Partial regularity for steady double phase fluids

Dedicated to Rosario Mingione for his 50th birthday.

1.   Introduction

Our research was inspired by two main contributions of Rosario: the regularity for non-Newtonian fluids ^[2] and the double phase problems, ^[4,13].

1.1.   The problem and the main assumptions

This article deals with nonlinear elliptic systems in divergence form, modeling double-phase non-Newtonian fluids in the stationary case:

Here $\Omega\subseteq {\mathbb{R}}^n$ denotes an open, bounded set, $n\geq3$, the vector-valued map ${\bf u}:\Omega\to {\mathbb{R}}^n$ can be interpreted as the stationary velocity field of a fluid, and the scalar function $\pi:\Omega\to {\mathbb{R}}$ plays the role of the pressure. The stress tensor will have double phase growth, as a function of the symmetric gradient $\mathcal{E}{\bf u}$.

For the nonlinear diffusion term ${\bf a}:\Omega\times {\mathbb{R}}_{\rm sym}^{n\times n}\to {\mathbb{R}}_{\rm sym}^{n\times n}$ we are considering a double phase growth condition of the type

where

and the modulating coefficient $\mu$ is non negative, bounded and Hölder continuous with exponent $\alpha$.

Notice that in the region $\{\mu = 0\}$, we have $H(x, t) = t^p$ so that $H$ has $p$-phase, while in the region $\{\mu > 0\}$ the function $H$ has $q$-phase. Moreover, for each $x\in\Omega$ such that $\mu(x) > 0$, the function $t\to H(x, t)$ is an $N$-function (see Section 2.2) complying with (2.1) where $g_1 = p$ and $g_2 = q$. The precise assumptions are:

for every $x\in\Omega$ and ${\mathit{\boldsymbol{\xi}}}\in {\mathbb{R}}_{\rm sym}^{n\times n}$, ${\mathit{\boldsymbol{\lambda}}} \in {\mathbb{R}}^{n\times n}$ and for some $0 < \nu\leq L$, where $H'(x, t)$ and $H''(x, t)$ denote the first and second derivative of $t\to H(x, t)$, respectively. Note that the second inequality in (1.4) implies

for every $x\in\Omega$ and ${\mathit{\boldsymbol{\xi}}}_1, {\mathit{\boldsymbol{\xi}}}_2\in {\mathbb{R}}_{\rm sym}^{n\times n}$. We further assume the existence of a nondecreasing and concave function $\omega:[0, \infty)\to[0, 1]$ with $\omega(0) = 0$ such that

and

for every $x, x_1, x_2\in\Omega$ and ${\mathit{\boldsymbol{\xi}}}, {\mathit{\boldsymbol{\xi}}}_1, {\mathit{\boldsymbol{\xi}}}_2\in {\mathbb{R}}_{\rm sym}^{n\times n}$.

For the force term $f$ we require that

for some $\beta > 0$.

1.2.   The main result

A pair $({\bf u}, \pi)\in W^{1, 1}(\Omega; {\mathbb{R}}^n)\times W^{1, 1}(\Omega, {\mathbb{R}})$ with $H(\cdot, | \mathcal{E}{\bf u}|)\in L^1_{\rm{loc}}(\Omega)$ is a weak solution to (1.1) if and only if ${\rm div}\, {\bf u} = 0$ in $\Omega$ in the sense of distributions and

holds for all ${\mathit{\boldsymbol{\varphi}}}\in C_0^\infty(\Omega, {\mathbb{R}}^n)$. If we test with divergence free vector fields

the pressure term in (1.9) vanishes and the system reduces to ${\rm div}\, {\bf u} = 0$ in $\Omega$ in the sense of distributions and

whenever ${\mathit{\boldsymbol{\varphi}}}\in C_{0, {\rm div}}^\infty(\Omega, {\mathbb{R}}^n)$. Within this setting, we say that ${\bf u} \in W^{1, 1}(\Omega; {\mathbb{R}}^n)$ with $H(\cdot, | \mathcal{E}{\bf u}|)\in L^1_{\rm{loc}}(\Omega)$ is a weak solution to (1.1) if and only if ${\rm div}\, {\bf u} = 0$ in $\Omega$ in the sense of distributions and (1.10) holds.

The main result of the paper is a partial Morrey regularity result:

Theorem 1.1. Let $H:\Omega\times[0, +\infty)\to[0, +\infty)$ be defined as in (1.2), with (1.3). Assume that the vector field ${\bf a}:\Omega\times {\mathbb{R}}_{\rm sym}^{n\times n}\to {\mathbb{R}}_{\rm sym}^{n\times n}$ complies with (1.4), (1.6), (1.7), and that (1.8) holds. Let $({\bf u}, \pi)\in W^{1, 1}(\Omega; {\mathbb{R}}^n)\times W^{1, 1}(\Omega, {\mathbb{R}})$ with $H(\cdot, | \mathcal{E}{\bf u}|)\in L^1_{\rm{loc}}(\Omega)$ be a weak solution to (1.1). Then there exists an open subset $\Omega_0 \subset \Omega$ such that

for every $\beta\in (0, 1)$. Moreover, $\Omega \setminus \Omega_0\subset \Sigma_1\cup\Sigma_2$ where

1.3.   Historical background

The study of PDEs/functionals with $(p, q)$ growth, corresponding to the case $\mu(x) \equiv1$, was initiated by Marcellini in the 90's ^[31] and has been generalized in many directions, see ^{[11,15,21,32,33,34]} and references therein. The main feature of these problems is a gap between coercitivity and growth condition, that can eventually lead even to unbounded minimizers/solutions. In order to prevent such irregular behaviour, one has to impose a bound on the ratio $\frac {q}{ p} < 1+\frac1{n}$. This bound has been improved recently by Bella and Schäffner to $\frac {q}{ p} < 1+\frac2{n-1}$, ^[5]. More recently, in ^[14,16], some tools from nonlinear potential theory have been employed to infer sharp partial regularity results for relaxed minimizers of degenerate/singular, nonuniformly elliptic quasiconvex functionals of $(p, q)$-growth.

As for the double phase problems, the ellipticity could eventually change from $p$-phase to $q$-phase, since the modulating coefficient could annihilate. Pioneer were Colombo and Mingione, Baroni, Colombo and Mingione who established Morrey estimates for local minimizers under the condition $\frac {q}{ p} < 1+\frac{\alpha}{n}$, where $\alpha$ is the modulus of continuity of the modulating coefficient, and also they considered borderline cases (of logarithmic type) and a unified approach to variable exponent, ^[4,13]. Their method relies on a refined alternative between $p$-phase and $(p, q)$-phase at every scale combined with an exit time argument. Their intrinsic approach was pushed further by Ok, who was able to find an easier proof in the superquadratic case, ^[36] and ^[37].

In the literature, nonlinear elliptic systems of (1.1) when the symmetric gradient has $p(x)$ growth are known as electrorheological fluids. The model was introduced by Rajagopal and Růžička, ^[39], while the existence theory was studied by Růžička, ^[40], see also ^[22]. Acerbi and Mingione, ^[2], first established regularity for solutions of such systems. They proved that the gradient is partially Hölder continuous on a set of full measure under suitable continuity assumptions on the exponent $p(x)$.

Our aim is to study regularity properties of solutions of the system (1.1) where the symmetric gradient presents a double phase growth. Featuring the double phase case considered by Colombo and Mingione, we cannot expect more than a partial Morrey regularity result. It is worth mentioning the paper ^[7] where the authors considered electrorheological fluids with discontinuous coefficients and proved Morrey partial regularity.

Recently, the uniqueness of small solutions for steady double phase fluids has been investigated in ^[1] for $\frac65 < p < 2 < q.$

1.4.   Strategy of the proof

We briefly explain the strategy of the proof of the main result. First, we stress the fact that our argument relies on two technical major ingredients which, to the best of our knowledge, appear for the first time in this paper.

The first one is the $\mathcal{A}$-Stokes approximation lemma for $\mathcal{A}$-Stokes systems with general growth (Theorem 2.10), which extends an analogous result for Stokes systems with $p$-growth provided by [10, Theorem 4.2]. The second one, contained in Lemma 2.11, is a higher integrability result for weak solutions to double phase Stokes systems (1.1) with lower order terms. Namely, we can find an exponent $s > 1$ depending on the data of the problem, such that

Analogous results were obtained for Stokes systems with $p(\cdot)$-growth, see [2, Theorem 4.2] and [7, Lemma 2.10].

When addressing the problem of partial regularity, the leading quantity is the Campanato-type excess functional

(see (3.1)). We are dealing with the non-degenerate setting; i.e., when

so that we can linearize the problem, via the $\mathcal{A}$-Stokes approximation, Theorem 2.10. The linearization procedure of Lemma 3.3 provides a suitable rescaling ${{\bf w}}$ of the solution ${\bf u}$ which is shown to be approximately $\mathcal{A}$-Stokes on a ball $B_\varrho(x_0)$, and the deviation from being $\mathcal{A}$-Stokes is measured in terms of the "hybrid" excess

Then, if (1.11) is small enough, Theorem 2.10 applies and ensures the existence of a smooth $\mathcal{A}$-Stokes function ${\bf h}$ which is close in an integral sense to ${\bf w}$ and for which classical decay estimates are available, see ^[23]. Scaling back from ${\bf w}$ to ${\bf u}$, this allows to prove an excess-decay estimate, which, in turn, permits the iteration of the rescaled excess $\frac{{\mathit{\Phi}}(x_0, \varrho)}{H(x_0, 1+|(\mathcal{E}{\bf u})_{x_0, \varrho}|)}$ and of a "Morrey-type" excess

at each scale.

Namely, there exists $\vartheta\in(0, 1)$ such that, if the smallness conditions

hold on some ball $B_\varrho(x_0)$, then

hold for every $m = 0, 1, \dots.$. Notice that, if on one hand $|(D{\bf u})_{x_0, \varrho}|$ might blow up in the iteration since we cannot expect $C^1$-regularity; on the other hand, the Morrey excess $\Theta(x_0, \vartheta^k\varrho)$ stays bounded, exactly as it should be for a $C^{0, \alpha}$-regularity result. The smallness of $\Theta$ at any level ensure Hölder continuity of ${\bf u}$ at $x_0$ provided the excess functionals ${\mathit{\Phi}}$ and $\Theta$ are small at some initial radius $\varrho$ (actually, this holds in a neighborhood of $x_0$, since these smallness conditions are open). Finally, it is then proven that such a smallness condition on the excesses is indeed satisfied on the complement of the set $\Sigma_1\cup\Sigma_2$ of Theorem 1.1. Notice that we were already using the Morrey type excess in ^[27], where we faced the partial regularity for discontinuous quasiconvex integrals with general growth and ^[38] where we considered the boundary partial regularity.

Outline of the paper. The paper is organized as follows. Section 2 collects some basic definitions and auxiliary results useful throughout the paper. Specifically, in Section 2.1 we fix some notation, Section 2.2 contains basic facts about Orlicz functions, while Sections 2.3–2.5 concern with typical auxiliary results in the study of systems depending on the symmetrized gradient, as a lemma of Bogowskiǐ and Sobolev-Korn-type inequalities. In Section 2.6 we formulate the $\mathcal{A}$-Stokes approximation lemma, which together with the self improving properties of weak solutions (Section 2.7), is a key tool in the linearization procedure. The problem of the partial regularity of weak solutions starts with Section 3. Namely, in Section 3.1 we prove the necessary Caccioppoli-type estimates, in Section 3.2 we show that a suitable scaling of the weak solution is almost $\mathcal{A}$-Stokes, while in Section 3.3 we get some excess decay estimates. Finally, Section 4 is entirely devoted to the proof of the main result, Theorem 1.1.

2.   Preliminaries and auxiliary results

2.1.   Basic notation

We denote by $\Omega$ an open bounded domain of ${\mathbb{R}}^n$. For $x_0\in {\mathbb{R}}^n$ and $r > 0$, $B_r(x_0)$ is the open ball of radius $r$ centred at $x_0$. In the case $x_0 = 0$, we will often use the shorthand $B_r$ in place of $B_r(x_0)$. If $f\in L^1(B_r(x_0))$, we denote the average of $f$ by

and we use the abbreviate notation $(f)_r$ for $(f)_{0, r}$. We denote by ${\mathbb{R}}^{n\times n}_{\rm sym}$ the set of all symmetric $n\times n$ matrices. For $x, y\in {\mathbb{R}}^n$, we denote their tensor product by $x\otimes y: = \{x_iy_j\}_{i, j}\in {\mathbb{R}}^{n^2}$ ad their tensor symmetric product by $x\odot y: = \frac{1}{2}(x\otimes y + y\otimes x)\in {\mathbb{R}}^{n\times n}_{\rm sym}$. For a function ${\bf u} = (u^i)\in L^1(\Omega)$, we denote by $\mathcal{E}{\bf u}$ and $\mathcal{W}{\bf u}$ its symmetric and skew-symmetric distributional derivative, respectively:

If $p > 1$, then $p': = \frac{p}{p-1}$ denotes the conjugate exponent of $p$. If $1 < p < n$, the number $p^*: = \frac{np}{n-p}$ stands for the Sobolev conjugate exponent of $p$, whereas $p^*$ is any real number if $p\geq n$.

2.2.   Some basic facts on $N$–functions

We recall here some elementary definitions and basic results about Orlicz functions. The following definitions and results can be found, e.g., in ^[3,6,28,30].

A real-valued function $G:[0, \infty)\to[0, \infty)$ is said to be an $N$-function if it is convex and satisfies the following conditions: $G(0) = 0$, $G$ admits the derivative $G'$ and this derivative is right continuous, non-decreasing and satisfies $G'(0) = 0$, $G'(t) > 0$ for $t > 0$, and $\lim_{t\to \infty} G'(t) = \infty$.

We assume that $G$ also satisfies

for some $1 < g_1\leq g_2 < \infty$. For instance, $G(t): = t^p$, $1 < p < \infty$, is an $N$-function complying with (2.1) where $g_1 = g_2 = p$. Also $G(t): = H(x_0, t)$, where $H$ is defined in (1.2), is an $N$-function satisfying (2.1) with $g_1 = p$ and $g_2 = q$. Moreover, it can be seen that if $G\in C^2(0, \infty)$ and satisfies

then $G$ is an $N$-function and (2.1) holds.

$G^\ast$ denotes the Young-Fenchel-Yosida conjugate function of $G$, given by $G^*(t) = \sup_{s \geq 0} (st - G(s))$. It is again an $N$-function; it satisfies (2.1) with $\frac{g_2}{g_2-1}$ and $\frac{g_1}{g_1-1}$ in place of $g_1$ and $g_2$, respectively. Also, $(G^\ast)^\ast = G$.

Proposition 1. Let $G:[0, \infty)\to[0, \infty)$ be an $N$-function complying with (2.1). Then

${\bf{(i)}}$ the mappings

are increasing and decreasing, respectively. In particular,

Moreover,

${\bf{(ii)}}$

${\bf{(iii)}}$ (Young's inequality) for any $\lambda\in(0, 1]$ it holds that

${\bf{(iv)}}$ there exists a constant $c = c(g_1, g_2) > 1$ such that

We say that $G$ satisfies the $\Delta_2$-condition if there exists $c > 0$ such that for all $t \geq 0$ holds $G(2t) \leq c\, G(t)$. We denote the smallest possible such constant by $\Delta_2(G)$. Since $G(t) \leq G(2t)$, the $\Delta_2$-condition is equivalent to $G(2t) \sim G(t)$, where "$\sim$" indicates the equivalence between $N$-functions.

In particular, from (2.3) it follows that both $G$ and $G^*$ satisfy the $\Delta_2$-condition with constants $\Delta_2(G)$ and $\Delta_2(G^*)$ determined by $g_1$ and $g_2$. We will denote by $\Delta_2({ G, G^\ast})$ constants depending on $\Delta_2(G)$ and $\Delta_2(G^*)$. Moreover, for $t > 0$ we have

The following inequalities hold for the inverse function $G^{-1}$:

for every $t\geq0$ with $0 < a\leq1$. The same result holds also for $a\geq1$ by exchanging the role of $g_1$ and $g_2$.

Another important set of tools are the shifted $N$-functions $\{ G_a \}_{a \ge 0}$ (see ^[17]). We define for $t\geq0$

We have the following relations:

The families $\{ G_a \}_{a \ge 0}$ and $\{(G_a)^* \}_{a \ge 0}$ satisfy the $\Delta_2$-condition uniformly in $a \ge 0$.

We recall also that, by virtue of [17, Lemma 30], uniformly in $\lambda\in[0, 1]$ and $a\geq0$ holds

The following lemma (see x[19, Corollary 26]) deals with the change of shift for $N$-functions.

Lemma 2.1 (change of shift). Let $G$ be an $N$-function with $\Delta_2(G), \Delta_2(G^*) < \infty$. Then for any $\eta > 0$ there exists $c_\eta > 0$, depending only on $\eta$ and $\Delta_2(G)$, such that for all ${\bf a}, {\bf b}\in {\mathbb{R}}^d$ and $t\geq0$

Let ${\bf{P}}_0, {\bf{P}}_1\in \mathbb{R}^{N\times n}$, $\theta\in[0, 1]$ and define ${\bf{P}}_{\theta}: = (1-\theta){\bf{P}}_0+\theta{\bf{P}}_1$. Then the following result holds (see [17, Lemma 20]).

Lemma 2.2. Let $G$ be a $N$-function with $\Delta_2(G, G^*) < \infty.$ Then uniformly for all ${\bf{P}}_0, {\bf{P}}_1\in \mathbb{R}^{N\times n}$ with $|{\bf{P}}_0|+|{\bf{P}}_1| > 0$ holds

where the constants only depend on $\Delta_2(G, G^*).$

In view of the previous considerations, the same proposition holds true for the shifted functions, uniformly in $a\ge 0$.

By $L^ G$ and $W^{1, G}$ we denote the classical Orlicz and Orlicz-Sobolev spaces, i.e., $f \in L^ G$ iff $\int G(|{f}|)\, dx < \infty$ and $f \in W^{1, G}$ iff $f, D f \in L^ G$. The space $W^{1, G}_0(\Omega)$ will denote the closure of $C^\infty_0(\Omega)$ in $W^{1, G}(\Omega)$. The following version of Sobolev-Poincaré inequality can be found in [17, Lemma 7].

Theorem 2.3 (Sobolev-Poincaré inequality). Let $G$ be an $N$-function with $\Delta_2(G, G^*) < +\infty$. Then there exist numbers $\theta = \theta(n, \Delta_2(G, G^*))\in(0, 1)$ and $K = K(n, N, \Delta_2(G, G^*)) > 0$ such that the following holds. If $B\subset {\mathbb{R}}^n$ is any ball with radius $R$ and ${{\bf w}}\in W^{1, G}(B, {\mathbb{R}}^N)$, then

where $({{\bf w}})_B: = \mathop {{\rlap{-} \smallint }}\limits_B {{\bf w}}(x)\, \mathrm{d}x$. Moreover, if ${{\bf w}}\in W^{1, G}_0(B, {\mathbb{R}}^N)$, then

where $K$ and $\theta$ have the same dependencies as before.

We conclude this section with the following useful lemma about an almost concave condition.

Lemma 2.4. Let $\Psi:[0, \infty)\to[0, \infty)$ be non-decreasing and such that $t\to \frac{\Psi(t)}{t}$ be non-increasing. Then there exists a concave function $\widetilde{\Psi}: [0, \infty)\to[0, \infty)$ such that

Proof. See [36, Lemma 2.2]. □

2.3.   A lemma of Bogowskiǐ

The following lemma is a key tool to deal with the constraint of divergence free vector fields, as we have to construct testing functions in divergence free form.

Lemma 2.5. Let $B_r(x_0)$ be a ball in ${\mathbb{R}}^n$ and $f\in L^\gamma(B_r(x_0))$ with $(f)_{x_0, r} = 0$, where $1 < \gamma_1\leq \gamma \leq \gamma_2 < +\infty$. Then there exists ${{\bf w}}\in W^{1, \gamma}_0(B_r(x_0), {\mathbb{R}}^n)$, weak solution to ${\rm div}\, {{\bf w}} = f$ in $B_r(x_0)$, such that

for every $t\in[\gamma_1, \gamma]$.

Proof. See ^[8] and [24, Chapter 3,Section 3]. □

Corollary 1. Let $B_r(x_0)$ be a ball in ${\mathbb{R}}^n$, and $G$ be an $N$-function such that $\Delta_2(G) < +\infty$ and $\Delta_2(G^*) < +\infty$. Let $G(|f|)\in L^1(B_r(x_0))$ with $(f)_{x_0, r} = 0$. Then there exists ${{\bf w}}\in W^{1, G}_0(B_r(x_0), {\mathbb{R}}^n)$, solution to ${\rm div}\, {{\bf w}} = f$ a.e. in $B_r(x_0)$, such that

Proof. See, e.g., [29, Corollary 4.2]. □}

2.4.   Affine functions

We define the space of traceless affine functions

In particular, each ${\mathit{\boldsymbol{\ell}}}\in \mathcal{T}$ satisfies ${\rm div}\, {\mathit{\boldsymbol{\ell}}} = 0$.

The set of rigid motions in ${\mathbb{R}}^n$ is defined as

the set of affine functions with skew-symmetric gradient. Let $x_0\in\Omega$ and $r > 0$. We define the affine function

Note that ${\mathit{\boldsymbol{\ell}}}_{x_0, r}\in \mathcal{T}({\mathbb{R}}^n)$ and

Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^n$ and let $x_0$ be its centroid; i.e.,

For every ${\bf u}\in L^1(\Omega; {\mathbb{R}}^n)$ and $x\in\Omega$ we set

where

As shown in [2, Proposition 2.6], ${\mathcal{P}}_\Omega {\bf u}\in\mathcal{R}$ for all ${\bf u}$; this implies, in particular, that $\mathcal{E} ({\mathcal{P}}_\Omega {\bf u}) = {\bf 0}$. Moreover, if ${\bf u}\in L^2(\Omega; {\mathbb{R}}^n)$ and $\Omega$ is a ball, then

Finally, we recall the following Korn-type inequality [2, Eq (2.21)]:

for every $t\in[1, p^*]$.

2.5.   Sobolev-Korn inequality

A standard tool in order to obtain local bounds of $D{\bf u}$ in terms of $\mathcal{E}{\bf u}$ on the scale of $L^p$ spaces is the following Sobolev-Korn inequality (see, e.g., ^[35]).

Lemma 2.6. Let $1 < p\leq r \leq q$ and ${\bf u}\in L^p(B_\varrho(x_0), {\mathbb{R}}^n)$ be such that $\mathcal{E}{\bf u}\in L^r(B_\varrho(x_0), {\mathbb{R}}^{n\times n}_{\rm sym})$. Then $D{\bf u}\in L^r(B_\varrho(x_0), {\mathbb{R}}^{n^2})$ and for some constant $c = c(n, p, q)$ it holds that

If, in addition, ${\bf u} = {\bf 0}$ on $\partial B_\varrho(x_0)$, then

We also need the following version of the Korn's inequality in Orlicz spaces (see [18, Lemma 3.3] and ^[9]).

Lemma 2.7. Let $B\subset {\mathbb{R}}^n$ be a ball. Let $G$ be an $N$-function such that both $G$ and $G^*$ satisfy the $\Delta_2$-condition. Then for all ${\bf u}\in W^{1, G}_0(B, {\mathbb{R}}^n)$ (with $\left(\mathcal{W}{\bf u}\right)_B = {\bf 0}$) the inequality

holds, and $c_{\rm Korn} = c_{\rm Korn}(\Delta_2(G, G^*))$.

The following lemma is useful to derive reverse Hölder estimates. It is a variant of the results by Gehring ^[25] and Giaquinta-Modica [26, Theorem 6.6].

Lemma 2.8. Let $B_0\subset {\mathbb{R}}^n$ be a ball, $f\in L^1(B_0)$, and $g\in L^{\sigma_0}(B_0)$ for some $\sigma_0 > 1$. Assume that for some $\theta\in(0, 1)$, $c_1 > 0$ and all balls $B$ with $2B\subset B_0$

Then there exist $\sigma_1 > 1$ and $c_2 > 1$ such that $g\in L^{\sigma_1}_{\rm loc}(B)$ and for all $\sigma_2\in[1, \sigma_1]$

2.6.   $\mathcal{A}$-Stokes functions and $\mathcal{A}$-Stokes approximation

Let $\mathcal{A}$ be a bilinear form on ${\mathbb{R}}_{\rm sym}^{n\times n}$. We say that $\mathcal{A}$ is strongly elliptic in the sense of Legendre-Hadamard if for all ${\mathit{\boldsymbol{\eta}}}, {\mathit{\boldsymbol{\zeta}}}\in {\mathbb{R}}_{\rm sym}^{n\times n}$ it holds that

for some $L_{\mathcal{A}}\geq {\kappa_{\!\mathcal{A}}} > 0$. The biggest possible constant ${\kappa_{\!\mathcal{A}}}$ is called the ellipticity constant of $\mathcal{A}$. By $\left| {\mathcal{A}} \right|$ we denote the Euclidean norm of $\mathcal{A}$. We say that a Sobolev function ${\bf w}$ on a ball $B_\varrho(x_0)$ is $\mathcal{A}$-Stokes on $B_\varrho(x_0)$ if it satisfies $- {\rm div}\, (\mathcal{A} \mathcal{E} {\bf w}) = 0$ in the sense of distributions; i.e.,

Now, we address the problem of the $\mathcal{A}$-Stokes approximation: given a Sobolev function ${\bf v}$ on a ball $B$, we want to find an $\mathcal{A}$-Stokes function ${{\bf h}}$ which is "close" the function ${\bf v}$. It will be the $\mathcal{A}$-Stokes function with the same boundary values as ${\bf v}$; i.e., a Sobolev function ${{\bf h}}$ which satisfies

in the sense of distributions. Setting ${{\bf w}} : = {{\bf h}} - {\bf v}$, then (2.21) is equivalent to finding a Sobolev function ${{\bf w}}$ which satisfies

in the sense of distributions. We formulate the $\mathcal{A}$-Stokes approximation result for Stokes systems with general growth. The proof for Stokes systems with $p$-growth can be found in [10, Theorem 4.2], whence the case of general growth can be inferred by using a duality argument as in ^[20], based on the following Lemma.

Lemma 2.9. Let $G$ be an N-function with $\Delta_2(G, G^*) < \infty$. Let $B \subset \mathbb{R}^n$ be a ball, and let $\mathcal{A}$ be strongly elliptic in the sense of Legendre-Hadarmard. Then the following variational estimates hold for all ${\bf v} \in W^{1, G}_{0, {\rm{dive}}}(B)$

The implicit constants in (2.23) only depend on $n$, ${\kappa_{\!\mathcal{A}}}$, $||{\mathcal{A}} ||$ and $\Delta_2(G, G^*)$.

Proof. This result can be deduced from [10, Lemma 4.1] along the lines of the proof of [20, Lemma 20]. We then omit the details. □

The $\mathcal{A}$-Stokes approximation can be stated as follows.

Theorem 2.10. Let $B \subset \mathbb{R}^n$ be a ball with radius $r_B$ and let $\widetilde{B} \subset \mathbb{R}^n$ denote either $B$ or $2B$. Let $G$ be an N-function with $\Delta_2(G, G^*) < \infty$ and let $s > 1$. Then for every $\kappa > 0$, there exists $\delta > 0$ only depending on $n$, $\kappa_A$, $\left| {\mathcal{A}} \right|$, $\Delta_2(G, G^*)$ and $s$ such that the following holds. If ${\bf v} \in W^{1, G}_{{\rm{dive}}}(\widetilde{B})$ is almost $\mathcal{A}$-Stokes on $B$ in the sense that

for all ${\mathit{\boldsymbol{\xi}}} \in C^\infty_{0, {\rm{dive}}}(B)$. Then the unique solution ${\bf w} \in (W^{1, G_s}_{0, {\rm{dive}}}(B))^n$ of (2.22) satisfies

It holds $\kappa = \kappa(G, s, \delta)$ and $\lim_{\delta\to 0} \kappa(G, s, \delta) = 0$. The function ${\bf h} = {\bf v}-{\bf w}$ is called the $\mathcal{A}$-Stokes approximation of ${\bf v}.$

Remark 1. We will exploit the previous approximation result in a slightly modified version. Indeed, under the additional assumption $(\mathcal{W}{\bf v})_{\tilde{B}} = {\bf 0}$ and

for some exponent $s > 1$ and for a constant $\epsilon > 0$, and (2.24) replaced by

by using Korn's inequality (Lemma 2.7) and following [12, Lemma 2.7], it can be seen that the unique solution ${{\bf w}} \in (W^{1, G_s}_{0, {\rm{dive}}}(B))^n$ of (2.22) satisfies

2.7.   Self-improving properties of weak solutions

We prove the self improving properties for double phase Stokes systems (1.1) with lower order terms. These are some higher integrability results which will play a crucial role in the sequel. They can be compared with analogous results obtained for Stokes systems with $p(\cdot)$-growth, see [2, Theorem 4.2] and [7, Lemma 2.10]. In this direction, our main difficulty is in handling the lower order terms without any "closeness" condition between the extreme exponents $p$ and $q$, which, on the contrary, can be ensured in the variable exponent setting by requiring some continuity assumption on $p(\cdot)$. For instance, in the main estimate (2.28), we are forced to control the average of the term $r^\frac{p}{2}|D{\bf u}|^p$ instead of the more natural $r^{p}|D{\bf u}|^p$ appearing in [7, Lemma 2.10]. Nevertheless, these issues can be overcome when dealing with the terms involving $H(x, t)$, just playing with the different behavior of $H$ according to the magnitude of $\mu$ inside a small ball.

Lemma 2.11. Let $H:\Omega \times [0, \infty)\to[0, \infty)$ be defined as in (1.2), and satisfying (1.3). Let ${\bf a}:\Omega\times {\mathbb{R}}_{\rm sym}^{n\times n}\to {\mathbb{R}}_{\rm sym}^{n\times n}$ be such that

for all $x\in\Omega$ and ${\mathit{\boldsymbol{\xi}}}\in {\mathbb{R}}_{\rm sym}^{n\times n}$, and for some $0 < \nu\leq L < \infty$, $\nu_0\in[0, 1]$. Let ${\bf u}\in W^{1, 1}(\Omega; {\mathbb{R}}^n)$ with $H(\cdot, | \mathcal{E}{\bf u}|)\in L^1(\Omega)$ be a weak solution to (1.1). Then there exists $s_0 > 1$, depending on $n, p, q, [\mu]_{C^\alpha}, \nu, L, \nu_0, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)}$, such that $H(\cdot, | \mathcal{E}{\bf u}|)\in L^{s_0}_{\rm loc}(\Omega)$ and, for every $s\in(1, s_0]$ and every $B_{2r}(x_0)\subset\subset\Omega$, $r < 1$,

for some $c = c(n, p, q, [\mu]_{C^\alpha}, \nu, L, \nu_0, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)}) > 0$.

Moreover, for each $t\in[0, 1]$ we have

for some constant $c_t = c_t(n, p, q, [\mu]_{C^\alpha}, \nu, L, \nu_0, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)}, t) > 0$.

Proof. Our argument partly follows that of [36, Theorem 3.4], devised for homogeneous double-phase systems depending on the gradient $D{\bf u}$, while the estimates for the right hand side's terms in (1.10) are essentially taken from Acerbi and Mingione [2, Theorem 4.2].

Let $\eta\in C_0^\infty(B_{2r}(x_0))$ be a cut-off function such that $\eta\equiv1$ on $B_{r}(x_0)$, $0\leq\eta\leq1$ and $|D\eta|\leq \frac{c}{2r}$, and let ${ \mathcal{P}}: = { \mathcal{P}}_{B_{2r}(x_0)}$ be the rigid displacement introduced in (2.16). We consider

where the function ${\bf w}$ is defined through Lemma 2.5 as a solution to

Such a ${\bf w}$ exists since ${\rm div}\, {\bf u} = 0$ and

and by (1.3) and the summability properties of ${\bf u}$ it holds that ${{\bf w}}\in W^{1, q}_0(B_{2r}(x_0); {\mathbb{R}}^n)$. We recall also that, by (2.13), we have the estimate

for every exponent $t$ for which the right hand side is finite. Taking ${\mathit{\boldsymbol{\psi}}}$ as a test function in (1.1) we get

By the second condition in (2.27) we have

while from the first one we get

Now, by using Young's inequality, for any $\kappa\in(0, 1)$ we obtain

whence

As for $J_3$, using again the first condition in (2.27) and Young's inequality, and Corollary 1, we have

We can write

Let $0 < \beta < \frac{1}{n+2}$. We set

and note that

since

By using Hölder's inequality, the Poincaré inequality for ${\bf u}-{ \mathcal{P}}$ and (2.31) we have

whence, with Young's inequality,

Again using Hölder's and Young's inequalities, (2.17) and the fact that (2.31) implies $1^*\leq (\frac{p}{\gamma})^*$, we obtain

for every $\kappa\in(0, 1)$. Using Hölder's inequality, recalling that ${{\bf w}}\in W^{1, q}_0(B_{2r}; {\mathbb{R}}^n)$, using Poincaré inequality, (2.30) and then Young's inequality we obtain the estimate

Arguing as above and with Korn's inequality we get

where in the last inequality we applied Young's inequality for some $\kappa\in(0, 1)$. Collecting the previous estimates and taking into account (2.31), we finally have

for every $\kappa\in(0, 1)$. Note that setting

we have $g\in L^\gamma(B_{2r})$, since by Korn's inequality and the Sobolev embedding theorem it holds that $|D{\bf u}|^{p}+|{\bf u}|^{p^*}\in L^1(\Omega)$. Now we claim that there exists $\theta = \theta(n, p, q)\in(0, 1)$ such that

where $c = c(q, p, \mu, \| \mathcal{E} {\bf u}\|_{L^p(\Omega)}) > 0$. We first note that, setting $q_*: = \frac{nq}{n+q}$, we have $q_* < p$ by (1.3). Then, by Korn's inequality it holds that

We distinguish between two cases. If $\sup_{B_{2r}}\mu(\cdot) \leq 4 [\mu]_{C^\alpha}r^\alpha$, using (2.34) and Hölder's inequality we then find

whence (2.33) follows by choosing $\theta: = \frac{q_*}{p} < 1$ and noting that, by Korn's inequality,

where in the latter we used $p_*\leq q_*$. If instead $\inf_{B_{2r}}\mu(\cdot) > 3[\mu]_{C^\alpha}r^\alpha$, then we can freeze $\mu$ at $x = x_0$ and apply Theorem 2.3 and Lemma 2.7 with $G(t): = H(x_0, t)$, since in this case $\frac{3}{4} H(x, t)\leq H(x_0, t)\leq \frac{4}{3} H(x, t)$ for every $x\in B_{2r}(x_0)$. This concludes the proof of the claim.

Now, combining (2.32) and (2.33) we can write

This estimate holds for every $\kappa\in(0, 1)$ and every ball $B_{2r}\subset\subset\Omega$, and the constants $c_\kappa$ only depend on the data. Thus, by virtue of a variant of Gehring's lemma (Lemma 2.8) we get (2.28). This concludes the proof. □

We define

where $s_0$ is that of Lemma 2.11. We now state an higher integrability estimate near the $p$-phase.

Lemma 2.12. Let $\varepsilon_0$ be as in (2.35). For $B_{2r}(x_0)\subset\subset\Omega$ with $r\leq1$, if

then

for some $c = c(n, p, q, [\mu]_{C^\alpha}, \nu, L, \nu_0, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)}, \|1+| \mathcal{E}{\bf u}|\|_{L^{ps_0}(B_{2r}(x_0))})$.

If, in addition,

then

Proof. The argument is similar to that of [37, Lemma 3.6]. Thus, we only sketch the proof.

We first note that under assumption (2.36), for every $x\in B_{2r}(x_0)$ we have $\mu(x)\leq 3[\mu]_{C^\alpha}r^{\alpha-\varepsilon_0}$. This fact together with (2.29), applied with $t = \frac{t_0}{2}\in(0, 1)$ where

implies

In (2.41), the second integral is finite since $(qt_0)/2\leq p$ by (2.40). Moreover, with Hölder's inequality and the fact that $(q-p)t_0\leq p$ again by (2.40), we have

Therefore, plugging (2.41) into (2.42) and using $r\leq1$, (1.3) and the definition of $\varepsilon_0$, we get

This concludes the proof of (2.37). In order to prove (2.39), we only have to check that under assumption (2.38), we have

Indeed, using the Poincaré inequality and the Sobolev-Korn inequality (2.18), we get

The proof is concluded. □

3.   Decay estimates for excess functionals

Let $x_0\in\Omega$, $B_\varrho(x_0)\subset\Omega$ and ${\mathit{\boldsymbol{\ell}}}_{x_0, \varrho}$ be the traceless affine function defined in (2.15). We introduce the following Campanato-type excess functionals, measuring the oscillations of $\mathcal{E}{\bf u}$:

where $H_{1+|(\mathcal{E}{\bf u})_{x_0, \varrho}|} (x_0, t)$ denotes the shifted $N$ function $H_a$ with shift $a: = 1+|(\mathcal{E}{\bf u})_{x_0, \varrho}|$.

3.1.   Caccioppoli-type estimates

The first key tool is the following "conditioned" Caccioppoli type estimate for ${\bf u}-{\mathit{\boldsymbol{\ell}}}_{x_0, r}$, which holds under suitable smallness assumptions, see (3.2) and (3.3) below.

Lemma 3.1. Let $B_{\varrho}(x_0)\subset\subset\Omega$, with $\varrho\leq1$. Assume that

and

Then we have

for some constant $c > 0$ depending on data, and $\bar{\gamma}: = \min\{\varepsilon_0, 1-\frac{1}{\beta+1}\}$.

Proof. We follow the argument of [36, Lemma 4.1]. We use the shorthands ${\mathit{\boldsymbol{\ell}}}_{\varrho}$ and $(\mathcal{E}{\bf u})_{\varrho}$ for ${\mathit{\boldsymbol{\ell}}}_{x_0, \varrho}$ and $(\mathcal{E}{\bf u})_{x_0, \varrho}$, respectively. We consider a cut-off function $\eta\in C_0^\infty(B_{\varrho}; [0, 1])$ such that $\eta\equiv1$ on $B_{\varrho/2}$ and $|D\eta|\leq \frac{c(n)}{\varrho}$. Correspondingly, we define the function ${\mathit{\boldsymbol{\psi}}}: = \eta^q({\bf u} - {\mathit{\boldsymbol{\ell}}}_{\varrho})\in W^{1, p}(B_{\varrho}; {\mathbb{R}}^n)$. Since

where we used that ${\rm div}\, {\bf u} = 0$ and ${\mathit{\boldsymbol{\ell}}}_{\varrho}\in\mathcal{T}({\mathbb{R}}^n)$, we conclude that ${\mathit{\boldsymbol{\psi}}}$ is not a divergence-free vector field. By virtue of Lemma 2.5 we can find ${{\bf w}}\in W^{1, q}_0(B_{\varrho}; {\mathbb{R}}^n)$ such that, setting

we have ${\rm div}\, {\mathit{\boldsymbol{\varphi}}} = 0$. Taking ${\mathit{\boldsymbol{\varphi}}}$ as a test function in (1.1) we get

This and the identity

imply that

Now, we proceed to estimate each term above separately. With (1.5) and (2.9) we get

To estimate $J_3$ we use the second inequality in (1.4), (2.13) with $G(t) = H_{1+|(\mathcal{E}{\bf u})_{\varrho}|}(x_0, t)$, Young's inequality for $G, G^*$, (2.5) and we get

We now treat separately the two cases where the inequality

holds true or not. Note that from (3.10), for every $x\in B_{\varrho}$ we get

Step 1: $p$-phase. Let $\mu$ comply with (3.10). Then, using Hölder's inequality, (1.3), the definition of $\epsilon_0$ and Lemma 2.12 we get

From this and (3.11) we deduce that

As for $J_2$, from (1.6), (3.12), (3.13), and using Young's inequality (2.4) for $H_{1+|(\mathcal{E}{\bf u})_{\varrho}|}(x_0, t)$ and its conjugate, (2.11), and (2.14),

we obtain

Plugging the estimates (3.8), (3.14) and (3.9) into (3.7) and reabsorbing some terms we obtain (3.4).

Step 2: $(p, q)$-phase. If (3.10) does not hold, then $\mu(x_0)\geq\inf_{x\in B_{\varrho}} \mu(x) > [\mu]_{C^\alpha}\varrho^{\alpha-\varepsilon_0}$, so that

Now, arguing as for Step 1 and using Young's inequality (2.4) for $H_{1+|(\mathcal{E}{\bf u})_{\varrho}|}(x_0, t)$ and its conjugate, and (2.11), we have

Now, we set $\gamma: = \left(\frac{p^*}{2}\right)'$. Note that if $p > \frac{3n}{n+2}$, then $\gamma < p$. Using Hölder's inequality we have

Estimate of $I_1$: using Poincaré inequality and (3.2), we get

Estimate of $I_2$: with the Sobolev-Korn's inequality (2.18) and (3.2) we obtain

Now, using Lemma 2.1 and (3.3),

whence, from Jensen's inequality for the function $\Psi(t): = t^\frac{p}{\gamma}+\mu(x_0)t^\frac{q}{\gamma}$ (convex, since $\gamma < p\leq q$) we obtain

Thus,

Estimate of $I_3$: we use Poincaré's inequality, Lemma 2.5 and Lemma 2.7, and we get

Now, using the change of shift formula, (3.3) and an analogous argument as for (3.16) with the convex function $\widetilde{\Psi}(t): = t +\mu(x_0)t^\frac{q}{p}$, we obtain

Collecting the previous estimates, we finally get

We are left to estimate $J_5$. Using Hölder's inequality, $p^* > \frac{n}{n-1}$ and the estimate of $I_3$ we have

The proof of (3.4) is then concluded. □

The following result can be obtained combining the argument of [36, Theorem 3.2] with Lemma 2.7.

Lemma 3.2. There exist $0 < \sigma_1 < 1$ depending on $n, p, q$ such that for any $B_\varrho(x_0)\subset\subset\Omega$ we have

for some constant $c = c(n, p, q) > 1$.

Proof. Let $a\geq0$ and denote by $H_a(x_0, t)$ the shifted $N$-function of $H(x_0, t)$ defined as in (2.8). Note that, by the definition of $H$ and (2.9), we have

where the $N$-functions $G_{p, a}$ and $G_{q, a}$ are equivalent to the corresponding shifted $N$-functions of $t^p$ and $t^q$, respectively. Note that

We first assume that

To enlighten the notation, we set

Note that $({\bf f}_{x_0, \varrho})_{x_0, \varrho} = {\bf 0}$.

Using the Sobolev-Poincaré inequality, together with (3.17), we get

Now, invoking the Korn's inequality of Lemma 2.7, we have

By Hölder's inequality, we get

where we have set $\sigma_1: = \frac{nq}{p(n+q)} < 1$. Combining with (3.18), we then obtain

If instead (3.17) does not hold, we can apply Theorem 2.3 and Lemma 2.7 with $G(t): = H_a(x_0, t)$. The proof is concluded. □

Now, we are in position to establish a higher integrability result for $H_{1+|(\mathcal{E}{\bf u})_{x_0, \varrho}|}(x_0, | \mathcal{E}{\bf u}-(\mathcal{E}{\bf u})_{x_0, \varrho}|)$. The result follows from Lemma 3.2 and Lemma 3.1 as a consequence of Gehring's lemma with increasing supports (Lemma 2.8):

Corollary 2. There exist a constant $c_{\rm high} = c_{\rm high}(data) > 0$ and $\sigma > 1$ such that

3.2.   Almost $\mathcal{A}$-Stokes functions

We can start with the linearization procedure for system (1.1). Let us set

Note that the bilinear form $\mathcal{A}$ satisfies the Legendre-Hadamard condition (2.20) by virtue of (1.4).

We aim to prove that ${{\bf w}}$ is approximately $\mathcal{A}$-Stokes. This fact, together with the higher integrability result (3.19) will allow us to apply the $\mathcal{A}$-Stokes approximation theorem.

Lemma 3.3. Let $B_{\varrho}(x_0)\subset\subset\Omega$. Assume that (3.2) and (3.3) hold, and let $\varrho\leq1$. Then there exists a constant $c_{\rm Stokes} > 0$ such that

for every ${\mathit{\boldsymbol{\varphi}}}\in C_{0, {\rm div}}^\infty(B_{\varrho/2}(x_0))$, where $\mathcal{R}(t): = [\omega(t^\frac{1}{2}) + t]^\frac{1}{2} \sqrt{t}$.

Proof. It will suffice to prove (3.21) for any fixed ${\mathit{\boldsymbol{\varphi}}}\in C_0^\infty(B_{\varrho/2}(x_0))$ with ${\rm div}\, {\mathit{\boldsymbol{\varphi}}} = 0$ such that $\|D{\mathit{\boldsymbol{\varphi}}}\|_{L^\infty(B_{\varrho/2}(x_0))}\leq1$, since the general case will follow by a standard normalization argument. To enlighten notation, we omit the explicit dependence on $x_0$.

From the definitions of $\mathcal{A}$ and ${{\bf w}}$ we have

We set $G(t): = H(x_0, t)$. From (1.7), (2.6), (2.2) and Lemma 2.2, we have

where $E: = \{x\in B_{\varrho/2}:\, \, | \mathcal{E}{\bf u}(x)-(\mathcal{E}{\bf u})_\varrho| > 1+|(\mathcal{E}{\bf u})_\varrho|\}$, and $F: = B_\varrho\backslash E$.

We start with the estimate of $J_{1, E}$. We have

For a.e. $x\in E$, we have

and taking into account (2.10), we obtain

Now, using that $\omega\leq1$, we finally get

whence

For what concerns $J_{1, F}$, arguing as for $J_{1, E}$ we get the preliminary estimate

For a.e. $x\in F$, we have

and with (2.6) and since $G'$ is increasing, we can write

Using the definition of $F$ again, the monotonicity of $G'$ and (2.6), for a.e. $x\in F$ we have

Using (2.9) we then get

Combining the previous estimates and using Hölder's inequality, the fact that $\omega\leq1$ and Jensen's inequality for the concave function $\omega(t^\frac{1}{2})$, we get

Collecting the estimates for $J_{1, E}$ and $J_{1, F}$, we then infer

In order to estimate $J_2$, we use (3.6), the definition of weak solution, (1.6) and we preliminarly obtain

Now, we have to distinguish between two cases, depending on whether condition (3.10) is satisfied or not.

Step 1: $p$-phase. Under assumption (3.10), we have

An analogous argument as for (3.12) and (3.13) shows that

whence, using the change of shift formula (2.12) and (3.3), we obtain

With this and (2.39) we then get

A similar estimate holds for the second summand in the right hand side of (3.25), with $\mu(x_0)$ in place of $\mu(x)$. Inserting these estimates in (3.25) and using the definition of $H_{1}(x_0, 1+|(\mathcal{E}{\bf u})_\varrho|)$ we finally obtain

Step 2: $(p, q)$-phase. If (3.10) does not hold, then (3.15) is in force. Now, using (3.3), the triangle inequality and the change of shift formula, we get

With this, Hölder's inequality and Jensen's inequality, we can estimate $J_{2, 1}$ as

which corresponds to (3.26). In order to estimate $J_{2, 2}$ and $J_{2, 3}$, we may argue as in the proof of Lemma 3.1, exploiting the smallness assumptions in (3.2), so we briefly sketch the proof.

Setting $\gamma: = \left(\frac{p^*}{2}\right)'$ again, and using Hölder's inequality we have

then the proof is similar. The only difference is the use of the Sobolev-Korn inequality (2.19) for ${\mathit{\boldsymbol{\varphi}}}$ with $| \mathcal{E}{\mathit{\boldsymbol{\varphi}}}|\leq1$. We then obtain

For what concerns $J_{2, 3}$, using Hölder's inequality, the fact that $p^* > \frac{n}{n-1}$ and the Sobolev-Korn inequality for ${\mathit{\boldsymbol{\varphi}}}$ with $| \mathcal{E}{\mathit{\boldsymbol{\varphi}}}|\leq1$, we have

Collecting the previous estimates, we then obtain

where $\widetilde{\gamma}: = \min\{\varepsilon_0, 1- \frac{1}{\beta+1}\}$. The final estimate (3.21) then follows inserting (3.23) and (3.27) into (3.22). □

3.3.   Excess decay estimates

In this section, we prove an excess improvement estimate for weak solutions to (1.1). This will be the content of Lemma 3.5. We start with a technical tool useful in the sequel.

Lemma 3.4. Let $\vartheta\in(0, 1)$. Assume that

where $c_{q}$ is the constant of the change of shift formula (2.12) with $\eta = \frac{1}{2^{q+1}}$. Then,

Proof. As a consequence of (2.12) for $\eta = \frac{1}{2^{q+1}}$ and with (3.28) we get

whence, passing to $[H(x_0, \cdot)]^{-1}$ and taking into account (2.7), we obtain

Now,

whence (3.29) follows by re-absorbing the first term of the right-hand side into the left. □

We are now in position to prove the excess improvement estimate.

Lemma 3.5. For any fixed $\theta\in(0, \frac{1}{8})$, there exists $\varepsilon_1 = \varepsilon_1(n, \nu, L, p, q, [\mu]_{C^\alpha}, \theta)\in(0, 1)$ such that if

then

for some constant $c_{\rm dec} = c_{\rm dec}(n, \nu, L, p, q, [\mu]_{C^\alpha}, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)})\geq1$.

Proof. Step 1: By virtue of (3.4) we have

Choosing

we obtain

Therefore, combining (3.30) and (3.33) we finally obtain

Inserting this estimate in (3.32) and using (3.34), we then obtain

Step 2: Following [12, Lemma 4.2], we prove that Theorem 2.10 and the subsequent remark can be applied to function

Setting

by using the fact that $G$ is indeed a shifted $N$-function it can be seen that

for some constant $\tilde{c} > 0$. Moreover, by Corollary 2 we have that

Now, setting

we can choose $\varepsilon_1$ such that $\varepsilon < 1$. Moreover, combining with (3.39) and (3.38), we get

Inserting (3.40) and (3.30) into (3.21), written for ${\bf v}$ and for some ${\mathit{\boldsymbol{\varphi}}}\in C_{0, {\rm div}}^\infty(B_{\varrho})$ with $\|D{\mathit{\boldsymbol{\varphi}}}\|_{L^\infty(B_{\varrho})}\leq1$, we obtain

Then, choosing $\varepsilon_1$ small enough, the assumptions of Theorem 2.10 and Remark 1 for function ${\bf v}$ are in force. We denote by ${{\bf h}}$ the $\mathcal{A}$-Stokes function in $B_\varrho$ such that ${{\bf h}} = {\bf v}$ on $\partial B_\varrho$. Assertion (2.26) gives

Moreover, we also have

whence

for some constant $\bar{c} > 0$.

Step 3: Now, we go back to the estimate of the right hand side of (3.35). As a consequence of Lemma 3.2, with (3.34) and (3.29), we get

Now, the integral in the right hand side above can be treated exactly as in [12, pp. 24-25] starting from (3.43), to obtain the estimate}

Thus, we omit further details. This concludes the proof of (3.31). □

Now, we prove that the previous excess improvement estimate can be iterated at each scale. For this, we introduce the Morrey-type excess

Lemma 3.6. Let ${\mathit{\Phi}}(x_0, \varrho)$ and $\Theta(x_0, \varrho)$ be defined as in (3.1) and (3.45), respectively. Then there exist constants $\delta_*$, $\varepsilon_*, \varrho_*\in(0, 1]$, $M\geq1$ and $\vartheta$ such that the following holds: if the conditions

hold on $B_\varrho(x_0)\subseteq\Omega$ for $\varrho\in(0, \varrho_*]$, then

for every $m = 0, 1, \dots.$ In particular, this would imply

Moreover, for any $\beta\in(0, 1)$ the following Morrey-type estimate holds:

for all $y\in B_{\varrho/2}(x_0)$ and $r\in(0, \varrho/2]$.

Proof. As usual, we omit the explicit dependence on $x_0$. Let $\vartheta\in(0, 1)$ be such that

where $c_{\rm dec}$ is the constant of Lemma 3.5 depending only on $n, \nu, L, p, q, [\mu]_{C^\alpha}, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)}$. Correspondingly, let $\varepsilon_1 = \varepsilon_1(n, \nu, L, p, q, [\mu]_{C^\alpha}, \| \mathcal{E}{\bf u}\|_{L^p(\Omega)}, \vartheta)$ be the constant of Lemma 3.5, applied with the choice $\varepsilon = \vartheta^{n+2}$. We choose $\varepsilon_* > 0$ such that

where $c_{\frac{1}{2}}$ is the constant in the change-shift formula (2.12) with $\eta = \frac{1}{2}$, while $c_q$ is obtained with $\eta = \frac{1}{2^{q+1}}$. Moreover, we choose a radius $0 < \varrho_*\leq1$ such that

As a consequence, $\varepsilon_*$ and $\varrho_*$ have the same dependencies as $\varepsilon_1$. We argue by induction on $m$. Since (3.47) are trivially true for $m = 0$ by assumption (3.46), our aim is to show that if (3.47) holds for some $m\geq1$, then the corresponding inequalities hold with $m+1$ in place of $m$. Setting

in order to prove the second inequalities in (3.47) it will suffice to show that

With (3.47) at step $m$, the shift-change formula (2.12) with $\eta = \frac{1}{2}$, (3.51) and Lemma 2.7, we have the estimate

which proves $(3.47)_2$ for $m+1$. With this at hand, since the function $G(t^\frac{1}{p}): = H(x_0, t^\frac{1}{p})$ is convex, from Jensen's inequality we get

whence, passing to $G^{-1}$, we obtain (3.48). Now, we prove by induction the first inequality in (3.47) for $m+1$. From (3.47) at step $k$ and the choice of $\varrho_*$ as in (3.52), we have

and

Then, by virtue of Lemma 3.5 and Lemma 3.4 applied with radius $\vartheta^m\varrho$ in place of $\varrho$, and recalling the choice of $\vartheta$ (3.50), we get

To conclude the proof, we are left to prove $(3.47)_3$ and (3.49). We use $(3.46)_3$, the Poincaré inequality with $(3.47)_2$, $\varrho\leq \varrho_*$ and (3.52), (3.48) for every $i = 0, \dots, m$ and we get

Finally, setting $G(t): = H(x_0, t)$, since the iteration starting from $m = 0$ of the estimate $G^{-1}(E(B_{\vartheta^{m+1}\varrho}))\leq 2^{\frac{q}{p}}G^{-1}(E(B_{\vartheta^{m}\varrho}))$, obtained by (3.54) and (2.7), with (3.50) yields

and this estimate a fortiori holds if we consider $E(B_{\vartheta^m\varrho}(y))$ for $y\in B_{\varrho/2}$ in place of $E(B_{\vartheta^m\varrho})$, we deduce the Morrey-type estimate

for all $y\in B_{\varrho/2}$ and $r\leq\varrho/2$, which is equivalent to (3.49). This concludes the proof. □

4.   Proof of Theorem 1.1

We are now in position to prove Theorem 1.1.

Proof. Let $\delta_*$, $\varepsilon_*, \varrho_*\in(0, 1]$ be the constants of Lemma 3.6. We define

where $U_{z_0}$ is an open neighborhood of $z_0$. Assuming that $x_0\in\Omega$ complies with

i.e., $x_0\in\Omega\setminus(\Sigma_1\cup\Sigma_2)$, we will prove that $x_0\in\Omega_0$. We fix $\beta\in(0, 1)$ and choose $t\in(0, 1)$ such that

where $s_0 > 1$ is the exponent of Lemma 2.11. We also set

where $c_*$ is the constant $c_\eta$ in the change of shift formula for $\eta_*: = \frac{\varepsilon_*}{2^{n+1} (M_{x_0}+1) c_H}$. By virtue of (4.1), we can find $0 < \varrho$ with

and $B_{3\varrho}(x_0)\subset\subset\Omega$ such that

Step 1: We first establish the higher integrability of $H(x_0, 1+|D{\bf u}|)$ on $B_\varrho(x_0)$. Namely, we show that

where $s_0 > 1$ is the exponent of Lemma 2.11.

If $\mu$ satisfies (2.36), since (4.5) implies (2.38), from (2.39) we infer

whence

Now, from Korn's inequality, Poincaré inequality, (4.5), (2.39) and $\varrho\leq1$ we have

Analogously, using also Hölder's inequality and $\varrho\leq1$, we have

Combining the previous estimates, we then obtain

If (2.36) doesn't hold true, then it can be shown that $H(x_0, t)\leq c H(x, t)$. In this case, with (4.5) and (2.28), (2.43), we get

whence assertion (4.7) can still be inferred. Now, using Korn's inequality, Poincaré inequality, (4.5), (4.9) we have

In a similar way, using also that $\mu\leq1$, we have

which combined with the previous one gives

Assertion (4.6) then follows combining (4.8) and (4.10) and choosing $\tilde{c}: = \max\{c_1, c_2\}$.

Step 2: From Hölder's inequality and the choice of $t$ in (4.2) we get

Now, using Jensen's inequality for the concave function $\tilde{\Psi}$ such that $\frac{1}{2}\tilde{\Psi}(t)\leq \Psi(t) : = [H(x_0, t)]^\frac{1}{q} \leq \tilde{\Psi}(t)$ (see Lemma 2.4) and (4.5), we have

On the other hand, using Jensen's inequality for the convex map $t\to [H(x_0, t)]^{s_0}$ we obtain

Combining the previous estimates with (4.6) we finally get

for a constant $\bar{c} = \bar{c}(n, \nu, L, p, q, [\mu]_{C^\alpha}, \|\mu\|_{L^\infty}, \|1+| \mathcal{E}{\bf u}|\|_{L^p})$. With the choice of $\varepsilon$ in (4.3), this implies

Now, using the change of shift formula with $\eta_*: = \frac{\varepsilon_*}{2^{n+1} (M_{x_0}+1) c_H}$ and $H(x_0, 1)\geq1$,

Moreover, with (4.6) and (4.4) we have

By the absolute continuity of the integral, we can find an open neighborhood $U_{x_0}$ of $x_0$ such that

for every $x\in U_{x_0}$. Then, taking into account also (4.1), we can apply Lemma 3.6 at each point of $U_{x_0}$. This provides a Morrey-type estimate as in (3.49) proving that ${\bf u}\in C^{0, \beta}(U_{x_0}, {\mathbb{R}}^n)$ for every $\beta\in(0, 1)$ (see, e.g., [27, Lemma 3.15]). Thus, $x_0\in\Omega_0$ and the proof is concluded. □

Acknowledgments

The authors are members of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM. G. Scilla and B. Stroffolini have been supported by the project STAR PLUS 2020 – Linea 1 (21‐UNINA‐EPIG‐172) "New perspectives in the Variational modeling of Continuum Mechanics".

Conflict of interest

The authors declare no conflict of interest.