On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1

Francesco Della Pietra; Carlo Nitsch; Francescantonio Oliva; Cristina Trombetti

doi:10.1515/acv-2021-0085

Article Open Access

On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1

Francesco Della Pietra , Carlo Nitsch , Francescantonio Oliva and Cristina Trombetti

Published/Copyright: June 29, 2022

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From the journal Advances in Calculus of Variations Volume 16 Issue 4

Abstract

In this paper, we study the Γ-limit, as p → 1 , of the functional

J p ⁢ ( u ) = ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p ,

where Ω is a smooth bounded open set in ℝ N , p > 1 and β is a real number. Among our results, for β > - 1 , we derive an isoperimetric inequality for

Λ ⁢ ( Ω , β ) = inf u ∈ BV ⁡ ( Ω ) , u ≢ 0 ⁡ | D ⁢ u | ⁢ ( Ω ) + min ⁡ ( β , 1 ) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u |

which is the limit as p → 1 + of λ ⁢ ( Ω , p , β ) = min u ∈ W 1 , p ⁢ ( Ω ) ⁡ J p ⁢ ( u ) . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ⁢ ( Ω , β ) when β ∈ ( - 1 , 0 ) and minimizes Λ ⁢ ( Ω , β ) when β ∈ [ 0 , ∞ ) .

Keywords: First Robin eigenvalue; isoperimetric inequalities

MSC 2010: 49R05; 35P30

1 Introduction

In the recent years, great attention has been devoted to the study of isoperimetric inequalities involving the following functional:

(1.1) λ ⁢ ( Ω , p , β ) = min u ∈ W 1 , p ⁢ ( Ω ) ⁡ ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p ,

where Ω is a smooth bounded open set in ℝ N , p > 1 and β is a real number. It is clear that the minimizers φ p of (1.1) are solutions to the following Robin boundary value problem:

{ - Δ p ⁢ φ p = λ ⁢ ( Ω , p , β ) ⁢ | φ p | p - 2 ⁢ φ p in ⁢ Ω , | ∇ ⁡ φ p | p - 2 ⁢ ∂ ⁡ φ p ∂ ⁡ ν + β ⁢ | φ p | p - 2 ⁢ φ p = 0 on ⁢ ∂ ⁡ Ω .

In the study of shape optimization problems for λ ⁢ ( Ω , p , β ) , there is a striking difference between the cases β < 0 and β ≥ 0 . We first aim to give an idea of the existing literature, which is huge for these problems. If β is nonnegative, it is widely known that the following Faber–Krahn inequality holds (see [5] for p = N = 2 , see [9] for p = 2 and N ≥ 2 , see [6, 8] for any 1 < p < + ∞ , and see also [10] for a more general nonlinear setting):

(1.2) λ ⁢ ( Ω , p , β ) ≥ λ ⁢ ( Ω # , p , β ) ,

where Ω # is the ball, centered at the origin, with the same volume of Ω; namely the ball minimizes λ in the class of sets with given volume.

On the contrary, when β < 0 , the situation becomes more delicate. Indeed, if p = 2 , it was conjectured in [4] that the ball maximizes λ ⁢ ( Ω , 2 , β ) among smooth bounded domains Ω of fixed volume. In that same paper, this property was initially proved when N = 2 , | β | is small enough and for nearly circular domains which are, roughly speaking, infinitesimal perturbations of a circle. Later, the conjecture has been disproved in [13], in any dimension, for | β | great enough and in the case of Ω as a spherical shell. By the way, for bounded planar domains of class C 2 , Freitas and Krejčiřík [13] show the existence of a value β * such that the ball maximizes the eigenvalue for any β ∈ [ β * , 0 ] ; this is mainly proved by a suitable asymptotic expansion in β for the first eigenvalue. Among other things, the former result has been extended in [17] for any N ≥ 3 and p > 1 when Ω is any simply connected bounded C 2 domain once again for | β | small enough. Moreover, it is worth mentioning that in [12] Ferone, Nitsch and Trombetti show the validity of (1.2) for any given β < 0 in the class of Lipschitz sets which are, in some sense, close to a ball in the Hausdorff sense. On the other hand, for any β ≤ 0 , λ ⁢ ( Ω , p , β ) is maximized by the ball when Ω belongs to a suitable class of sets with the same perimeter, in particular C 2 planar bounded domains [2] or convex bounded domains in ℝ N (see [7]).

In the current paper, we deal with the case p → 1 in problem (1.1); as one should expect here, the natural associated space is BV ⁡ ( Ω ) . As we will see, λ ⁢ ( Ω , p , β ) converges to

(1.3) Λ ⁢ ( Ω , β ) = inf u ∈ BV ⁡ ( Ω ) , u ≢ 0 ⁡ | D ⁢ u | ⁢ ( Ω ) + min ⁡ ( β , 1 ) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | ,

where | D ⁢ u | ⁢ ( Ω ) stands for the total variation of the distributional gradient of u in Ω. We first prove that problem (1.3) admits a minimum for any β > - 1 (see Theorem 3.5 below); this result is strengthened by the fact that in the case β < - 1 the infimum of the associated functional in (1.1) is - ∞ . The latter assertion, roughly speaking, depends on the fact that, while the denominator may tend to zero, the numerator remains strictly negative (see Remark 3.4 for the precise computations). However, the main result concerns the validity of an isoperimetric inequality for Λ ⁢ ( Ω , β ) when β > - 1 (see Theorem 5.3 below). In particular, we show that, among all bounded, sufficiently smooth open sets with given volume, the ball maximizes Λ ⁢ ( Ω , β ) when β ∈ ] - 1 , 0 ] and it minimizes Λ ⁢ ( Ω , β ) when β ≥ 0 .

The proof of Theorem 5.3 follows as an application of Proposition 5.2 in which the computation of Λ ⁢ ( Ω , β ) is made explicit when Ω is a ball. Here the proof’s main ingredient is given by the Γ-convergence of J p towards J (defined in (3.1) and (3.3) below) and then the convergence of λ ⁢ ( Ω , p , β ) to Λ ⁢ ( Ω , β ) as p → 1 + (see Theorem 4.2 and Corollary 4.3 below).

The plan of this paper is as follows: in Section 3, we present the minimization problem for p ≥ 1 , while in Section 4 we deal with the Γ-convergence result. In Section 5, we state and prove the isoperimetric inequalities for Λ ⁢ ( Ω , β ) . We conclude with Section 6 by showing a Cheeger-type result.

2 Notations and preliminaries

Throughout this paper, Ω is a bounded open and connected set of ℝ N ( N ≥ 2 ) with sufficiently smooth boundary (see Remark 3.4 below). For a given set A, we denote by χ A its characteristic function, by | A | its Lebesgue measure, and by A # the ball centered at the origin such that | A # | = | A | . Moreover, ℋ N - 1 ⁢ ( E ) will be the ( N - 1 ) -dimensional Hausdorff measure of a set E, and B R will be the ball centered at the origin with radius R.

Let us now recall some basic facts on functions of bounded variation. For a detailed treatment of the subject, we refer the reader, for example, to [1, 11]. The total variation in Ω of a function u ∈ L 1 ⁢ ( Ω ) is

| D ⁢ u | ⁢ ( Ω ) = sup ⁡ { ∫ Ω u ⁢ div ⁡ ϕ : ϕ ∈ C 0 1 ⁢ ( Ω , ℝ N ) , ∥ ϕ ∥ L ∞ ≤ 1 } .

Then u is a function of bounded variation in Ω, and we write u ∈ BV ⁡ ( Ω ) , if | D ⁢ u | ⁢ ( Ω ) is finite. In this case, the distributional derivative of u is a finite Radon measure in Ω, that is,

∫ Ω u ⁢ ∂ ⁡ ϕ ∂ ⁡ x i ⁢ 𝑑 x = - ∫ Ω ϕ ⁢ 𝑑 D i ⁢ u for all ⁢ ϕ ∈ C 0 ∞ ⁢ ( Ω ) , i = 1 , … , N ,

for some vector-valued measure D ⁢ u = ( D 1 ⁢ u , … , D N ⁢ u ) . The set BV ⁡ ( Ω ) is a Banach space if endowed with the norm

∥ u ∥ BV ⁡ ( Ω ) = ∥ u ∥ L 1 ⁢ ( Ω ) + | D ⁢ u | ⁢ ( Ω ) .

If the characteristic function χ E of a set E ⊂ ℝ N has bounded variation, we say that E has finite perimeter in Ω, and denote the relative perimeter of E by

P Ω ⁢ ( E ) = | D ⁢ χ E | ⁢ ( Ω ) .

Finally, we will write P ⁢ ( E ) instead of P ℝ N ⁢ ( E ) .

The coarea formula states that if u ∈ BV ⁡ ( Ω ) , then

| D u | ( Ω ) = ∫ - ∞ + ∞ P Ω ( u > t ) d t .

We finally recall that a sequence u n in BV ⁡ ( Ω ) weak * converges to u ∈ BV ⁡ ( Ω ) if u n → u in L 1 ⁢ ( Ω ) and

lim n → + ∞ ⁡ ∫ Ω ϕ ⁢ 𝑑 D i ⁢ u n = ∫ Ω ϕ ⁢ 𝑑 D i ⁢ u for all ⁢ ϕ ∈ C 0 ⁢ ( Ω ) , i = 1 , … , N .

Let us recall the definition of the Cheeger constant h ⁢ ( Ω ) for Ω ⊂ ℝ N , which is given by

(2.1) h ⁢ ( Ω ) := inf E ⊆ Ω ⁡ P ⁢ ( E ) | E | = inf u ∈ BV ⁡ ( Ω ) ⁡ | D ⁢ u | ⁢ ( ℝ N ) ∫ Ω | u | .

It is well known that there is at least a minimum for problem (2.1) when Ω has Lipschitz boundary. A set for which the minimum is attained is called Cheeger set for Ω. Finally, for subsequent use, we recall that the ball is self-Cheeger, that is, if Ω = B R for some R > 0 , then the Cheeger set of Ω is B R itself and then

(2.2) h ⁢ ( B R ) = P ⁢ ( B R ) | B R | .

In what follows, we will use a trace inequality for BV -functions: if Ω is a Lipschitz bounded open set, then

(2.3) ∫ ∂ ⁡ Ω | v | ≤ c 1 ⁢ | D ⁢ v | ⁢ ( Ω ) + c 2 ⁢ ∫ Ω | v |

for any v ∈ BV ⁡ ( Ω ) , with c 1 = 1 + L 2 and L is the Lipschitz constant of ∂ ⁡ Ω . Actually, if Ω has C 1 boundary, then c 1 can be chosen as 1 + ε for any ε > 0 , with c 2 depending on ε as well. Furthermore, if ∂ ⁡ Ω has mean curvature bounded from above, then we may choose c 1 = 1 . We refer the reader to [3, 15]. To conclude this section, we recall the following lower semicontinuity result.

Proposition 2.1 ([21]).

For any β ≥ - 1 , the functional

F ⁢ ( u ) = | D ⁢ u | ⁢ ( Ω ) + min ⁡ { β , 1 } ⁢ ∫ ∂ ⁡ Ω | u |

is lower semicontinuous on BV ⁡ ( Ω ) with respect to the topology of L 1 ⁢ ( Ω ) .

3 The first eigenvalue problem with Robin boundary conditions

In this section, we highlight some well-known features on the Robin first eigenvalue problem in both cases p > 1 and p = 1 .

3.1 The case p > 1

Let us consider the functional

(3.1) J p ⁢ ( u ) := ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p , u ∈ W 1 , p ⁢ ( Ω ) , u ≢ 0 ,

and let

(3.2) λ ⁢ ( Ω , p , β ) = inf φ ∈ W 1 , p ⁢ ( Ω ) , φ ≢ 0 ⁡ J p ⁢ ( φ ) .

The following classical result concerning problem (3.2) holds.

Theorem 3.1.

Let Ω be a bounded Lipschitz open set, and let p > 1 . For any β ∈ R , there exists a minimum of problem (3.2). Moreover, if Ω is connected, then λ ⁢ ( Ω , p , β ) is simple, that is, it admits a unique minimizer, up to a multiplicative constant. In this case, the minimizers are positive or negative in Ω and they are in C 1 , α ⁢ ( Ω ) . Furthermore, they satisfy the following Robin boundary value problem:

{ - Δ p ⁢ φ p = λ ⁢ ( Ω , p , β ) ⁢ | φ p | p - 2 ⁢ φ p in ⁢ Ω , | ∇ ⁡ φ p | p - 2 ⁢ ∂ ⁡ φ p ∂ ⁡ ν + β ⁢ | φ p | p - 2 ⁢ φ p = 0 on ⁢ ∂ ⁡ Ω .

Proof.

Regarding the existence of minimizers, it follows by a well-known argument of Calculus of Variations (see, for example, [17]). For the simplicity of the eigenvalue and the regularity of the eigenfunctions, we can refer the reader, for example, to [6, 20]. ∎

Remark 3.2.

We explicitly observe that the case β = 0 is trivial, being λ ⁢ ( Ω , p , β ) = 0 for any p > 1 and the minimizers are the constant functions. Moreover, when β → + ∞ , then λ ⁢ ( Ω , p , β ) tends to the first Dirichlet eigenvalue of the p-Laplacian.

3.2 The case p = 1

Here we are mainly interested in the study of the functional

(3.3) J ⁢ ( u ) := | D ⁢ u | ⁢ ( Ω ) + min ⁡ ( β , 1 ) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | , u ∈ BV ⁡ ( Ω ) , u ≢ 0 ,

which in the case β ≥ 1 is nothing else than (when u is assumed to be identically zero outside Ω)

J ⁢ ( u ) = | D ⁢ u | ⁢ ( ℝ N ) ∫ Ω | u | , u ∈ BV ⁡ ( Ω ) , u ≢ 0 ,

and we deal with the related minimization problem

(3.4) Λ ⁢ ( Ω , β ) = inf φ ∈ BV ⁡ ( Ω ) , φ ≢ 0 ⁡ J ⁢ ( φ ) .

Remark 3.3.

If β > 1 , one could ask what happens when studying the problem

λ ⁢ ( Ω , 1 , β ) = inf u ∈ BV ⁡ ( Ω ) , u ≢ 0 ⁡ | D ⁢ u | ⁢ ( Ω ) + β ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | .

Actually, in this case

λ ⁢ ( Ω , 1 , β ) = Λ ⁢ ( Ω , β ) = h ⁢ ( Ω ) ,

where h ⁢ ( Ω ) is the Cheeger constant defined in (2.1). Indeed, it is easy to see that λ ⁢ ( Ω , 1 , β ) ≥ h ⁢ ( Ω ) . Moreover, if v is a minimum for h ⁢ ( Ω ) , then [18, Theorem 3.1] assures the existence of a sequence v k ∈ C c ∞ ⁢ ( Ω ) which converges to v in L q ⁢ ( Ω ) for any q ≤ N N - 1 and such that ∥ ∇ ⁡ v k ∥ L 1 ⁢ ( Ω ) converges to | D ⁢ v | ⁢ ( ℝ N ) as k → ∞ . Hence,

λ ⁢ ( Ω , 1 , β ) ≤ lim k → + ∞ ⁡ ∫ Ω | ∇ ⁡ v k | ∫ Ω | v k | = h ⁢ ( Ω ) .

Let us observe that the study of the minimization problem given in (3.4) can be dealt with by limiting u only to characteristic functions.

Thus, for E ⊆ Ω , evaluating J on the characteristic function of E, one has

J ⁢ ( χ E ) = P Ω ⁢ ( E ) + min ⁡ ( β , 1 ) ⁢ ℋ N - 1 ⁢ ( ∂ ⁡ Ω ∩ ∂ ⁡ E ) | E | ,

and from here on we set

R ⁢ ( E , β ) := J ⁢ ( χ E ) .

Now, we consider the minimization problem given by

(3.5) ℓ ⁢ ( Ω , β ) := inf E ⊆ Ω ⁡ R ⁢ ( E , β ) .

Depending on β, the infimum in (3.4) can be proven to be achieved under different conditions on the regularity of the set Ω.

Therefore, before stating the existence of a minimum for (3.4), we need to highlight the regularity of ∂ ⁡ Ω , which is required throughout this paper.

Remark 3.4 (Assumptions on Ω and on β).

The boundary of Ω is required to have Lipschitz regularity when β is nonnegative. When β is negative, the regularity of ∂ ⁡ Ω is mainly related to a suitable use of the trace inequality. In particular, throughout this paper, in the case - 1 < β < 0 we assume ∂ ⁡ Ω ∈ C 1 . For the case β ≤ - 1 , let us observe that the minimum problem for J is not well posed; namely one can show that, for any fixed Lipschitz bounded open set Ω, if β < 1 , then

inf u ∈ BV ⁡ ( Ω ) , u ≢ 0 ⁡ J ⁢ ( u ) = - ∞ .

Indeed, there exists a sequence Ω ε ⊂ Ω such that P Ω ⁢ ( Ω ε ) ≤ P ⁢ ( Ω ) + ε for any ε > 0 and such that | Ω ∖ Ω ε | → 0 as ε → 0 (see, for example, [22, Theorem 1.1] as well as the references therein).

Hence, one obtains

J ⁢ ( χ Ω ∖ Ω ε ) = P Ω ⁢ ( Ω ε ) + β ⁢ P ⁢ ( Ω ) | Ω ∖ Ω ε | ≤ ( 1 + β ) ⁢ P ⁢ ( Ω ) + ε | Ω ∖ Ω ε | ,

which goes to - ∞ as ε → 0 .

In the case β = - 1 , when Ω has bounded mean curvature, then Λ ⁢ ( Ω , - 1 ) = ℓ ⁢ ( Ω , - 1 ) and they are finite.

We observe that the functional

J ⁢ ( χ E ) = P Ω ⁢ ( E ) - ℋ N - 1 ⁢ ( ∂ ⁡ Ω ∩ ∂ ⁡ E ) | E |

describes the free energy of a liquid which occupies a region E inside a container Ω. If the energy functional is minimized under a volume constraint | E | = m , then a minimizer exists (see [19, Theorem 19.5]). Moreover, such a minimizer, denoted by F, has constant distributional mean curvature in Ω, and by the Young law it satisfies (see [19, Theorems 19.7 and 19.8]) 〈 ν F , ν Ω 〉 = - 1 on ∂ ⁡ ( Ω ∩ ∂ ⁡ F ) . The above considerations ensure that if one considers a rounded square in the plane, the minimum in (3.5) is not achieved; see Figure 1.

Figure 1

When β = - 1 and Ω is a rounded square of the plane, the sequence of minimizers of Λ ⁢ ( Ω , - 1 ) is given by the sets bounded by a circle of radius R contained in Ω and tangent to the square, and one of the rounded corners (the grey domain is one of such sets). When the radius of this circle approaches the radius of the circle in the corner, J approaches λ.

Now, we are in a position to state the existence result for (3.4).

Theorem 3.5.

Let β > - 1 . Then there exists a minimum to problem (3.4). In particular, it holds

(3.6) Λ ⁢ ( Ω , β ) = ℓ ⁢ ( Ω , β ) .

Moreover, if v ∈ BV ⁡ ( Ω ) is a minimizer of (3.4), then

(3.7) Λ ( Ω , β ) = R ( { v > t } , β )

for some t ∈ R .

Proof.

Let u n ∈ BV ⁡ ( Ω ) be a minimizing sequence to (3.4), with ∥ u n ∥ L 1 ⁢ ( Ω ) = 1 .

Let β > 0 . In this case, u n is bounded in BV ⁡ ( Ω ) . Then u n converges (up to a subsequence) weak * in BV and strongly in L 1 ⁢ ( Ω ) to u ∈ BV ⁡ ( Ω ) . Since u n and u are assumed to be identically zero outside Ω, in the case β ≥ 1 , we have J ⁢ ( u n ) = | D ⁢ u n | ⁢ ( ℝ N ) and the lower semicontinuity of the total variation gives

J ⁢ ( u ) ≤ lim inf n ⁡ J ⁢ ( u n ) .

Hence, u is a minimum for the functional J.

When 0 < β < 1 , given δ > 0 and Ω δ = { x ∈ Ω : d ⁢ ( x , ∂ ⁡ Ω ) > δ } , where d ⁢ ( ⋅ , ∂ ⁡ Ω ) is the distance function to the boundary of Ω, we have

J ⁢ ( u n ) ≥ | D ⁢ u n | ⁢ ( Ω δ ) + β ⁢ [ | D ⁢ u n | ⁢ ( Ω ∖ Ω δ ) + ∫ ∂ ⁡ Ω | u n | ] .

By lower semicontinuity, we obtain

lim inf n ⁡ J ⁢ ( u n ) ≥ | D ⁢ u | ⁢ ( Ω δ ) + β ⁢ | D ⁢ u | ⁢ ( ℝ N ∖ Ω δ ) .

Since u ∈ BV ⁡ ( Ω ) for δ → 0 , we obtain

lim inf n ⁡ J ⁢ ( u n ) ≥ J ⁢ ( u ) ,

and u is a minimum for J.

Now, let us suppose - 1 < β < 0 . We have J ⁢ ( u n ) ≤ C . Then, using (2.3) with c 1 = 1 + ε , for ε sufficiently small we have

J ⁢ ( u n ) ≥ ( 1 + β + β ⁢ ε ) ⁢ | D ⁢ u n | ⁢ ( Ω ) + c 2 ⁢ β ≥ c 2 ⁢ β .

Moreover, using again (2.3), we get

| D ⁢ u n | ⁢ ( Ω ) ≤ C - β ⁢ ( 1 + ε ) ⁢ | D ⁢ u n | ⁢ ( Ω ) - β ⁢ c 2 ,

and for ε small this implies that u n is bounded in BV ⁡ ( Ω ) .

Finally, the functional J is lower semicontinuous by Proposition 2.1. This allows to conclude that u is a minimum of J.

Now we show (3.6). Obviously, we have

Λ ⁢ ( Ω , β ) ≤ ℓ ⁢ ( Ω , β ) .

In order to show the reverse inequality, let v ∈ BV ⁡ ( Ω ) be a minimizer of (3.4). Then by the coarea formula

| D v | ( Ω ) = ∫ - ∞ + ∞ P Ω ( { v > t } ) d t ,

one obtains

Λ ⁢ ( Ω , β ) = J ⁢ ( v ) = ∫ - ∞ + ∞ P Ω ( { v > t } ) d t + min ( β , 1 ) ∫ - ∞ + ∞ ℋ n - 1 ( { v > t } ∩ ∂ Ω ) d t ∫ - ∞ + ∞ | { v > t } | d t

= ∫ - ∞ + ∞ R ( { v > t } , β ) | { v > t } | d t ∫ - ∞ + ∞ | { v > t } | d t

≥ inf E ⊆ Ω ⁡ R ⁢ ( E , β )

(3.8) = ℓ ⁢ ( Ω , β ) ,

which shows that

(3.9) Λ ⁢ ( Ω , β ) = ℓ ⁢ ( Ω , β ) .

In particular, combining (3.8) and (3.9), we have that

∫ - ∞ + ∞ { R ( { v > t } , β ) - ℓ ( Ω , β ) } | { v > t } | d t = 0 ,

and the integrand is nonnegative by the definition of ℓ . Since v ≢ 0 , equation (3.7) holds. ∎

4 Γ-convergence

In this section, we are concerned with the convergence and the relevant insight that one gains by taking p → 1 + in the functional J p defined in (3.1). Once that J p is extended to ∞ for functions belonging to BV ⁡ ( Ω ) ∖ W 1 , p ⁢ ( Ω ) , we will show that the Γ-limit as p → 1 + for J p is the functional J.

For the sake of completeness, we firstly precise the notion of Γ-convergence in our setting.

Definition 4.1.

The functional J p Γ-converges to J as p → 1 + in the weak * topology of BV ⁡ ( Ω ) if, for any u ∈ BV ⁡ ( Ω ) , the following conditions hold:

liminf inequality: for any sequence u p ∈ BV ⁡ ( Ω ) which converges to u weak * in BV ⁡ ( Ω ) as p → 1 + ,

(4.1) lim inf p → 1 + ⁡ J p ⁢ ( u p ) ≥ J ⁢ ( u ) .
limsup inequality: there exists a sequence u p ∈ W 1 , p ⁢ ( Ω ) which converges to u weak * in BV ⁡ ( Ω ) as p → 1 + such that

(4.2) lim sup p → 1 + ⁡ J p ⁢ ( u p ) ≤ J ⁢ ( u ) .

Theorem 4.2.

Let β > - 1 . Then J p Γ-converges to the functional J in the sense of Definition 4.1.

Proof.

For any u ∈ BV ⁡ ( Ω ) , we have to show inequalities (4.1) and (4.2) where, without loss of generality, one can assume that ∥ u p ∥ L p ⁢ ( Ω ) = 1 . It is clear that in order to show (4.1) we can assume that the lim inf is finite (otherwise the inequality is trivially satisfied). Hence, we can suppose that u p is in W 1 , p ⁢ ( Ω ) for any p > 1 .

We split the proof in several cases depending to the value of β.

Case β ≥ 1 . We first show (4.2). Let us note that [18, Theorem 3.1] assures the existence of a sequence u k ∈ C c ∞ ⁢ ( Ω ) which converges to u in L q ⁢ ( Ω ) for any q ≤ N N - 1 and such that ∥ ∇ ⁡ u k ∥ L 1 ⁢ ( Ω ) converges to | D ⁢ u | ⁢ ( ℝ N ) as k → ∞ . Hence, for any k > 0 , one clearly has that ∥ ∇ ⁡ u k ∥ L p ⁢ ( Ω ) converges to ∥ D ⁢ u k ∥ L 1 ⁢ ( Ω ) . This also implies the existence of a subsequence p k → 1 + as k → ∞ such that ∥ ∇ ⁡ u k ∥ L p k ⁢ ( Ω ) p k converges to | D ⁢ u | ⁢ ( ℝ N ) as k → ∞ . This is sufficient to take k → ∞ in J p k ⁢ ( u k ) deducing (4.2).

Now, let us focus on (4.1). Then u p ∈ W 1 , p ⁢ ( Ω ) is any given sequence which converges to u weak * in BV ⁡ ( Ω ) as p → 1 + . The Hölder inequality, a convexity argument and the fact that β ≥ 1 yield

( ∫ Ω | ∇ ⁡ u p | + ∫ ∂ ⁡ Ω | u p | ) p ≤ ( ( ∫ Ω | ∇ ⁡ u p | p ) 1 p ⁢ | Ω | 1 - 1 p + ( ∫ ∂ ⁡ Ω | u p | p ) 1 p ⁢ | ∂ ⁡ Ω | 1 - 1 p ) p

≤ 2 p - 1 ⁢ ( ∫ Ω | ∇ ⁡ u p | p ⁢ | Ω | p - 1 + β ⁢ ∫ ∂ ⁡ Ω | u p | p ⁢ | ∂ ⁡ Ω | p - 1 )

≤ 2 p - 1 ⁢ max ⁡ ( | Ω | p - 1 , | ∂ ⁡ Ω | p - 1 ) ⁢ J p ⁢ ( u p ) .

Now, since 2 p - 1 ⁢ max ⁡ ( | Ω | p - 1 , | ∂ ⁡ Ω | p - 1 ) tends to one as p → 1 + , one deduces from lower semicontinuity that

lim inf p → 1 + ⁡ J p ⁢ ( u p ) ≥ | D ⁢ u | ⁢ ( Ω ) + ∫ ∂ ⁡ Ω | u | = | D ⁢ u | ⁢ ( ℝ N ) = J ⁢ ( u ) ,

which is (4.1).

Case 0 ≤ β < 1 . In order to show (4.2), we observe that it follows from [1, Theorem 3.9] that there exists a sequence u k ∈ C ∞ ⁢ ( Ω ) strictly converging to u in the BV -norm, namely u k converges strongly to u in L 1 ⁢ ( Ω ) as k → ∞ and ∥ ∇ ⁡ u k ∥ L 1 ⁢ ( Ω ) converges to | D ⁢ u | ⁢ ( Ω ) as k → ∞ . Moreover, it follows from [1, Theorem 3.88] that u k converges to u in L 1 ⁢ ( ∂ ⁡ Ω , ℋ N - 1 ) . Reasoning as for the case β ≥ 1 , one deduces that, as p k → 1 + , ∥ ∇ ⁡ u k ∥ L p k ⁢ ( Ω ) p k converges to | D ⁢ u | ⁢ ( Ω ) as k → ∞ and

∥ u k ∥ L p k ⁢ ( ∂ ⁡ Ω , ℋ N - 1 ) p k

converges to

∥ u ∥ L 1 ⁢ ( ∂ ⁡ Ω , ℋ N - 1 )

as k → ∞ . This means that u k is the sequence which allows to deduce (4.2).

To obtain (4.1), let us observe that it follows from the Young inequality that

J p ⁢ ( u p ) = ∫ Ω | ∇ ⁡ u p | p + β ⁢ ∫ ∂ ⁡ Ω | u p | p ≥ ∫ Ω | ∇ ⁡ u p | + β ⁢ ∫ ∂ ⁡ Ω | u p | - p - 1 p ⁢ ( | Ω | + | ∂ ⁡ Ω | ) ,

where u p is any sequence converging weak * to u in BV ⁡ ( Ω ) as p → 1 + . Then inequality (4.1) is a direct application of [21, Proposition 1.2].

Case - 1 < β < 0 . In this case, inequality (4.2) can be deduced as for the case 0 ≤ β < 1 . To show (4.1), let u p ∈ W 1 , p ⁢ ( Ω ) be a sequence converging to u weak * in BV ⁡ ( Ω ) as p → 1 + and let us set

Ω δ := { x ∈ Ω : d ⁢ ( x , ∂ ⁡ Ω ) > δ } , Ω δ ′ := Ω ∖ Ω δ .

Let ψ be a smooth function which is zero on Ω δ , one on ∂ ⁡ Ω and such that | ∇ ⁡ ψ | ≤ c ⁢ δ - 1 for some positive constant c. An application of (2.3) to v = ( u - | u p | p - 1 ⁢ u p ) ⁢ ψ gives that

∫ ∂ ⁡ Ω | u - | u p | p - 1 ⁢ u p | ≤ c 1 ⁢ | D ⁢ ( u - | u p | p - 1 ⁢ u p ) | ⁢ ( Ω δ ′ ) + ( c 2 + c 1 ⁢ c ⁢ δ - 1 ) ⁢ ∫ Ω δ ′ | u - | u p | p - 1 ⁢ u p |

(4.3) ≤ c 1 | D u | ( Ω δ ′ ) + c 1 ∫ Ω δ ′ | ∇ | u p | p - 1 u p | + ( c 2 + c 1 c δ - 1 ) ∫ Ω δ ′ | u - | u p | p - 1 u p | ,

where δ has been chosen such that | D ⁢ ( u - | u p | p - 1 ⁢ u p ) | ⁢ ( ∂ ⁡ Ω δ ) = 0 for any p > 1 , which is admissible since u - | u p | p - 1 ⁢ u p ∈ BV ⁡ ( Ω ) .

Now, by using that | a - b | ≥ | | a | - | b | | , it follows from (4.3) that

J ⁢ ( u ) - J p ⁢ ( u p ) = | D ⁢ u | ⁢ ( Ω ) - ∫ Ω | ∇ ⁡ u p | p + β ⁢ ∫ ∂ ⁡ Ω ( | u | - | u p | p )

≤ | D u | ( Ω ) - ∫ Ω | ∇ u p | p + | β | c 1 | D u | ( Ω δ ′ ) + | β | c 1 ∫ Ω δ ′ | ∇ | u p | p - 1 u p |

(4.4) + | β | ( c 2 + c 1 c δ - 1 ) ∫ Ω δ ′ | u - | u p | p - 1 u p | = : A .

It follows from Remark 3.4 and from the discussion following the trace inequality (2.3) on the value of the constants c 1 and c 2 that one can fix c 1 = | β | - 1 . Hence, a simple calculation yields

A = A - | D u | ( Ω δ ) + | D u | ( Ω δ ) - ∫ Ω δ | ∇ | u p | p - 1 u p | + ∫ Ω δ | ∇ | u p | p - 1 u p |

≤ 2 | D u | ( Ω δ ′ ) + ( | D u | ( Ω δ ) - ∫ Ω δ | ∇ | u p | p - 1 u p | ) + ∫ Ω | ∇ | u p | p - 1 u p |

(4.5) - ∫ Ω | ∇ ⁡ u p | p + | β | ⁢ ( c 2 + c 1 ⁢ c ⁢ δ - 1 ) ⁢ ∫ Ω δ ′ | u - | u p | p - 1 ⁢ u p | .

Let us observe that from the Young inequality one has that

∫ Ω | ∇ | u p | p - 1 u p | = ∫ Ω p | u p | p - 1 | ∇ u p |

(4.6) ≤ ∫ Ω | ∇ ⁡ u p | p + ( p - 1 ) ⁢ ∫ Ω | u p | p .

Therefore, gathering (4.5) and (4) in (4.4) yields

J ( u ) - J p ( u p ) ≤ 2 | D u | ( Ω δ ′ ) + | D u | ( Ω δ ) - ∫ Ω δ | ∇ | u p | p - 1 u p |

(4.7) + ( p - 1 ) ⁢ ∫ Ω | u p | p + | β | ⁢ ( c 2 + c 1 ⁢ c ⁢ δ - 1 ) ⁢ ∫ Ω δ ′ | u - | u p | p - 1 ⁢ u p | .

Now, observe that, since from the compact embedding one has that u p converges to u in L q ⁢ ( Ω ) for any q < N N - 1 , it also holds that | u p | p - 1 ⁢ u p converges to u in L 1 ⁢ ( Ω ) as p → 1 + . Then, taking p → 1 + in (4.7) (recall that the second term on the right-hand side of (4.7) is lower semicontinuous with respect to the L 1 -convergence), one gains

lim sup p → 1 + ⁡ ( J ⁢ ( u ) - J p ⁢ ( u p ) ) ≤ 2 ⁢ | D ⁢ u | ⁢ ( Ω δ ′ ) .

Finally, taking δ → 0 + , one obtains (4.1).∎

Corollary 4.3.

Let β > - 1 . Then

(4.8) lim p → 1 + ⁡ λ ⁢ ( Ω , p , β ) = Λ ⁢ ( Ω , β ) .

Moreover, if u p ∈ W 1 , p ⁢ ( Ω ) are minimizers of (3.2), with ∥ u p ∥ L p ⁢ ( Ω ) = 1 , then

u p → u 𝑤𝑒𝑎𝑘 * ⁢ in ⁢ BV ⁡ ( Ω ) ,

where u is a minimizer of (3.4).

Proof.

Let u ¯ ∈ BV ⁡ ( Ω ) be a minimizer of (3.4). We have that by Theorem 4.2 there exists w p converging to u ¯ which satisfies (4.2); on the other hand, we claim that u p weak* converges in BV ⁡ ( Ω ) to a function u ∈ BV ⁡ ( Ω ) . Indeed, since

lim sup p → 1 + ⁡ J p ⁢ ( u p ) ≤ lim sup p → 1 + ⁡ J p ⁢ ( w p ) ≤ J ⁢ ( u ¯ ) = Λ ⁢ ( Ω , β ) ,

it holds that J p ⁢ ( u p ) ≤ C for any p > 1 where C does not depend on p. This implies that u p is bounded in BV ⁡ ( Ω ) . This is obvious if β ≥ 0 , while if β < 0 it is a consequence of trace inequality (2.3) that

∫ Ω | ∇ ⁡ u p | p ⁢ 𝑑 x ≤ C - β ⁢ ∫ Ω | ∇ ⁡ ( u p p ) | - β ⁢ c 2 ,

and by (4) it holds

( 1 + β ) ⁢ ∫ Ω | ∇ ⁡ u p | p ⁢ 𝑑 x ≤ C - β ⁢ C 2 - β ⁢ ( p - 1 ) .

Hence, by compactness we get the claim.

Then it holds, by Theorem 4.2 and the fact that λ ⁢ ( Ω , p , β ) = J p ⁢ ( u p ) , that

Λ ⁢ ( Ω , β ) = J ⁢ ( u ¯ ) ≤ J ⁢ ( u ) ≤ lim inf p → 1 + ⁡ J p ⁢ ( u p ) ≤ Λ ⁢ ( Ω , β ) .

Hence, (4.8) holds; moreover, u is a minimizer of (3.4). ∎

5 An isoperimetric inequality for Λ ⁢ ( Ω , β )

In this section, we deal with the shape optimization problem; the essential tool is the study of the radial case. In particular, we need an explicit computation for λ in the case of a ball of radius R. We first focus on some qualitative properties of the first eigenfunction associated to Λ ⁢ ( Ω , β ) for any p > 1 . We state the following classical result.

Proposition 5.1.

Let p > 1 , β ∈ R and let Ω = B R be the ball in R N centered at the origin with radius R. Let v p ∈ W 1 , p ⁢ ( B R ) be the first positive eigenfunction corresponding to λ ⁢ ( B R , p , β ) . Then

v p = ψ p ⁢ ( r ) ∈ C ∞ ⁢ ( ( 0 , R ) ) ∩ C 1 , α ⁢ ( [ 0 , R ] )

is a radially symmetric function, which solves

{ - | ψ p ′ ⁢ ( r ) | p - 2 ⁢ [ ( p - 1 ) ⁢ ψ p ′′ ⁢ ( r ) + N - 1 r ⁢ ψ p ′ ⁢ ( r ) ] = λ ⁢ ( B R , p , β ) ⁢ ψ p p - 1 ⁢ ( r ) in ] 0 , R [ , ψ p ′ ⁢ ( 0 ) = 0 , | ψ p ′ ⁢ ( R ) | = | β | 1 p - 1 ⁢ ψ p ⁢ ( R ) ,

with ψ p ′ < 0 in ] 0 , R ] if β > 0 , and ψ p ′ > 0 in ] 0 , R ] if β < 0 .

Let us state and prove an explicit computation for Λ ⁢ ( Ω , β ) in case Ω is a ball. The proof uses classical tools and the reduction of the study of the minimum in (3.4) to the characteristic functions in the case of positive β. When β is negative, the proof is more delicate and it strongly relies on the Γ-convergence proven in Theorem 4.2. Let us recall that, in the sequel, h ⁢ ( Ω ) is the Cheeger constant for Ω as defined in (2.1).

Proposition 5.2.

If β > - 1 , then

(5.1) Λ ⁢ ( B R , β ) = β ^ ⁢ h ⁢ ( B R ) = β ^ ⁢ N R ,

where β ^ = min ⁡ { β , 1 } .

Proof.

We split the proof into two cases with respect to the sign of β.

If β ≥ 0 , equation (5.1) follows directly from (3.6) and from the isoperimetric inequality. Indeed, if E ⊆ B R , then

R ⁢ ( E , β ) = P B R ⁢ ( E ) + β ^ ⁢ ℋ N - 1 ⁢ ( ∂ ⁡ B R ∩ ∂ ⁡ E ) | E |

(5.2) ≥ β ^ ⁢ P ⁢ ( E ) | E | ≥ β ^ ⁢ P ⁢ ( E # ) | E # | ≥ β ^ ⁢ P ⁢ ( B R ) | B R | = β ^ ⁢ N R ,

where the last relation holds from the definition and the ball’s property of being self-Cheeger given by (2.1) and (2.2). This shows that Λ ⁢ ( B R , β ) ≥ β ^ ⁢ N R . For the reverse inequality, one can simply choose E = B R in (3.5), which gives that (5.1) holds.

The proof of the case - 1 < β < 0 is more involved and it makes use of the Γ-convergence of the functional and Proposition 5.1.

Let v p ∈ W 1 , p ⁢ ( B R ) be a positive minimizer of (3.2). Then it follows from Corollary 4.3 that

Λ ⁢ ( B R , β ) = lim p → 1 + ⁡ J p ⁢ ( v p ) .

By Proposition 5.1, the v p ⁢ ( x ) = ψ p ⁢ ( r ) , r = | x | , are radially increasing functions. Without loss of generality, we may suppose ψ p ⁢ ( R ) = 1 .

Moreover, v p converges strongly in L 1 ⁢ ( B R ) to v ∈ BV ⁡ ( B R ) , almost everywhere in B R and weak * in BV ⁡ ( B R ) as p → 1 + . Since v p is radially increasing, v is nondecreasing. Hence, its superlevel sets { v > t } are N-dimensional spherical shells { r < | x | < R } , and by (3.7) it holds that

Λ ⁢ ( B R , β ) = N R ⋅ ( r R ) N - 1 + β 1 - ( r R ) N

for some r ∈ [ 0 , R [ . Then, by minimizing the function

f ( t ) = t N - 1 + β 1 - t N , t ∈ [ 0 , 1 [ ,

it is easy to see that the minimum is attained at t = 0 , and this gives (5.1). ∎

Now, we are in a position to state and prove the above cited inequalities. The proof consists in a suitable use of the explicit computation of λ given in Proposition 5.2. We give the following result.

Theorem 5.3.

It holds

(5.3) Λ ⁢ ( Ω # , β ) ≤ Λ ⁢ ( Ω , β ) if ⁢ β ≥ 0 ,

and

(5.4) Λ ⁢ ( Ω # , β ) ≥ Λ ⁢ ( Ω , β ) if - 1 < β < 0 .

Proof.

We split the proof depending on the sign of β. In the case β ≥ 0 , arguing as in (5) and if E ⊆ Ω , one can write

R ⁢ ( E , β ) ≥ Λ ⁢ ( Ω # , β ) = β ^ ⁢ P ⁢ ( Ω # ) | Ω # |

from which (5.3) simply follows.

If - 1 < β < 0 , then one can use the isoperimetric inequality and (5.1) in order to deduce

Λ ⁢ ( Ω , β ) ≤ β ⁢ P ⁢ ( Ω ) | Ω | ≤ β ⁢ P ⁢ ( Ω # ) | Ω # | = Λ ⁢ ( Ω # , β ) ,

that is, (5.4). ∎

6 A Cheeger-type inequality for p-Laplace with Robin conditions

As final remark, we derive an estimate for positive β (see also [14] for a similar inequality). This is a generalization, for Robin boundary conditions, of the well-known Cheeger inequality in the Dirichlet ( β = + ∞ ) case (see, e.g., [16]).

Lemma 6.1.

Let β > 0 , let p > 1 and let Ω be a Lipschitz bounded open set. Then

λ ⁢ ( Ω , p , β ) ≥ Λ ⁢ ( Ω , β ) ⁢ β ~ - ( p - 1 ) ⁢ β ~ p p - 1 ,

where β ~ = max ⁡ { 1 , β } . Moreover, if β ≥ ( h ⁢ ( Ω ) p ) p - 1 , then

λ ⁢ ( Ω , p , β ) ≥ ( h ⁢ ( Ω ) p ) p .

Proof.

Let φ p be a positive minimizer of (3.2) and let us set

U t = { x ∈ Ω : φ p ⁢ ( x ) > t } ,

S t = { x ∈ Ω : φ p ⁢ ( x ) = t } ,

Γ t = { x ∈ ∂ ⁡ Ω : φ p ⁢ ( x ) > t } .

For a continuous function ψ, we also define the following functional:

H ⁢ ( U t , ψ ) = 1 | U t | ⁢ ( ∫ S t ψ ⁢ 𝑑 ℋ N - 1 + ∫ Γ t β ⁢ 𝑑 ℋ N - 1 - ( p - 1 ) ⁢ ∫ U t ψ p p - 1 ) .

From [6, Proposition 2.3 and Theorem 3.2], it follows that there exists a set S ⊆ ( 0 , 1 ) of positive measure such that

(6.1) λ ⁢ ( Ω , p , β ) ≥ H ⁢ ( U t , ψ )

for all t ∈ S . We also mention that the choice

ψ = ( | ∇ ⁡ φ p | φ p ) p - 1

gives the equality in (6.1).

Then, choosing ψ = β ~ in (6.1), one gets

λ ⁢ ( Ω , p , β ) ≥ H ⁢ ( U t , ψ ) ≥ { h ⁢ ( Ω ) ⁢ β - ( p - 1 ) ⁢ β p p - 1 if ⁢ β ≥ 1 , min E ⊆ Ω ⁡ P Ω ⁢ ( E ) + β ⁢ ℋ n - 1 ⁢ ( ∂ ⁡ E ∩ ∂ ⁡ Ω ) | E | - ( p - 1 ) if ⁢ β < 1 ,

where h ⁢ ( Ω ) is the Cheeger constant for Ω. Recalling that if β ≥ 1 , then Λ ⁢ ( Ω , β ) = h ⁢ ( Ω ) , we have

λ ⁢ ( Ω , p , β ) ≥ λ ⁢ ( Ω , 1 , β ) ⁢ β ~ - ( p - 1 ) ⁢ β ~ p p - 1 .

Furthermore, if β ≥ ( h ⁢ ( Ω ) p ) p - 1 , one gets that

λ ⁢ ( Ω , p , β ) ≥ H ⁢ ( U t , h ⁢ ( Ω ) p - 1 p p - 1 ) ≥ h ⁢ ( Ω ) p p p - 1 - ( p - 1 ) ⁢ h ⁢ ( Ω ) p p p = ( h ⁢ ( Ω ) p ) p ,

which concludes the proof. ∎

Communicated by Juha Kinnunen

Funding statement: This work has been partially supported by PRIN project 2017JPCAPN grant: “Qualitative and quantitative aspects of nonlinear PDEs”, PON Ricerca e Innovazione 2014–2020, by FRA Project (Compagnia di San Paolo and Università degli studi di Napoli Federico II) 000022-ALTRI_CDA_75_2021_FRA_PASSARELLI, and by GNAMPA of INdAM.

References

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, Oxford, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar

[2] P. R. S. Antunes, P. Freitas and D. Krejčiřík, Bounds and extremal domains for Robin eigenvalues with negative boundary parameter, Adv. Calc. Var. 10 (2017), no. 4, 357–379. 10.1515/acv-2015-0045Search in Google Scholar

[3] G. Anzellotti and M. Giaquinta, BV functions and traces, Rend. Semin. Mat. Univ. Padova 60 (1978), 1–21. Search in Google Scholar

[4] M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal. 8 (1977), no. 2, 280–287. 10.1137/0508020Search in Google Scholar

[5] M.-H. Bossel, Membranes élastiquement liées inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimétrique de Rayleigh–Faber–Krahn, Z. Angew. Math. Phys. 39 (1988), no. 5, 733–742. 10.1007/BF00948733Search in Google Scholar

[6] D. Bucur and D. Daners, An alternative approach to the Faber–Krahn inequality for Robin problems, Calc. Var. Partial Differential Equations 37 (2010), no. 1–2, 75–86. 10.1007/s00526-009-0252-3Search in Google Scholar

[7] D. Bucur, V. Ferone, C. Nitsch and C. Trombetti, A sharp estimate for the first Robin–Laplacian eigenvalue with negative boundary parameter, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 4, 665–676. 10.4171/RLM/866Search in Google Scholar

[8] Q.-Y. Dai and Y.-X. Fu, Faber–Krahn inequality for Robin problems involving p-Laplacian, Acta Math. Appl. Sin. Engl. Ser. 27 (2011), no. 1, 13–28. 10.1007/s10255-011-0036-3Search in Google Scholar

[9] D. Daners, A Faber–Krahn inequality for Robin problems in any space dimension, Math. Ann. 335 (2006), no. 4, 767–785. 10.1007/s00208-006-0753-8Search in Google Scholar

[10] F. Della Pietra and N. Gavitone, Faber-Krahn inequality for anisotropic eigenvalue problems with Robin boundary conditions, Potential Anal. 41 (2014), no. 4, 1147–1166. 10.1007/s11118-014-9412-ySearch in Google Scholar

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar

[12] V. Ferone, C. Nitsch and C. Trombetti, On a conjectured reverse Faber–Krahn inequality for a Steklov-type Laplacian eigenvalue, Commun. Pure Appl. Anal. 14 (2015), no. 1, 63–82. 10.3934/cpaa.2015.14.63Search in Google Scholar

[13] P. Freitas and D. Krejčiřík, The first Robin eigenvalue with negative boundary parameter, Adv. Math. 280 (2015), 322–339. 10.1016/j.aim.2015.04.023Search in Google Scholar

[14] N. Gavitone and L. Trani, On the first Robin eigenvalue of a class of anisotropic operators, Milan J. Math. 86 (2018), no. 2, 201–223. 10.1007/s00032-018-0286-0Search in Google Scholar

[15] E. Giusti, The equilibrium configuration of liquid drops, J. Reine Angew. Math. 321 (1981), 53–63. 10.1515/crll.1981.321.53Search in Google Scholar

[16] B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin. 44 (2003), no. 4, 659–667. Search in Google Scholar

[17] H. Kovařík and K. Pankrashkin, On the p-Laplacian with Robin boundary conditions and boundary trace theorems, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 49. 10.1007/s00526-017-1138-4Search in Google Scholar

[18] S. Littig and F. Schuricht, Convergence of the eigenvalues of the p-Laplace operator as p goes to 1, Calc. Var. Partial Differential Equations 49 (2014), no. 1–2, 707–727. 10.1007/s00526-013-0597-5Search in Google Scholar

[19] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Stud. Adv. Math. 135, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139108133Search in Google Scholar

[20] S. Martinez and J. D. Rossi, Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition, Abstr. Appl. Anal. 7 (2002), no. 5, 287–293. 10.1155/S108533750200088XSearch in Google Scholar

[21] L. Modica, Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 5, 487–512. 10.1016/s0294-1449(16)30360-2Search in Google Scholar

[22] T. Schmidt, Strict interior approximation of sets of finite perimeter and functions of bounded variation, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2069–2084. 10.1090/S0002-9939-2014-12381-1Search in Google Scholar

Received: 2021-10-29

Revised: 2022-02-02

Accepted: 2022-02-24

Published Online: 2022-06-29

Published in Print: 2023-10-01

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https://doi.org/10.1515/acv-2021-0085

Keywords for this article

First Robin eigenvalue; isoperimetric inequalities

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