUP D53D23005880006).

Conflict of interest

All authors declare no conflicts of interest in this paper.

Appendix

For reader's convenience and aiming to the precise formulation of unilateral constraint, in this section we encompass some results about capacity theory which are essential to achieve the results of present paper and somehow present in the literature, though they are spread in several different contexts and not easy to find as stated in this form: In particular Proposition A.1 and Eq (A.2) can be proven as like as Propositions 5.8.3 and 5.8.4 in ^[7] although the results seem slightly different.

Proposition A.1. Let $G$ an open bounded subset of ${{\mathbb R}}^N$. Then

The above property can be generalized to every bounded subset of ${{\mathbb R}}^N$ by the following:

Proposition A.2. Let $E$ a bounded subset of ${{\mathbb R}}^N$. Then

We state and prove some results which play a crucial role in the proof of our main theorem.

In the sequel $\Omega$ will denote an open bounded subset of ${{\mathbb R}}^N$ with Lipschitz boundary and $E$ will denote a subset of $\overline \Omega$ such that ${\mathop{{{\rm{cap}}}}\nolimits} E > 0$.

Lemma A.3. Let $u\in H^1(\Omega), \ u\ge 0$ a.e. in $\Omega$ such that $u^*({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in D$, where $D$ is a closed subset of $\overline \Omega$. Then there is an extension $v\in H^1({{\mathbb R}}^N)$ of $u$ such that ${\mathop{{{\rm{spt}}}}\nolimits} v$ is compact, $v\ge 0\ {{a.e.\; in}}\ {{\mathbb R}}^N$ and

for q.e. ${\mathbf{x}}\in D$.

Proof. We recall that $u^*$ is defined as

for q.e. ${\mathbf{x}}\in \overline \Omega$, where $w$ is any Sobolev extension of $u$. Therefore the claim follows easily by choosing a cut off function $\varphi\in C^{\infty}_0({{\mathbb R}}^N), \ \varphi\equiv 1$ on $\Omega$ and by setting $v: = w^+\varphi$ which is a Sobolev extension of $u$ with compact support since $v = u$ a.e. in $\Omega$, and ${\mathop{{{\rm{spt}}}}\nolimits} v\subset {\mathop{{{\rm{spt}}}}\nolimits} \varphi$. □

Lemma A.4. Let $u\in H^1(\mathbb R^N)$ with compact support such that $u\ge 0\ \mathit{\text{a.e. in}}\ {{\mathbb R}}^N$ and $u^*({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E$. Then there exists a sequence $u_j\in C^1(\mathbb R^N)\cap H^1(\mathbb R^N), \ u_j\ge 0\ \mathit{\text{in}}\ {{\mathbb R}}^N$ such that $u_j({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E$ and $u_j\to u$ in $H^1(\mathbb R^3)$.

Proof. For every $j\in \mathbb N, \ j\ge 1$ let $\overline u_j: = \min\{u, j^{\frac{1}{4}}\}\in H^1(\mathbb R^N)$. By Theorem 3.11.6 and Remark 3.11.7 of ^[57], there exists $v_j\in C^1(\mathbb R^N)\cap H^1(\mathbb R^N)$ with ${\mathop{{{\rm{spt}}}}\nolimits} v_j\subset \{{{\bf x}}: d({{\bf x}}, {\mathop{{{\rm{spt}}}}\nolimits} {\overline u}_j) \le j^{-1}\}$ such that if $F_j: = \{{\mathbf{x}}: v_j({\mathbf{x}})\neq {\overline u}_j({\mathbf{x}})\}$ then

so (2.8) entails

By recalling that ${\mathop{{{\rm{spt}}}}\nolimits} {\overline u}_j$ is compact we get that $F_j$ is bounded so, by taking account of (A.2), there exists $w_j\in C^{\infty}_0(\mathbb R^N; [0, 1]), \ w_j\equiv 1$ in a neighbourhood $U_j$ of $F_j$ such that

We define $u_j: = (1-w_j)v_j$: it is readily seen that $u_j\in C^1(\mathbb R^N)\cap H^1(\mathbb R^N), \ u_j\ge 0\ \text{in}\ {{\mathbb R}}^N$, that $u_j({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E\cap U_j$ and that $u_j\equiv \overline u_j(1-w_j)$ outside $U_j$, hence, by recalling that ${\overline u_j}^*({\mathbf{x}}) = u^*({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E$, we get $u_j({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E\setminus U_j$ that is $u_j({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E$. We claim that $u_j\to u$ in $H^1(\mathbb R^N)$: to this aim, by noticing that $u_j - u = v_j- \overline u_j+\overline u_j-u-v_jw_j$ and that $\overline u_j\to u$ in $H^1(\mathbb R^N)$, thanks to (A.4) we have only to show that $v_jw_j\to 0$ in $H^1(\mathbb R^N)$. We first notice that by setting

then by (A.6) and Tchebichev inequality we get $|B_j|\le \beta j^{-\frac{1}{2}}$, therefore

since

Analogously by recalling (A.6), that $w_j\equiv 1$ on $U_j$, that $v_j\equiv \overline u_j$ outside $U_j$ and $\|\overline u_j\|_{\infty}^2\le \sqrt j$, we get

as in (A.7) thus proving the lemma. □

Lemma A.5. Let $u\in H^1(\Omega)$ such that $u^*({\mathbf{x}})\ge 0$ for q.e. ${\mathbf{x}}\in E$. Then there exists a sequence $u_j\in C^1(\overline \Omega)$ such that $u_j({\mathbf{x}})\ge 0$ for q.e. ${\mathbf{x}}\in E$ and $u_j\to u$ in $H^1(\Omega)$.

Proof. We recall that by Remark 2.3 $u^*({\mathbf{x}})\ge 0$ for q.e. ${\mathbf{x}}\in E$ if and only if $(u^-)^*({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E_{ess}$ and that $E_{ess}$ is a closed subset of $\overline \Omega$. By Lemma A.3 there exists a Sobolev extension $v$ of $u^-$ such that ${\mathop{{{\rm{spt}}}}\nolimits} v$ is compact, $v\ge 0\ \text{a.e. in}\ {{\mathbb R}}^3$ and

for q.e. ${\mathbf{x}}\in E_{ess}$, so by Lemma A.4, there exists a sequence $v_j\in C^1(\mathbb R^N)\cap H^1(\mathbb R^N), \ v_j\ge 0\ \text{in}\ {{\mathbb R}}^N$ such that $v_j({\mathbf{x}}) = 0$ for q.e. ${\mathbf{x}}\in E_{ess}$ and $v_j\to v$ in $H^1(\mathbb R^N)$. Let now $w$ be a Sobolev extension of $u^+$. We may assume without loss of generality that $w\ge 0$ a.e. in ${{\mathbb R}}^N$ and if $\rho_j$ is a sequence of smooth mollifiers then $w_j: = w*\rho_j\ge 0$ and $w_j\to w$ in $H^1(\mathbb R^N)$. Therefore by setting $u_j: = w_j-v_j$, we have $u_j\in C^1(\overline \Omega)$, $u_j({\mathbf{x}})\ge 0$ for q.e. ${\mathbf{x}}\in E$ and $u_j\to u$ in $H^1(\Omega)$ thus proving the lemma. □

Remark A.6. If $E$ is a non empty subset of $\overline \Omega$ and $u\in H^1(\Omega)$ we say that $u\ge 0$ on $E$ in the sense of $H^1(\Omega)$ if there exists a sequence $u_j\in C^1(\overline \Omega)$ such that $u_j\ge 0$ on $E$ and $u_j\to u$ in $H^1(\Omega)$ (according to [31, Definition 5.1]). We claim that $(u^-)^* = 0$ q.e. in $E$ (or equivalently $u^*\ge 0$ q.e. in $E$) if and only if $u\ge 0$ on $E$ in the sense of $H^1(\Omega)$: Indeed if $(u^-)_* = 0$ q.e. in $E$ then Lemma A.5 provides a sequence $u_j\in C^1(\overline \Omega)$ such that $u_j\to u$ in $H^1(\Omega), \ u_j\ge 0$ on $E$, while the converse follows easily by recalling that if $u_j\to u$ in $H^1(\Omega)$ then, up to subsequences, $u_j({{\bf x}})\to u^{*}({{\bf x}})$ for q.e. ${\mathbf{x}}\in E$. $