Feasible robust Liu estimator to combat outliers and multicollinearity effects in restricted semiparametric regression model

Dedicatoria. Al Ingenioso Hidalgo Don Ireneo.

1.   Introduction

In this paper we consider a nonlinear operator arising from the superposition of a classical $p$-Laplace operator and a fractional $p$-Laplace operator, of the form

with $s\in(0, 1)$ and $p\in [2, +\infty)$. Here, as usual, $\Delta_pu = \mathrm{div}(|\nabla u|^{p-2}\nabla u)$, while the fractional $p$-Laplace operator is defined (up to a multiplicative constant that we neglect) as

where $\mathrm{p.v.}$ stands for the principal value notation.

Given a bounded open set $\Omega\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem for the operator $\mathcal{L}_{p, s}$ with homogeneous Dirichlet boundary conditions (i.e., the eigenfunctions are prescribed to vanish in the complement of $\Omega$). In particular, we define $\lambda_1(\Omega)$ to be the smallest of such eigenvalues and $\lambda_2(\Omega)$ to be the second smallest one (in the sense made precise in ^[8,29]).

The main result that we present here is a version of the Hong–Krahn–Szegö inequality for the second Dirichlet eigenvalue $\lambda_2(\Omega)$, according to the following statement:

Theorem 1.1. Let $\Omega\subseteq{\mathbb{R}}^n$ be a bounded open set. Let $B$ be any Euclidean ball with volume $|\Omega|/2$. Then,

Furthermore, equality is never attained in $(1.2)$; however, the estimate is sharp in the following sense: if $\{x_j\}_j, \, \{y_j\}_j\subseteq{\mathbb{R}}^n$ are two sequences such that

and if we define $\Omega_j : = B_r(x_j)\cup B_r(y_j)$, then

To the best of our knowledge, Theorem 1.1 is new even in the linear case $p = 2$. Also, an interesting consequence of the fact that equality in (1.2) is never attained is that, for all $c > 0$, the shape optimization problem

does not admit a solution.

Remark 1.2. We stress that in this paper we deal with the case $p\geq 2$. As a matter of fact, as we shall see in Section 4, a key tool for the proof of Theorem 1.1 is the interior regularity of the ${\mathcal{L}}_{p, s}$-Dirichlet eigenfunctions (see Section 2 for the relevant definitions); we establish this regularity result by adapting an idea already exploited by Brasco, Lindgren and Schikorra ^[12] in the purely non-local case, which requires the bound $p\geq 2$.

On the other hand, after this paper was completed, the manuscript ^[28] appeared in the literature, in which the Authors mention a result implying the global Hölder regularity of the ${\mathcal{L}}_{p, s}$-Dirichlet eigenfunctions for every $p > 1$; see, precisely, [28,Remark 2.4]. Using this result, one could possibly drop the assumption $p\geq 2$ and prove Theorem 1.1 for every $p > 1$.

Before diving into the technicalities of the proof of Theorem 1.1, we devote Section 1.1 to showcase the available results on the shape optimization problems related to the first and the second eigenvalues of several elliptic operators.

1.1.   Shape optimization problems for the first and second eigenvalues in the context of elliptic (linear and nonlinear, classical and fractional) equations

1.1.1.   The case of the Laplacian

One of the classical shape optimization problems is related to the detection of the domain that minimizes the first eigenvalue of the Laplacian with homogeneous boundary conditions. This is the content of the Faber–Krahn inequality ^[24,32], whose result can be stated by saying that among all domains of fixed volume, the ball has the smallest first eigenvalue.

In particular, as a physical application, one has that that among all drums of equal area, the circular drum possesses the lowest voice, and this somewhat corresponds to our intuition, since a very elongated rectangular drum produces a high pitch related to the oscillations along the short edge.

Another physical consequence of the Faber–Krahn inequality is that among all the regions of a given volume with the boundary maintained at a constant temperature, the one which dissipates heat at the slowest possible rate is the sphere, and this also corresponds to our everyday life experience of spheres minimizing contact with the external environment thus providing the optimal possible insulation.

From the mathematical point of view, the Faber–Krahn inequality also offers a classical stage for rearrangement methods and variational characterizations of eigenvalues.

In view of the discussion in Section A, the subsequent natural question investigates the optimal shape of the second eigenvalue. This problem is addressed by the Hong–Krahn–Szegö inequality ^[31,33,37], which asserts that among all domains of fixed volume, the disjoint union of two equal balls has the smallest second eigenvalue.

Therefore, for the case of the Laplacian with homogeneous Dirichlet data, the shape optimization problems related to both the first and the second eigenvalues are solvable and the solution has a simple geometry.

It is also interesting to point out a conceptual connection between the Faber–Krahn and the Hong–Krahn–Szegö inequalities, in the sense that the proof of the second typically uses the first one as a basic ingredient. More specifically, the strategy to prove the Hong–Krahn–Szegö inequality is usually:

● Use that in a connected open set all eigenfunctions except the first one must change sign,

● Deduce that $\lambda_2(\Omega) = \max\{\lambda_1(\Omega_+), \lambda_1(\Omega_-)\}$, for suitable subdomain $\Omega_+$ and $\Omega_-$ which are either nodal domains for the second eigenfunction, if $\Omega$ is connected, or otherwise connected components of $\Omega$,

● Utilize the Faber–Krahn inequality to show that $\lambda_1(\Omega_\pm)$ is reduced if we replace $\Omega_\pm$ with a ball of volume $|\Omega_\pm|$,

● Employ the homogeneity of the problem to deduce that the volumes of these two balls are equal.

That is, roughly speaking, a cunning use of the Faber–Krahn inequality allows one to reduce to the case of disjoint balls, which can thus be addressed specifically.

1.1.2.   The case of the $p$-Laplacian

A natural extension of the optimal shape results for the Laplacian recalled in Section 1.1.1 is the investigation of the nonlinear operator setting and in particular the case of the $p$-Laplacian. This line of research was carried out in ^[10] in which a complete analogue of the results of Section 1.1.1 have been established for the $p$-Laplacian. In particular, the first Dirichlet eigenvalue of the $p$-Laplacian is minimized by the ball and the second by any disjoint union of two equal balls.

We stress that, in spite of the similarity of the results obtained, the nonlinear case presents its own specific peculiarities. In particular, in the case of the $p$-Laplacian one can still define the first eigenvalue by minimization of a Rayleigh quotient, in principle the notion of higher eigenvalues become more tricky, since discreteness of the spectrum is not guaranteed and the eigenvalues theory for nonlinear operators offers plenty of open problems at a fundamental level. For the second eingevalue however one can obtain a variational characterization in terms of a mountain-pass result, still allowing the definition of a spectral gap between the smallest and the second smallest eigenvalue.

1.1.3.   The cases of the fractional Laplacian and of the fractional $p$-Laplacian

We now consider the question posed by the minimization of the first and second eigenvalues in a nonlocal setting.

The optimal shape problems for the first eigenvalue of the fractional Laplacian with homogeneous external datum was addressed in ^[3,9,11,41], showing that the ball is the optimizer.

As for the nonlinear case, the spectral properties of the fractional $p$-Laplacian possess their own special features, see ^[26], and they typically combine the difficulties coming from the nonlocal world with those arising from the theory of nonlinear operators. In ^[11] the optimal shape problem for the first Dirichlet eigenvalue of the fractional $p$-Laplacian was addressed, by detecting the optimality of the ball as a consequence of a general Pólya–Szegö principle.

For the second eigenvalue, however, the situation in the nonlocal case is quite different from the classical one, since in general nonlocal energy functionals are deeply influenced by the mutual position of the different connected components of the domain, see ^[35].

In particular, the counterpart of the Hong–Krahn–Szegö inequality for the fractional Laplacian and the fractional $p$-Laplacian was established in ^[13] and it presents significant differences with the classical case: in particular, the shape optimizer for the second eigenvalue of the fractional $p$-Laplacian with homogeneous external datum does not exist and one can bound such an eigenvalue from below by the first eigenvalue of a ball with half of the volume of the given domain (and this is the best lower bound possible, since the case of a domain consisting of two equal balls drifting away from each other would attain such a bound in the limit).

1.1.4.   The case of mixed operators

The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant ^{[1,4,5,6,8,16,17,20,21,23,27,39]}, and of their importance in practical applications such as the animal foraging hypothesis ^[22,36].

In regard to the shape optimization problem, a Faber–Krahn inequality for mixed local and nonlocal linear operators when $p = 2$ has been established in ^[7], showing the optimality of the ball in the minimization of the first eigenvalue. The corresponding inequality for the nonlinear setting presented in (1.1) will be given here in the forthcoming Theorem 4.1.

The inequality of Hong–Krahn–Szegö type for mixed local and nonlocal linear operators presented in (1.1) would thus complete the study of the optimal shape problems for the first and second eigenvalues of the operator in (1.1).

1.2.   Plan of the paper

The rest of this paper is organized as follows. Section 2 sets up the notation and collects some auxiliary results from the existing literature.

In Section 3 we discuss a regularity theory which, in our setting, plays an important role in the proof of Theorem 1.1 in allowing us to speak about nodal regions for the corresponding eigenfunction (recall the bullet point strategy presented on page 3). In any case, this regularity theory holds in a more general setting and can well come in handy in other situations as well.

Section 4 introduces the corresponding Faber–Krahn inequality for the operator in (1.1) and completes the proof of Theorem 1.1.

In Appendix A we also discuss the importance of first and second eigenvalues in general problems of applied mathematics (not necessarily related to partial differential equations, nor to integro-differential equations).

2.   Preliminaries

To deal with the nonlinear and mixed local/nonlocal operator in (1.1), given an open and bounded set $\Omega\subseteq{\mathbb{R}}^n$, it is convenient to introduce the space

defined as the closure of $C_0^{\infty}(\Omega)$ with respect to the global norm

We highlight that, since $\Omega$ is bounded, $\mathcal{X}^{1, p}_0(\Omega)$ can be equivalently defined by taking the closure of $C_0^\infty(\Omega)$ with respect to the full norm

however, we stress that $\mathcal{X}_0^{1, p}(\Omega)$ is different from the usual space $W^{1, p}_0(\Omega)$, which is defined as the closure of $C_0^{\infty}(\Omega)$ with respect to the norm

As a matter of fact, while the belonging of a function $u$ to $W^{1, p}_0(\Omega)$ only depends on its behavior inside of $\Omega$ (actually, $u$ does not even need to be defined outside of $\Omega$), the belonging of $u$ to $\mathcal{X}_0^{1, p}(\Omega)$ is a global condition, and it depends on the behavior of $u$ on the whole space ${\mathbb{R}^n }$ (in particular, $u$ has to be defined on ${\mathbb{R}}^n$). Just to give an example of the difference between these spaces, let $u\in C^\infty_0({\mathbb{R}}^n)\setminus\{0\}$ be such that

Since $u\equiv 0$ inside of $\Omega$, we clearly have that $u\in W^{1, p}_0(\Omega)$; on the other hand, since $u\not\equiv 0$ in ${\mathbb{R}}^n\setminus\Omega$, one has $u\notin \mathcal{X}_0^{1, p}(\Omega)$ (even if $u\in W^{1, p}({\mathbb{R}}^n)$).

Although they do not coincide, the spaces $\mathcal{X}_0^{1, p}(\Omega)$ and $W^{1, p}_0(\Omega)$ are related: to be more precise, using [14,Proposition 9.18] and taking into account the definition of $\mathcal{X}_0^{1, p}(\Omega)$, one can see that

$({\rm{i}})$ if $u\in W_0^{1, p}(\Omega)$, then $u\cdot{\bf{1}}_\Omega\in \mathcal{X}_0^{1, p}(\Omega)$;

$({\rm{ii}})$ if $u\in \mathcal{X}_0^{1, p}(\Omega)$, then $u\big|_{\Omega}\in W_0^{1, p}(\Omega)$.

Moreover, we can actually characterize $\mathcal{X}_0^{1, p}(\Omega)$ as follows:

The main issue in trying to use (ⅰ)–(ⅱ) to identify $W_0^{1, p}(\Omega)$ with $\mathcal{X}_0^{1, p}(\Omega)$ is that, if $u$ is globally defined and $u\in W^{1, p}({\mathbb{R}}^n)$, then

however, we cannot say in general that $u = u\cdot{\bf{1}}_\Omega$. Even if they cannot allow to identify $\mathcal{X}_0^{1, p}(\Omega)$ with $W_0^{1, p}(\Omega)$, assertions (ⅰ)–(ⅱ) can be used to deduce several properties of the space $\mathcal{X}_0^{1, p}(\Omega)$ starting from their analog in $W_0^{1, p}(\Omega)$; for example, we have the following fact, which shall be used in the what follows:

Remark 2.1. In the particular case when the open set $\Omega$ is of class $C^1$, it follows from [14,Proposition 9.18] that, if $u\in W^{1, p}({\mathbb{R}}^n)$ and $u = 0$ a.e. in ${\mathbb{R}}^n\setminus\Omega$, then

As a consequence, we have

This fact shows that, when $\Omega$ is sufficiently regular, $\mathcal{X}_0^{1, p}(\Omega)$ coincides with the space $\mathbb{X}_p(\Omega)$ introduced in ^[5] (for $p = 2$) and in ^[8] (for a general $p > 1$).

For future reference, we introduce the following set

After these preliminaries, we can turn our attention to the Dirichlet problem for the operator ${\mathcal{L}}_{p, s}$. Throughout the rest of this paper, to simplify the notation we set

Moreover, we define

Definition 2.2. Let $q\geq (p^*)'$, and let $f\in L^q(\Omega)$. We say that a function $u\in W^{1, p}({\mathbb{R}}^n)$ is a weak solution to the equation

if, for every $\phi\in \mathcal{X}_0^{1, p}(\Omega)$, the following identity is satisfied

Moreover, given any $g\in W^{1, p}({\mathbb{R}}^n)$, we say that a function $u\in W^{1, p}({\mathbb{R}}^n)$ is a weak solution to the $({\mathcal{L}_{p, s})}$-Dirichlet problem

if $u$ is a weak solution to (2.3) and, in addition,

Remark 2.3. (1) We point out that the above definition is well-posed: indeed, if $u, v\in W^{1, p}(\Omega)$, by Hölder's inequality and [19,Proposition 2.2] we get

Moreover, since $f\in L^q(\Omega)$ and $q\geq (p^*)'$, again by Hölder's inequality and by the Sobolev Embedding Theorem (applied here to $v\in W^{1, p}({\mathbb{R}}^n)$), we have

(2) If $W^{1, p}({\mathbb{R}}^n)$ is is a weak solution to the $({\mathcal{L}}_{p, s})$-Dirichlet problem (2.5), it follows from the definition of $\mathcal{X}_0^{1, p}(\Omega)$ that

Thus, $\mathcal{X}_0^{1, p}(\Omega)$ is the 'right space' for the weak formulation of (2.5).

With Definition 2.2 at hand, we now introduce the notion of Dirichlet eigenvalue/eigenfunction for the operator ${\mathcal{L}}_{p, s}$.

Definition 2.4. We say that $\lambda\in{\mathbb{R}}$ is a Dirichlet eigenvalue for ${\mathcal{L}}_{p, s}$ if there exists a solution $u\in W^{1, p}(\Omega)\setminus\{0\}$ of the $({\mathcal{L}}_{p, s})$-Dirichlet problem

In this case, we say that $u$ is an eigenfunction associated with $\lambda$.

Remark 2.5. We note that Definition 2.4 is {well-posed}. Indeed, if $u$ is any function in $W^{1, p}({\mathbb{R}}^n)$, by the Sobolev Embedding Theorem we have

then, a direct computation shows that $q : = p^*/(p-1)\geq (p^*)'$. As a consequence, the notion of weak solution for (2.6) agrees with the one contained in Definition 2.2. In particular, if $u$ is an eigenfunction associated with some eigenvalue $\lambda$, then

and thus $u\big|_\Omega\in W_0^{1, p}(\Omega)$ and $u = 0 \; {\rm{a.e}}.\, {\rm{in}} \; {\mathbb{R}}^n\setminus\Omega$.

After these definitions, we close the section by reviewing some results about eigenvalues/eigenfucntions for ${\mathcal{L}}_{p, s}$ which shall be used here below.

To begin with, we recall the following result proved in [8,Proposition 5.1] which establishes the existence of the smallest eigenvalue and detects its basic properties.

Proposition 2.6. The smallest eigenvalue $\lambda_1(\Omega)$ for the operator ${\mathcal{L}}_{p, s}$ is strictly positive and satisfies the following properties:

$1)$ $\lambda_1(\Omega)$ is simple;

$2)$ the eigenfunctions associated with $\lambda_1(\Omega)$ do not change sign in ${\mathbb{R}}^n$;

$3)$ every eigenfunction associated to an eigenvalue

is nodal, i.e., sign changing.

Moreover, $\lambda_1(\Omega)$ admits the following variational characterization

where $\mathcal{M}(\Omega)$ is as in $(2.1)$. The minimum is always attained, and the eigenfunctions for ${\mathcal{L}}_{p, s}$ associated with $\lambda_1(\Omega)$ are precisely the minimizers in $(2.7)$.

We observe that, on account of Proposition 2.6, there exists a unique non-negative eigenfunction $u_0\in\mathcal{M}(\Omega)\subseteq\mathcal{X}_0^{1, p}(\Omega)$ associated with $\lambda_1(\Omega)$; in particular, $u_0$ is a minimizer in (2.7), so that

We shall refer to $u_0$ as the principal eigenfunction of ${\mathcal{L}}_{p, s}$.

The next result was proved in [29,Section 5] and concerns the second eigenvalue for ${\mathcal{L}}_{p, s}$.

Theorem 2.7. We define:

where $\mathcal{K} : = \{f:S^1\to\mathcal{M}(\Omega):\, {{f \; is \; continuous \; and \; odd}}\}$, with $\mathcal{M}(\Omega)$ as in (2.1).

Then:

$1)$ $\lambda_2(\Omega)$ is an eigenvalue for ${\mathcal{L}}_{p, s}$;

$2)$ $\lambda_2 (\Omega) > \lambda_1(\Omega)$;

$3)$ If $\lambda > \lambda_1(\Omega)$ is an eigenvalue for ${\mathcal{L}}_{p, s}$, then $\lambda \geq \lambda_2(\Omega)$.

In the rest of this paper, we shall refer to $\lambda_1(\Omega)$ and $\lambda_2(\Omega)$ as, respectively, the first and second eigenvalue of ${\mathcal{L}}_{p, s}$ (in $\Omega$). We notice that, as a consequence of (2.7)–(2.9), both $\lambda_1(\cdot)$ and $\lambda_2(\cdot)$ are translation-invariant, that is,

To proceed further, we now recall the following global boundedness result for the eigenfunctions of ${\mathcal{L}}_{p, s}$ (associated with any eigenvalue $\lambda$) established in [8,Theorem 4.4].

Theorem 2.8. Let $u\in \mathcal{X}_0^{1, p}(\Omega)\setminus\{0\}$ be an eigenfunction for ${\mathcal{L}}_{p, s}$, associated with an eigenfunction $\lambda \geq \lambda_1(\Omega)$. Then, $u\in L^\infty({\mathbb{R}}^n)$.

Remark 2.9. Actually, in [8,Theorem 4.4] it is proved the global boundedness of any non-negative weak solution to the more general Dirichlet problem

where $f:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory function satisfying the properties

$(a)$ $f(\cdot, t)\in L^\infty(\Omega)$ for every $t\geq 0$;

$(b)$ There exists a constant $c_p > 0$ such that

However, by scrutinizing the proof of the theorem, it is easy to check that the same argument can be applied to our context, where we have

but we do not make any assumption on the sign of $u$ (see also [40,Proposition 4]).

Finally, we state here an algebraic lemma which shall be useful in the forthcoming computations.

Lemma 2.10. Let $1 < p < +\infty$ be fixed. Then, the following facts hold.

$1)$ For every $a, b\in \mathbb{R}$ such that $ab\leq 0$, it holds that

$2)$ There exists a constant $c_p > 0$ such that

3.   Interior regularity of the eigenfunctions

In this section we prove the interior Hölder regularity of the eigenfunctions for ${\mathcal{L}}_{p, s}$, which is a fundamental ingredient for the proof of Theorem 1.1. As a matter of fact, on account of Theorem 2.8, we establish the interior Hölder regularity for any bounded weak solution of the non-homogeneous equation (2.3), when

In what follows, we tacitly understand that

moreover, $\Omega\subseteq{\mathbb{R}}^n$ is a bounded open set and $f\in L^\infty(\Omega)$.

Remark 3.1. The reason why we restrict ourselves to consider $2\leq p\leq n$ follows from the definition of weak solution to (2.3).

Indeed, if $u$ is a weak solution to (2.3), then by definition we have $u\in W^{1, p}({\mathbb{R}}^n)$; as a consequence, if $p > n$, by the classical Sobolev Embedding Theorem we can immediately conclude that $u\in C^{0, \gamma}({\mathbb{R}}^n)$, where $\gamma = 1-n/p$.

In order to state (and prove) the main result of this section, we need to fix a notation: for every $z\in{\mathbb{R}}^n, \, \rho > 0$ and $u\in L^p({\mathbb{R}}^n)$, we define

The quantity $\mathrm{Tail}(u, z, \rho)$ is referred to as the $({\mathcal{L}_{p, s})}$-tail of $u$, see e.g., ^[18,34].

Theorem 3.2. Let $f\in L^\infty(\Omega)$, and let $u\in W^{1, p}({\mathbb{R}}^n)\cap L^\infty({\mathbb{R}}^n)$ be a weak solution to $(2.3)$. Then, there exists some $\beta = \beta(n, s, p)\in (0, 1)$ such that $u\in C^{0, \, \beta}_{{\mathrm{loc}}}(\Omega)$.

More precisely, for every ball $B_{R_0}(z)\Subset\Omega$ we have the estimate

where

and $C > 0$ is a constant independent of $u$ and $R_1$.

In order to prove Theorem 3.2, we follow the approach in ^[12]; broadly put, the main idea behind this approach is to transfer to the solution $u$ the oscillation estimates proved in ^[27] for the ${\mathcal{L}_{p, s}}$-harmonic functions.

To begin with, we establish the following basic existence/uniqueness result for the weak solutions to the $({\mathcal{L}}_{p, s})$-Dirichlet problem (2.5).

Proposition 3.3. Let $f\in L^\infty(\Omega)$ and $g\in W^{1, p}({\mathbb{R}}^n)$ be fixed. Then, there exists a unique solution $u = u_{f, \, g}\in W^{1, p}({\mathbb{R}}^n)$ to the Dirichlet problem (2.5).

Proof. We consider the space

and the functional $J:\mathbb{W}(g)\to{\mathbb{R}}$ defined as follows:

On account of [12,Remark 2.13], we have that $J$ is strictly convex; hence, by using the Direct Methods in the Calculus of Variations, we derive that $J$ has a unique minimizer $u = u_{f, \, g}$ on $\mathbb{W}(g)$, which is the unique weak solution to (2.5).

Thanks to Proposition 3.3, we can prove the following result. Throughout the rest of this paper, if $u\in L^1_{\mathrm{loc}}(\Omega)$ and if $A\subseteq \Omega$ is a measurable set with positive measure, we adopt the classical notation

In particular, if $A = B(x_0, r)$, we set

Lemma 3.4. Let $f\in L^\infty(\Omega)$ and let $u\in W^{1, p}({\mathbb{R}}^n)$ be a weak solution to $(2.3)$. Moreover, let $B$ be a given Euclidean ball such that $B\Subset\Omega$, and let $v\in W^{1, p}({\mathbb{R}}^n)$ be the unique weak solution to the Dirichlet problem

Then, there exists a constant $C = C(n, s, p) > 0$ such that

In particular, we have

Proof. We observe that the existence of $v$ is ensured by Proposition 3.3. Then, taking into account that $u$ is a weak solution to (2.3) and $v$ is the weak solution to (3.2), for every $\phi\in \mathcal{X}^{1, p}_0(B)$ we get

Choosing, in particular, $\phi : = u-v$ (notice that, since $v$ is a weak solution of (3.2), by definition we have $v-u\in\mathcal{X}_0^{1, p}(\Omega)$), we obtain

where $t_1: = u(x)-u(y), \, t_2 : = v(x)-v(y)$ and

Now, an elementary computation based on Cauchy-Schwarz's inequality gives

Moreover, since $p\geq 2$, by exploiting [12,Remark A.4] we have

where $C = C(p) > 0$ is a constant only depending on $p$. Thus, by combining (3.5), (3.6) and (3.7), we obtain the following estimate:

where we have also used the Hölder's inequality and $p^*_s > 1$ is the so-called fractional critical exponent, that is,

Finally, by applying the fractional Sobolev inequality to $\phi = u-v$ (notice that $\phi$ is compactly supported in $B$), we get

and this readily yields the desired (3.3). To prove (3.4) we observe that, by using the Hölder inequality and again the fractional Sobolev inequality, we have

thus, estimate (3.4) follows directly from (3.3).

Using Lemma 3.4, we can prove the following excess decay estimate.

Lemma 3.5. Let $f\in L^\infty(\Omega)$ and let $u\in W^{1, p}({\mathbb{R}}^n)$ be a weak solution to $(2.3)$. Moreover, let $x_0\in\Omega$ and let $R \in (0, 1)$ be such that $B_{4R}(x_0)\Subset \Omega$.

Then, for every $0 < r\leq R$ we have the estimate

where $C, \, \gamma$ and $\alpha$ are positive constants only depending on $n$, $s$ and $p$.

Proof. Let $v\in W^{1, p}({\mathbb{R}}^n)$ be the unique weak solution to the problem

We stress that the existence of $v$ is guaranteed by Proposition 3.3. We also observe that, for every $r\in(0, R]$, we have that

As a consequence, we obtain

where $\kappa = \kappa_p > 0$ is a constant only depending on $p$.

Now, since $B_{3R}(x_0)\Subset \Omega$ and $v$ is the weak solution to (3.9), by Lemma 3.4 we have

On the other hand, since $v\in W^{1, p}({\mathbb{R}}^n)$ and $v$ is ${\mathcal{L}}_{p, s}$-harmonic in $B_{3R}(x_0)$ (that is, ${\mathcal{L}}_{p, s}v = 0$ in the weak sense), we can apply [27,Theorem 5.1], obtaining

where $C$ and $\alpha$ are positive constants only depending on $n$, $s$ and $p$. By combining estimates (3.11)-(3.12) with (3.10), we then get

where we have set

To complete the proof of (3.8) we observe that, since $u \equiv v$ a.e. on ${\mathbb{R}}^n\setminus B_{3R}(x_0)$ (and $0 < R \leq 1$), by definition of $\mathrm{Tail}(v, x_0, R)$ we have

Moreover, by using again Lemma 3.4, we get

Thus, by inserting (3.15)-(3.16) into (3.13), we obtain the desired (3.8).

By combining Lemmata 3.4 and 3.5, we can provide the

Proof of Theorem 3.2. The proof follows the lines of [12,Theorem 3.6]. First, we consider a ball $B_{R_0}(z) \subset\subset \Omega$ and we define the quantities

Thus, we can choose a point $x_0 \in B_{R_0}(z)$ and the ball $B_{4R}(x_0)$, where $R < \min\{1, \tfrac{d}{8}\}$. In particular, this implies that $B_{4R}(x_0)\subset B_{R_1}(z)$. Since $R < 1$, we can then apply Lemma 3.5: this gives, for every $0 < r \leq R$,

where $\gamma > 0$ is as in (3.14). Now, we notice that for every $x \notin B_{R_1}(z)$ it holds that

Therefore, we have

for a constant $C$ depending on $n$, $s$ and $p$. We recall that in the last estimate we exploited that

Consequently, continuing the estimate started with (3.18), we find that

We can now define the positive number

and take $r: = R^{\theta}$ in (3.19), which yields

where we have set

This shows that $u \in \mathcal{L}^{p, n+\beta\gamma}(B_{R_0}(z))$, the Campanato space isomorphic to the Hölder space $C^{0, \, \beta}(\overline{B_{R_0}(z)})$. This completes the proof of Theorem 3.2.

By gathering together Theorems 2.8 and 3.2, we can easily prove the needed interior Hölder regularity of the eigenfunctions of ${\mathcal{L}}_{p, s}$.

Theorem 3.6. Let $\lambda\geq \lambda_1(\Omega)$ be an eigenvalue of ${\mathcal{L}}_{p, s}$, and let $\phi_\lambda\in \mathcal{X}_0^{1, p}(\Omega)\setminus\{0\}$ be an eigenfunction associated with $\lambda$. Then, $\phi_\lambda\in C(\Omega)$.

Proof. On account of Theorem 2.8, we know that $\phi_\lambda\in L^\infty({\mathbb{R}}^n)$. As a consequence, $\phi_\lambda$ is a globally bounded weak solution to (2.3), with

We are then entitled to apply Theorem 3.2, which ensures that $\phi_\lambda\in C^{0, \, \beta}_{{\mathrm{loc}}}(\Omega)$ for some $\beta = \beta(n, s, p)\in (0, 1)$. This ends the proof of Theorem 3.6.

4.   The Hong-Krahn-Szegö inequality for ${\mathcal{L}}_{p, s}$

In this last section of the paper we provide the proof of Theorem 1.1. Before doing this, we establish two preliminary results.

First of all, we prove the following Faber-Krahn type inequality for ${\mathcal{L}_{p, s} }$.

Theorem 4.1. Let $\Omega\subseteq{\mathbb{R}}^n$ be a bounded open set, and let $m: = |\Omega|\in (0, \infty)$. Then, if $B^{(m)}$ is any Euclidean ball with volume $m$, one has

Moreover, if the equality holds in $(4.1)$, then $\Omega$ is a ball.

Proof. The proof is similar to that in the linear case, see [7,Theorem 1.1]; however, we present it here in all the details for the sake of completeness.

To begin with, let $\widehat{B}^{(m)}$ be the Euclidean ball with centre $0$ and volume $m$. Moreover, let $u_0\in\mathcal{M}(\Omega)$ be the principal eigenfunction for ${\mathcal{L}}_{p, s}$. We recall that, by definition, $u_0$ is the unique non-negative eigenfunction associated with the first eigenvalue $\lambda_1(\Omega)$; in particular, we have (see (2.8))

Then, we define $u_0^\ast:{\mathbb{R}}^n\to{\mathbb{R}}$ as the (decreasing) Schwarz symmetrization of $u_0$. Now, since $u_0\in\mathcal{M}(\Omega)$, from the well-known inequality by Pólya and Szegö (see e.g., ^[38]) we deduce that

Furthermore, by [2,Theorem 9.2] (see also [25,Theorem A.1]), we also have

Gathering all these facts and using (4.2), we get

From this, since $\lambda_{1}(\cdot)$ is translation-invariant, we derive the validity of (4.1) for every Euclidean ball $B^{(m)}$ with volume $m$.

To complete the proof of Theorem 4.1, let us suppose that

for some (and hence, for every) ball $B^{(m)}$ with $|B^{(m)}| = m$. By (4.5) we have

In particular, by (4.3) and (4.4) we get

We are then in the position to apply once again [25,Theorem A.1], which ensures that $u_0$ must be proportional to a translation of a symmetric decreasing function. As a consequence of this fact, we immediately deduce that

must be a ball (up to a set of zero Lebesgue measure). This completes the proof of Theorem 4.1.

Then, we establish the following lemma on nodal domains.

Lemma 4.2. Let $\lambda > \lambda_1(\Omega)$ be an eigenvalue of ${\mathcal{L}}_{p, s}$, and let $\phi_\lambda\in \mathcal{X}_0^{1, p}(\Omega)\setminus\{0\}$ be an eigenfunction associated with $\lambda$. We define the sets

Then $\lambda > \max\left\{\lambda_1(\Omega^{+}), \lambda_1(\Omega^-)\right\}$.

The proof of Lemma 4.2 takes inspiration from [13,Lemma 6.1] (see also [29,Lemma 4.2]).

Proof of Lemma 4.2. First of all, on account of Theorem 3.6 we have that the sets $\Omega^+$ and $\Omega^-$ are open, and therefore the eigenvalues $\lambda_{1}(\Omega^{\pm})$ are well–defined.

Moreover, thanks to Proposition 2.6, we know that $\phi_\lambda$ changes sign in $\Omega$, and therefore it is convenient to write $\phi_{\lambda} = \phi_{\lambda}^+ - \phi_{\lambda}^-,$ where $\phi_{\lambda}^+$ and $\phi_{\lambda}^-$ denote, respectively, the positive and negative parts of $\phi_{\lambda}$, with the convention that both the functions $\phi_{\lambda}^+$ and $\phi_{\lambda}^-$ are non-negative.

Let us now prove that $\lambda > \lambda_{1}(\Omega^+)$. By using the fact that $\phi_{\lambda}$ is an eigenfuction of ${\mathcal{L}}_{p, s}$ corresponding to $\lambda$, it follows that

In consideration of the fact that $\phi_{\lambda}^{+} \in \mathcal{X}_{0}^{1, p}(\Omega)$, we can take $v = \phi_{\lambda}^+$ as a test function.

Now, since

we easily get that

Moreover, since both $\Omega_+$ and $\Omega_-$ are non-void open set (remind that $\phi_\lambda$ is continuous on $\Omega$ and it changes sign in $\Omega$), we have

and

We can therefore exploit Lemma 2.10-(1) with

obtaining (remind that, by assumption, $p\geq 2$)

where we used the variational characterization of $\lambda_{1}(\Omega^+)$, see (2.7). In particular, this gives that $\lambda > \lambda_{1}(\Omega^+)$. With a similar argument (see e.g., [13,Lemma 6.1]), one can show that $\lambda > \lambda_{1}(\Omega^-)$ as well, and this closes the proof of Lemma 4.2.

By virtue of Theorem 4.1 and Lemma 4.2, we can provide the

Proof of Theorem 1.1. We split the proof into two steps.

Step Ⅰ: In this step, we prove inequality (1.2). To this end, let $\phi\in\mathcal{M}(\Omega)$ be a $L^p$-normalized eigenfunction associated with $\lambda_2(\Omega)$ (recall the definition of the space $\mathcal{M}(\Omega)$ in (2.1)). On account of Theorem 2.7, we know that $\phi\in C(\Omega)$.

Moreover, since $\phi$ changes sign in $\Omega$ (see Proposition 2.6), we can define the non-void open sets

Then, by combining Lemma 4.2 with Theorem 4.1, we get

where $B_+$ is a Euclidean ball with volume equal to $|\Omega_+|$ and $B_-$ is a Euclidean ball with volume $|\Omega_-|$.

Now, since $\Omega_+\cup\Omega_- = \Omega$, we have

Taking into account this inequality, we claim that

being $B$ a ball of volume $|\Omega|/2$. In order to prove (4.7), we distinguish three cases.

$({\rm{i}})$ $|B_+|, \, |B_-|\leq m/2$. In this case, since $\lambda_1(\cdot)$ is translation-invariant, we can assume without loss of generality that $B\subseteq B_+, \, B_-$; as a consequence, since $\lambda_1(\cdot)$ is non-increasing, we obtain

and this proves the claimed (4.7).

$({\rm{ii}})$ $|B_-| < m/2 < |B_+|$. In this case, we can assume that $B_-\subseteq B\subseteq B_+$; from this, since $\lambda_1(\cdot)$ is non-increasing, we obtain

and this immediately implies the claimed (4.7).

$({\rm{iii}})$ $|B_+| < m/2 < |B_-|$. In this last case, it suffices to interchange the roles of the balls $B_-$ and $B_+$, and to argue exactly as in case $({\rm{ii}})$.

Gathering (4.6) and (4.7), we obtain the claim in (1.2).

Step Ⅱ: Now we prove the sharpness of (1.2). To this end, according to the statement of the theorem, we fix $r > 0$ and we define

where $\{x_j\}_j, \, \{y_j\}_j\subseteq{\mathbb{R}}^n$ are two sequences satisfying

On account of (4.8), we can assume that

Let now $u_0\in \mathcal{M}(B_r)$ be a $L^p$-normalized eigenfunction associated with $\lambda_1(B_r)$ (here, $B_r = B_r(0)$). For every natural number $j\geq 1$, we set

Since $\lambda_1(\cdot)$ is translation-invariant, it is immediate to check that $\phi_j$ and $\psi_j$ are normalized eigenfunctions associated with $\lambda_1(B_r(x_j))$ and $\lambda_1(B_r(y_j))$, respectively.

Moreover, taking into account (4.9), it is easy to see that

and $\phi_j\psi_j\equiv 0$ on ${\mathbb{R}}^n$.

We then consider the function $f$ defined as follows:

Taking into account that $B_r(x_j), \, B_r(y_j)\subseteq\Omega_j$ and $u_0\in \mathcal{M}(B_r)$, it is readily seen that $f(S_1)\subseteq \mathcal{X}_0^{1, p}(\Omega_j)$.

Furthermore, the function $f$ is clearly odd and continuous. Also, using (4.9) and the fact that $\phi\equiv 0$ out of $B_r$, one has

We are thereby entitled to use $f$ in the definition of $\lambda_2(\Omega)$, see (2.9): setting $a_j: = \phi_j(x)-\phi_j(y)$ and $b_j: = \psi_j(x)-\psi_j(y)$ to simplify the notation, this gives, together with (4.9) and (4.11), that

On the other hand, by applying Lemma 2.10-(2), we get

where we have also used that $u_0$ is a normalized eigenfunction associated with the first eigenvalue $\lambda_1(B_r)$.

Summarizing, we have proved that

We now set

and we claim that $\mathcal{R}_j\to 0$ as $j\to+\infty$.

Indeed, since $\phi_j\psi_j\equiv 0$ on ${\mathbb{R}}^n$, we have that

As a consequence, recalling (4.11)

Taking into account (4.8), we thereby conclude that

Gathering together (4.12) and (4.13), we obtain the desired result in (1.3).

Acknowledgments

The authors are members of INdAM. S. Biagi is partially supported by the INdAM-GNAMPA project Metodi topologici per problemi al contorno associati a certe classi di equazioni alle derivate parziali. S. Dipierro and E. Valdinoci are members of AustMS. S. Dipierro is supported by the Australian Research Council DECRA DE180100957 PDEs, free boundaries and applications. E. Valdinoci is supported by the Australian Laureate Fellowship FL190100081 Minimal surfaces, free boundaries and partial differential equations. E. Vecchi is partially supported by the INdAM-GNAMPA project Convergenze variazionali per funzionali e operatori dipendenti da campi vettoriali.

Conflict of interest

The authors declare no conflict of interest.