An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

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An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

Lorenzo Brasco, Giovanni Franzina

To cite this version:

Lorenzo Brasco, Giovanni Franzina. An anisotropic eigenvalue problem of Stekloff type and weighted

Wulff inequalities. 2012. �hal-00731074�

AND WEIGHTED WULFF INEQUALITIES

LORENZO BRASCO AND GIOVANNI FRANZINA

Abstract. We study the Stekloff eigenvalue problem for the so-called pseudop−Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalueσ2,p, providing various equivalent characterizations for it. We also prove an upper bound forσ2,p, in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock-Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appears to be interesting in themselves.

Contents

1. Introduction 1

1.1. The problem 1

1.2. One step back: the linear case 2

1.3. Results and style of the paper 3

1.4. A note on the regularity of eigenfunctions 4

1.5. Plan of the work 4

2. Technical machinery 5

3. The Stekloff spectrum of the pseudo p-Laplacian 7

4. Existence of an unbounded sequence 10

5. The first nontrivial eigenvalue 14

6. Further properties 20

7. An upper bound for σ

24

Appendix A. Weighted Wulff Inequalities 26

References 30

1. Introduction

1.1. The problem. In this paper, we are concerned with some spectral properties of the so-called pseudo p−Laplacian operator (see [4]), defined by

∆ e

u :=

N

X

i=1

∂

∂x

∂u

∂x

p−2

∂u

∂x

! .

2010Mathematics Subject Classification. 35P30, 47A75, 34B15.

Key words and phrases. Stekloff eigenvalue problem, pseudop−Laplacian, Wulff inequality.

1

For p = 2 this operator coincides with the usual Laplacian, but for p 6= 2 it considerably differs from the more familiar p−Laplace operator, given by

∆

u :=

N

X

i=1

∂

∂x

|∇u|

∂u

∂x

.

More precisely, we are concerned with investigating the Stekloff spectrum of ∆ e

on a generic open bounded Lipschitz set Ω ⊂ R

, i.e. the set of those real numbers σ such that the problem

(1.1)



 

 

∆ e

u = 0, in Ω,

N

X

i=1

|u

|

u

ν

= σ|u|

u, on ∂Ω,

admits nontrivial W

(Ω) weak solutions, where ν

= (ν

, . . . , ν

) is the outer normal versor. It is easily seen that these numbers σ can be equivalently characterized as the critical points of the restriction of the anisotropic p−Dirichlet integral

(1.2) Φ

(u) =

N

X

j=1

Z

Ω

|u

(x)|

dx, u ∈ W

(Ω), to the manifold {u ∈ W

(Ω) : kuk

= 1}.

1.2. One step back: the linear case. In order to smoothly present the topics and the aims of this paper, it could be useful to recall some basic facts from the linear case, i.e.

we consider p = 2 in (1.1). In this case, for every open bounded Lipschitz set Ω ⊂ R

, its Stekloff eigenvalues of the Laplacian are the numbers σ such that

(1.3)

∆u = 0, in Ω,

h∇u, ν

i = σ u, on ∂Ω,

admits nontrivial W

(Ω) solutions. By exploiting the compactness of the embedding W

(Ω) , → L

(∂Ω), it is easy to see that these eigenvalues form a discrete non-decreasing sequence diverging at ∞, i.e. σ

(Ω) ≤ σ

(Ω) ≤ σ

(Ω) ≤ . . . , with σ

(Ω) = 0 corresponding to constant solutions of (1.3). Again, these eigenvalues can be characterized as the critical values of the Dirichlet integral on the L

(∂Ω) unitary sphere. The corresponding critical points are the Stekloff eigenfunctions of ∆ on Ω, normalized by the condition on the L

(∂Ω) norm. Also, they turn out to form an orthonormal basis of L

(∂Ω). In particular, the first nontrivial eigenvalue σ

(Ω) coincides with the value of the best constant in the following Poincar´ e-Wirtinger trace inequality

c

Z

∂Ω

|u(x) − u

|

dH

≤ Z

Ω

|∇u(x)|

dx, u ∈ W

(∂Ω),

where H

stands for the (N −1)−dimensional Hausdorff measure and u

is the average of (the trace of) u on ∂Ω.

In analogy with the well-known Dirichlet and Neumann cases (see [24, Chapters 3 and 7]), one may be interested in the spectral optimization problem of maximizing

σ

under volume constraint. A well-known result asserts that the (unique) solutions to this problem are given by balls. This is the so-called Brock-Weinstock inequality (see [10, 33]). For ease of completeness, it is worth mentioning that Weinstock’s result (valid only in dimension

1On the contrary, it is not difficult to see that the problem of minimizingσ2 is always trivial.

N = 2) is even stronger, since it asserts that disks are still maximizers among simply connected set of given perimeter. By observing that σ

scales like a length to the power

−1 and that σ

(B

) = R

for a ball of radius R, the Brock-Weinstock inequality can be written in scaling invariant form as follows

(1.4) σ

(Ω) ≤

ω

|Ω|

N

,

where ω

is the measure of the N −dimensional ball of radius 1. Moreover, equality in (1.4) can hold if and only if Ω itself is a ball. Recently, a sharp quantitative stability estimate for this inequality has been given in [8]. We recall that such an “isoperimetric” property of the ball is in turn a consequence of the fact that the ball (centered at the origin) is the only solution of

min Z

∂Ω

|x|

dH

: Ω ⊂ R

, |Ω| = c

.

This result has been proved in [6] (see also [8] for a different proof). In other words, we have the following weighted isoperimetric inequality

(1.5)

Z

∂Ω

|x|

dH

≥ N ω

|Ω|

, with equality if and only if Ω is a ball centered at the origin.

1.3. Results and style of the paper. Of course, problem (1.1) for p 6= 2 is a com- pletely different story. We are now facing a nonlinear eigenvalue problem and no general results guarantee, for example, infiniteness and discreteness of the spectrum. Here we show that these eigenvalues forms at least a countably infinite sequence of positive num- bers σ

(Ω) ≤ σ

(Ω) ≤ σ

(Ω) ≤ . . . , diverging at infinity (Theorem 4.2). For this, we use some standard minimax methods from Nonlinear Analysis, but we will avoid to refer to the so-called Ljusternik-Schnirelmann theory and rather we will employ an alternative (and elegant) procedure introduced by Drab´ ek and Robinson in [16] (see the next section).

Also, we will show that the first eigenvalue (again, we have σ

(Ω) = 0 and it corresponds to constant eigenfunctions) is isolated in the spectrum, the latter being a close set (Propo- sition 3.6), and that the first nontrivial eigenvalue σ

(Ω) can still be characterized as the best constant in some Poincar´ e-Wirtinger trace inequality (Theorem 5.6 and Remark 5.7).

Namely, σ

(Ω) coincides with the best constant in c

min

Z

∂Ω

|u + t|

% dH

≤

N

X

i=1

Z

Ω

|u

|

dx, u ∈ W

(Ω).

We will also address the issue of generalizing the Brock-Weinstock inequality, for the first nontrivial Stekloff eigenvalue of the pseudo p−Laplacian. Indeed, by adapting Brock’s method of proof, we are able to prove (Theorems 7.2 and 7.3)

(1.6) σ

(Ω) ≤

|B

|

|Ω|

N

,

where B

is the N −dimensional `

unit ball, i.e. B

= {x ∈ R

: |x

|

+ · · · + |x

|

< 1}.

The previous inequality can be seen as a nonlinear counterpart of (1.4). Unfortunately,

we are not able to detect the cases of equality in (1.6). In other words, while it is rather

easy to see that equality in (1.6) implies Ω to coincide with (a translated and scaled copy

of) B

, we can not guarantee that equality can really hold. This is due to the fact that

we are not able to determine σ

(B

) and this in turn is intimately linked to the lack of information on the shape of the nodal line of eigenfunctions corresponding to σ

(B

) (see Remark 7.4). It is useful to recall at this point that even for the second eigenvalue of the p−Laplacian with Dirichlet boundary conditions, it is still an open problem to decide whether the nodal line in the Euclidean ball is given by a diameter (like in the linear case p = 2) or not.

The proof of (1.6) is based on the anisotropic version of (1.5), which is here derived and appears to be new. Namely, we prove the following weighted Wulff inequality

(1.7)

Z

∂Ω

V (kxk) kν

k

dH

≥ N |K|

|Ω|

,

where k · k and k · k

are norms dual to each other, while K is the unit ball of k · k centered at the origin and V is an increasing weight function, satisfying suitable assumptions. Here again, equality can hold if and only if Ω = K, up to a scaling factor. Inequality (1.7) is the other main contribution of this paper, the proof of (1.7) being an adaptation of the technique used in [8]. Since this result is not directly related with the study of the Stekloff spectrum, we added it in the Appendix at the end of the paper.

About the style of the paper, we point out that we tried to keep technicalities at a low level, in order to make the paper suitable for a wide audience, which can be possi- bly interested also in Isoperimetric Inequalities and Shape Optimization. Having this in mind, all the proofs, in despite of presenting many similarities with the vast literature on Nonlinear Spectral Theory, are self-contained. Further, where possible, our arguments are simpler and more direct. This is the case for example of the mountain pass characteriza- tion of the first nontrivial eigenvalue (Proposition 5.4), which is based on some peculiar convexity properties of the functional Φ

in (1.2) (the so-called hidden convexity, see [5, 9]).

Our method of proof still works for the standard p−Laplacian with Dirichlet or Neumann boundary conditions, thus it can be seen as alternative to that of [12, Theorem 3.1].

1.4. A note on the regularity of eigenfunctions. In passing, it is useful to say some- thing about the regularity of Stekloff eigenfunctions of the pseudo p−Laplacian. We recall that in the case of the standard p−Laplacian operator, it is well-known that solutions to

∆

u = 0 are C

(see [15, 27]). Then one could expect such a result to hold for ∆ e

as well.

However, we point out that the by now classical results of [15, 27] do not apply directly to the case of ∆ e

, since the type of degeneracy is quite different. Low regularity (like L

or C

) is standard routine (see [23, Theorem 7.6]), due to the fact that pseudo p−harmonic functions are local minimizers of a functional having p−growth in the gradient variable.

On the contrary, higher regularity is not clear.

For example, to the best of our knowledge, even the Lipschitz character of solutions seems not to be fully understood. We mention [32] where this is proved for the case p > 3 and [7], where an “almost Lipschitz” estimate is proven for p ≥ 2 (which is valid for fairly more degenerate equations).

On the other hand, in the case 1 < p < 2 the existence of second weak derivatives for pseudo p−harmonic functions can still be proved (see [18]), thus paralleling the case of the p−Laplacian (see [1]).

1.5. Plan of the work. We start with Section 2, where we recall some technical facts that

will be useful throughout the whole paper. Then in Section 3, we introduce the Stekloff

eigenvalue problem we are interested in and prove some first properties. We refine this

study in Section 4, where the existence of unbounded sequence of eigenvalues is exhibited.

The subsequent Section 5 is devoted to the first nontrivial Stekloff eigenvalue. We prove some equivalent characterizations for it, which imply in particular that the first eigenvalue is always isolated in the spectrum. We further analyze the first nontrivial eigenspace in Section 6, proving some nodal domains properties of eigenfunctions and giving yet another characterization of the first nontrivial eigenvalue, this time in terms of some eigenvalue problems with mixed boundary conditions. Finally, in Section 7 we provide some upper bounds for the first nontrivial eigenvalue, which can be seen as the nonlinear analogue of (1.4). As already said, a self-contained appendix devoted to weigthed Wulff inequalities complements the paper.

2. Technical machinery

Given 1 < p < ∞, for every x ∈ R

we will denote by kxk

its `

norm, i.e. the quantity kxk

=

N

X

i=1

|x

|

!

.

For an open set Ω ⊂ R

, the symbol W

(Ω) will stand for the usual Sobolev space, endowed with the norm

(2.1) kuk

=

Z

Ω

|u(x)|

dx + Z

Ω

|∇u(x)|

.

As always, the symbol W

(Ω) will stand for the closure of C

(Ω) with respect to the norm k · k

. It is useful to recall that when Ω ⊂ R

has a Lipschitz boundary, the embedding W

(Ω) , → L

(Ω) is compact. Moreover, each function in W

(Ω) has a trace belonging to the fractional Sobolev spaces W

(∂Ω). Then the embedding W

(Ω) , → L

(∂Ω) is compact as well.

Lemma 2.1. Let 1 < p < ∞ and Ω be an open bounded Lipschitz set in R

. Then k∇uk

+ kuk

, u ∈ W

(Ω),

defines a norm on the Sobolev space W

(Ω), which is equivalent to (2.1).

Proof. It is straightforward to check that the above quantity defines a norm. Then the thesis is a consequence of the trace inequality

kuk

≤ c

kuk

, and of the following Poincar´ e inequality

kuk

≤ e c

k∇uk

+ kuk

,

which in turn follows by a standard contradiction argument, exploiting the compact em- beddings W

(Ω) , → L

(Ω) and W

(Ω) , → L

(∂Ω).

The function t 7→ |t|

t enjoys the following monotonicity properties, which will be useful in the sequel. For the proof, the reader is referred to [28, Section 10].

Lemma 2.2. Let 2 ≤ p < ∞. Then

(2.2) (|t|

t − |s|

s) (t − s) ≥ 2

|t − s|

, t, s ∈ R.

For 1 < p < 2, we have

(2.3) (|t|

t − |s|

s) (t − s) ≥ (p − 1) |t − s|

(1 + |s|

+ |t|

)

, t, s ∈ R .

As an application of the previous Lemma, we obtain the following fact, which we will use repeatedly.

Proposition 2.3. Let {u

}

⊂ L

(Ω) be such that

(2.4) lim

k→∞

Z

Ω

|u

|

u

− |u|

u

(u

− u) dx = 0, for some function u ∈ L

(Ω). Then {u

}

converges in L

(Ω) to u.

Proof. We have two distinguish between two cases. If p ≥ 2, then by the first inequality of Lemma 2.2, it follows directly that

k→∞

lim Z

Ω

|u

− u|

dx = 0.

For 1 < p < 2, we start observing that the hypothesis implies (2.5)

Z

Ω

|u

|

dx ≤ C, for every k ∈ N.

Indeed, by means of Young inequality we can infer Z

Ω

|u

|

u

− |u|

u

(u

− u) dx ≥

1 − ε

p − ε

q Z

Ω

|u

|

dx

+

1 − ε

p − ε

q Z

Ω

|u|

dx,

where we set q = p/(p − 1). By taking ε small enough and using (2.4), we then obatin (2.5). We now use inequality (2.3) of Lemma 2.2, raised to the power p/2. Thus we get

Z

Ω

|u

− u|

dx ≤ C Z

Ω

(1 + |u

|

+ |u|

)

(2−p)p 4

h

|u

|

u

− |u|

u

(u

− u) i

dx.

An application of H¨ older inequality with exponents 2/(2 − p) and 2/p yields Z

Ω

|u

− u|

dx ≤ C Z

Ω

(1 + |u

|

+ |u|

)

dx

× Z

Ω

|u

|

u

− |u|

u

(u

− u) dx

.

The conclusion is now an easy consequence of (2.4) and (2.5).

To make the paper as self-contained as possible, we also recall the definition of Palais- Smale condition.

Definition 2.4. Let Φ : X → R be a C

functional defined on some Banach space X and let M ⊂ X be a C

manifold. Given x ∈ M, let us denote by T

M the tangent space at M in the point x and by DΦ(x)

the differential in x of the restriction of Φ to M .

Then Φ is said to satisfy the Palais-Smale condition on M at level λ, if for every sequence {x

}

⊂ M such that

n→∞

lim Φ(x

) = λ and lim

n→∞

kDΦ(x

)

k

= 0,

we have that x

is strongly convergent, possibly up to the extraction of a subsequence. If this condition is verified for every λ, we will simply say that Φ satisfies the Palais-Smale condition on M.

In order to find critical points of functionals restricted to manifolds, in this paper we will use the following minimax procedure, introduced by Dr´ abek and Robinson in [16]

and further studied by Cuesta in [11]. This is alternative to the so-called Ljusternik- Schnirelmann procedure, which relies on the concept of Krasnoselskii genus (see below).

Our choice permits to produce a sufficiently large set of eigenvalues, avoiding at the same time unnecessary technicalities. The general result that we need is the following, due to Cuesta (see [11, Proposition 2.7] or [18] for the proof).

Theorem 2.5. Let X be a uniformly convex Banach space and M ⊂ X a C

manifold.

By indicating with S

the unit sphere in R

, we set

C

(S

; M) = {f : S

→ M : f is continuous and odd}.

Suppose that Φ : X → R is an even C

functional and define λ

= inf

f∈Co(S^k−1;M)

max

u∈f(S^k−1)

Φ(u),

If Φ verifies the Palais-Smale condition on M at level λ

, then λ

is a critical value of Φ on M, i.e. there exists x

∈ M such that

Φ(x

) = λ and DΦ(x

)

0M

= 0.

Remark 2.6. For reader’s convenience, we recall that the Krasnoselskii genus of a compact, nonempty and symmetric subset A ⊂ X of a Banach space is defined by

γ(A) = inf n

k ∈ N : ∃ a continuous odd mapp f : A → S

o

,

with the convention that γ(A) = +∞, if no such an integer k exists. Using the Krasnoselskii genus, an infinite sequence of critical values of Φ is usually produced as follows (see [22, 31])

e λ

= inf

γ(A)≥k

max

u∈A

Φ(u), k ∈ N .

It is an interesting open problem to decide whether or not the two previous minimax procedures give the same sets of critical values.

3. The Stekloff spectrum of the pseudo p -Laplacian

Let Ω be a bounded Lipschitz open set in R

and % : ∂Ω → R be a measurable function satisfying

(3.1) 0 < c

≤ %(x) ≤ c

< ∞, H

− a.e. on ∂Ω.

For every 1 < p < ∞, we consider the pseudo p-Laplacian, i.e. the nonlinear operator

∆ e

u =

N

X

j=1

|u

|

u

xj

.

Definition 3.1. A real number σ is said to be a Stekloff eigenvalue of the pseudo p- Laplacian in Ω if the boundary value problem

(3.2)



 

 

− ∆ e

u = 0, in Ω,

N

X

i=1

|u

|

u

ν

= σ|u|

u%, on ∂Ω,

admits a nontrivial solution u. If this is the case, we say that u is a Stekloff eigenfunction corresponding to σ. We also set

S

(Ω) = {σ ∈ R : σ is a Stekloff eigenvalue}, to denote the Stekloff spectrum of the pseudo p−Laplacian on Ω.

Remark 3.2. Since the behaviour of the spectrum under varying weights is not investi- gated here, the notation does not account for the choice of the function % : ∂Ω → R.

The solutions u of the problem (3.2) are always understood in the weak sense, i.e.

u ∈ W

(Ω) and (3.3)

N

X

i=1

Z

Ω

|u

|

u

ϕ

dx = σ Z

∂Ω

|u|

u ϕ % dH

, for every ϕ ∈ W

(Ω).

Observe that the integral on the right-hand side is well-defined, since the trace of a function in W

(Ω) belongs to L

(∂Ω).

We start with the following basic result.

Lemma 3.3. Let 1 < p < ∞, Ω be a bounded open Lipschitz set and % : ∂Ω → R be such that (3.1) holds. There exists a least eigenvalue, given by σ = 0 and corresponding to constant eigenfunctions. Moreover, any other eigenfunction whose trace does not change sign on ∂Ω, is constant in Ω.

Proof. By testing ϕ = u, equation (3.3) implies Z

Ω

k∇uk

dx = σ Z

∂Ω

|u|

% dH

,

so that every eigenvalue must be positive. Moreover, it is easily seen that σ = 0 is an eigenvalue and by the previous equality any corresponding eigenfunction is constant.

Let us now prove the second part of the statement. Let u 6= 0 have a constant sign on the boundary and assume, arguing by contradiction, that it corresponds to an eigenvalue σ 6= 0. Inserting a constant test function in (3.3) we then obtain

Z

∂Ω

|u|

% dH

= 0,

where we also used that u does not change sign on ∂Ω. Thus, u has a null trace on ∂Ω and it solves in a weak sense the problem

∆ ˜

u = 0 in Ω, u = 0 on ∂Ω.

Solutions to the latter problem are minimizers of the strictly convex energy v 7→

Z

Ω

k∇vk

dx,

on W

(Ω). Since the unique minimizer is given by the zero constant function, there must hold u ≡ 0, a contradiction. Therefore, σ = 0 and u is a constant eigenfunction.

Definition 3.4. If u is a Stekloff eigenfunction, we will call nodal domains the connected components of {x ∈ Ω : u(x) 6= 0}. Observe that the latter is an open set, since each pseudo p−harmonic function is locally H¨ older continuous, as a local minimizer of R

Ω

k∇vk

(see [23, Theorem 7.6]). We also observe that each nodal domain is itself an open set. This

follows from the fact that the connected components of an open sets are open as well.

We record the following property of eigenfunctions, which will be useful in the next section.

Lemma 3.5. Let u ∈ W

(Ω) be a Stekloff eigenfunction, with eigenvalue σ > 0. Then u has at least two nodal domains, both touching the boundary.

Proof. The fact that u has to change sign follows from Lemma 3.3. Let us now take u

(x) = max{u(x), 0} and u

(x) = max{0, −u(x)},

and let Ω

, Ω

, . . . , be the nodal domains of u. Suppose that for some j we have Ω

b Ω.

Then the restriction of u to Ω

belongs to W

(Ω

) and it solves

− ∆ e

u = 0, on Ω

,

in the weak sense. This implies that u ≡ 0 on Ω

, hence contradicting the definition of

nodal domain.

We prove that the whole collection of Stekloff eigenvalues forms a closed set.

Proposition 3.6. Let 1 < p < ∞, Ω ⊂ R

a bounded Lipschitz domain and % : ∂Ω → R be a function such that (3.1) holds. Then S

(Ω) is a non empty closed subset of [0, ∞).

Proof. The fact that the collection of all the Stekloff eigenvalues is non empty and consists of nonnegative numbers is due to Lemma 3.3. In order to prove the second part of the statement, we take a sequence of eigenvalues {σ

}

⊂ S

(Ω) converging to some positive number σ and we let {u

}

⊂ W

(Ω) be a sequence of corresponding eigenfunctions, normalized by the condition

Z

∂Ω

|u

|

% dH

(x) = 1, k ∈ N . This implies in particular that

N

X

i=1

Z

Ω

|u

i

|

dx = σ

, k ∈ N ,

so that the sequence {u

}

is bounded in W

(Ω), thanks to Lemma 2.1. Thus, by the compactness of the embedding W

(Ω) , → L

(∂Ω), the sequence weakly converges (up to a subsequence) to some limit function u in W

(Ω). Moreover, this convergence is strong in L

(∂Ω). We have to show that u is an eigenfuction with eigenvalue σ: testing the equations solved by u

with ϕ = u

− u, we then obtain

N

X

i=1

Z

Ω

|u

i

|

u

− |u

|

u

(u

− u

) dx

= σ

Z

∂Ω

|u

|

u

− |u|

u

% dH

−

N

X

i=1

Z

Ω

|u

|

u

u

− u

dx.

Then, by the strong convergence of {u

}

⊂ L

(∂Ω), sending k to infinity yields

k→∞

lim

N

X

i=1

Z

Ω

|u

i

|

u

− |u

|

u

(u

− u

) dx = 0.

Thanks to Proposition 2.3, the previous gives the strong convergence of ∇u

to ∇u in L

(Ω). Since {u

}

converges to u strongly in L

(∂Ω), we also have

(3.4) lim

k→∞

Z

∂Ω

|u

− u|

% dH

= 0.

Thanks to these informations, we can now pass to the limit in the equation (3.3) satisfied by u

, so to obtain that u is an eigenfunction as well, with eigenvalue σ. This shows that

σ ∈ S

(Ω), which is then closed.

4. Existence of an unbounded sequence

In this section, we will show that S

(Ω) contains an infinite sequence of eigenvalues, diverging at ∞. At this aim, we observe that the elements of S

(Ω) are the critical values of the functional

(4.1) Φ(u) =

Z

Ω

k∇u(x)k

dx, u ∈ W

(Ω), on the manifold M , defined by

(4.2) M =

u ∈ W

(Ω) : G(u) = 1 , where G(u) = Z

∂Ω

|u|

% dH

. Then, we will use the minimax procedure of Theorem 2.5. For this, we have to check that Φ verifies the Palais-Smale condition on M . This is the content of the next result.

Lemma 4.1. Let Ω ⊂ R

be an open bounded Lipschitz set and %

: ∂Ω → R be a function such that (3.1) holds. Then Φ satisfies the Palais-Smale condition on the manifold M . Proof. First of all, we observe that M is a C

manifold. It is sufficient to verify that 1 is a regular value for G, i.e. DG(u) 6= 0, for every u ∈ M . This is easily verified, since for every u ∈ M, we have

DG(u)[u] = p Z

∂Ω

|u(x)|

%(x) dH

(x) = p 6= 0.

We now verify that Φ satisfies the Palais-Smale condition on M. At this aim, we need to show that for every C > 0 and every {u

}

⊂ M , the following implication holds true (4.3)

Φ(u

) ≤ C lim

DΦ(u

)

= 0







= ⇒ {u

}

converges in W

(Ω).

We first observe that for every u ∈ M, the tangent space to M at the point u is given by T

M =

ϕ ∈ W

(Ω) : Z

∂Ω

|u|

u ϕ % dH

= 0

. Hence, the hypothesis in (4.3) reads

0 = lim

n→∞

DΦ(u

)

= lim

n→∞













 sup

ϕ∈T_unM ϕ6=0

N

X

i=1

Z

Ω

|u

i

|

u

ϕ

dx kϕk















.

By this assumption, we can infer the existence of an infinitesimal sequence {ε

}

of strictly positive numbers, such that

(4.4)

N

X

i=1

Z

Ω

|u

i

|

u

ϕ

dx

≤ ε

kϕk

, for every ϕ ∈ T

M.

We now observe that the sequence {u

}

is bounded in W

(Ω). Indeed, using (3.1) we have

ku

k

≤ c

G(u

) = c

and k∇u

k

≤ c Φ(u

) ≤ C, e and the boundedness follows by Lemma 2.1.

Thus, by possibly passing to a subsequence, we can assume that {u

}

converges weakly to a function u in W

(Ω), and that the convergence is strong both in L

(Ω) and L

(∂Ω). In particular, such a limit function u still belongs to M. Using the strong convergence in L

(∂Ω) again, we have

(4.5) lim

n→∞

δ

= 0, where we set δ

= Z

∂Ω

|u

|

u

(u

− u) % dH

. Using the fact that u

∈ M , it is easy to check that the function v

:= (1 − δ

) u

− u belongs to the tangent space T

M . Thus, it is possible to choose ϕ = v

in (4.4), so as to obtain

N

X

i=1

Z

Ω

|u

i

|

u

(1 − δ

) u

− u

dx

≤ ε

(1 − δ

) u

− u

≤ ε

2 ku

k

+ kuk

, where we used that |1 − δ

| ≤ 2 by (4.5), provided n is large enough. By passing to the limit as n → ∞, we then get

n→∞

lim

N

X

i=1

Z

Ω

|u

i

|

u

(1 − δ

) u

− u

= 0, and so

(4.6) lim

n→∞

N

X

i=1

Z

Ω

|u

i

|

u

u

− u

dx = 0.

Since u

weakly converges to u in W

(Ω), we also have

(4.7) lim

n→∞

N

X

i=1

Z

Ω

|u

|

u

u

− u

dx = 0.

Subtracting (4.7) to (4.6) and using again Proposition 2.3 on each component u

, we then get

n→∞

lim Z

Ω

|∇(u

− u)|

dx = 0.

(4.8)

Owing to Lemma 2.1, finally the strong convergence of {u

}

in W

(Ω) follows by (4.8). This shows the validity of the implication (4.3), thus concluding the proof.

We can then assure that S

(Ω) contains an unbounded sequence. In the following, we

keep the same notation as in Section 2.

Theorem 4.2. Given 1 < p < ∞, let Ω ⊂ R

be an open bounded connected set, having Lipschitz boundary. Let also % : ∂Ω → R be a function such that (3.1) holds. For every k ∈ N , we define

(4.9) σ

(Ω) = inf

f∈C_o(S^k−1;M)

max

u∈f(S^k−1)

Z

Ω

k∇uk

dx.

Then each σ

(Ω) is a Stekloff eigenvalue of the pseudo p-Laplacian on Ω. Moreover, (4.10) 0 = σ

(Ω) < σ

(Ω) ≤ . . . ≤ σ

(Ω) ≤ . . .

and σ

(Ω) → ∞ as k → ∞.

Proof. For the sake of readability, we divide the proof in various steps.

Eigenvalues. We already observed that the Stekloff eigenvalues of the pseudo p-Laplacian are precisely the critical values of the functional (4.1) on the C

manifold M . It is then suf- ficient to observe that the assumptions of Theorem 2.5 are satisfied, thanks to Lemma 4.1.

Hence the first part of the statement is proved.

The sequence is non-decreasing. Let k ∈ N and f : S

→ M be an odd continuous mapping.

Then, we take a k-dimensional vector subspace E ⊂ R

and we consider the restriction g

of f to the intersection S

∩ E ' S

. Whence

u∈f(

max

Z

Ω

k∇uk

dx ≥ max

u∈f(S^k∩E)

Z

Ω

k∇uk

dx

= max

u∈g_E(S^k−1)

Z

Ω

k∇uk

dx

≥ inf

g∈Co(S^k−1;M)

max

u∈g(S^k−1)

Z

Ω

k∇uk

dx = σ

(Ω).

Since the mapping f was arbitrarly chosen, passing to the infimum we get σ

(Ω) ≥ σ

(Ω),

as desired.

The first element is zero. We now observe that the first element of the sequence (4.9) is in fact zero. To this end, we note that any continuous odd mapping from S

= {1, −1} to M can be identified with the choice of an antipodal pair u, −u on the symmetric manifold M.

This and the fact that the functional is even imply that if k = 1 formula (4.9) gives the minimum of (4.1) on M. The latter is of course zero, corresponding to constant functions.

Existence of a gap. We prove that there is a gap between zero and the second term of the sequence, i.e. σ

(Ω) > 0. At this aim, let us argue by contradiction and suppose that

inf

f∈Co(S¹;M)

max

u∈f(S¹)

Z

Ω

k∇uk

dx = 0,

so that, for all n ∈ N, there exists an odd continuous mapping f

from S

to M such that

(4.11) max

u∈f_n(S¹)

Z

Ω

k∇uk

dx ≤ 1 n . We now observe that

c = Z

∂Ω

%(x) dH

(x)

p

defines the unique (modulo the choice of the sign) constant function belonging to M . Let 0 < ε < 1/2 and consider the two neighborhoods

B

= {u ∈ M : ku − ck

< ε}, B

= {u ∈ M : ku − (−c)k

< ε}, which are disjoint, by construction. Here we set for brevity

kϕk

= Z

∂Ω

|ϕ(x)|

%(x) dH

(x)

.

Since the mapping f

is odd and continuous, for every n ∈ N the image f

( S

) is symmetric and connected, then it can not be contained in B

∪ B

, the latter being symmetric and disconnected. So we can pick an element

(4.12) u

∈ f

( S

) \

B

∪ B

.

This yields a sequence {u

}

⊂ M, which is bounded in W

(Ω) by (4.11). Hence, there exists a function v ∈ M such that {u

}

converges to v weakly in W

(Ω) and strongly in L

(∂Ω), possibly by passing to a subsequence. By the weak convergence it follows that

Z

Ω

k∇vk

dx ≤ lim inf

n→∞

Z

Ω

k∇u

k

dx ≤ lim

n→∞

1 n = 0.

This in turn shows that v ∈ M is constant, so that either v = c or v = −c. Let us assume that v = c, for example: using the strong convergence in L

(∂Ω), we get

0 = kv − ck

= lim

n→∞

ku

− ck

≥ ε,

where in the last inequality we used (4.12). This gives a contradiction and thus σ

(Ω) > 0.

Unboundedness. It remains to prove that this sequence of eigenvalues is unbounded. To this aim, we suitably modified the argument of [22, Proposition 5.4] (for a different proof, avoiding the use of Schauder bases, one could adapt the argument of [19, Theorem 5.2]).

We recall that the Sobolev space W

(Ω) admits a Schauder basis (see [20, 29]). Hence, there exists an ordered countable set of elements {e

}

⊂ W

(Ω) with the property that for all u ∈ W

(Ω), we have

u =

∞

X

n=1

α

e

,

for a (uniquely determined) sequence of scalars {α

}

. Here the converge of the previous series has to be understood in the sense of the norm topology. Clearly, if we now denote by

E

= Vect({e

, . . . , e

}),

the linear envelope of the first n elements of the basis, then the union S

n∈N

E

is dense in W

(Ω). Let us also set

F

= Vect({e

}

),

which is the topological supplement of the finite-dimensional vector space E

, and define the new sequence

τ

(Ω) = inf

f∈Co(S^k−1;M)

max

u∈f(S^k−1)∩Fk−1

Z

Ω

k∇u(x)k

dx, k ∈ N .

At first, we verify that such a sequence is actually well defined. Indeed, let f be an odd

and continuous map from the unit sphere S

to M and assume that the intersection

f (S

) ∩ F

is empty: this implies that for every ω ∈ S

, the element f (ω) always

has at least a nontrivial component on E

. By composing f with the continuous odd operator

P

: W

(Ω) → E

,

given by the natural projection on the linear space E

, the map P

◦f is odd, continuous and P

◦ f (ω) 6= 0, for every ω ∈ S

. That is, we constructed an odd continuous map from S

to E

\ {0} ' R

\ {0}, which is impossible thanks to the Borsuk-Ulam Theorem

. Hence the image of any f ∈ C

( S

; M ) has to intersect F

, for every k ∈ N .

Since obviously τ

(Ω) ≤ σ

(Ω), it sufficies now to show that

k→∞

lim τ

(Ω) = +∞.

At this aim, assume by contradiction that τ

(Ω) < τ, for all k ∈ N . Then, for every k ∈ N , we can take a mapping f ∈ C

( S

; M ) and u

∈ f S

∩ F

such that (4.13)

Z

Ω

k∇u

k

dx < τ.

Since u

∈ M for all k ∈ N, equation (4.13) implies that the sequence {u

}

is bounded in W

(Ω) and weakly converges (up to a subsequence) to some limit function u ∈ M .

On the other hand, by definition of Schauder basis, the functionals defined on W

(Ω) by

φ

(u) = α

, if u =

+∞

X

n=1

α

e

∈ W

(Ω), k ∈ N ,

are linear and they turn out to be continuous, thanks to [3, page 83]. By the weak conver- gence of the sequence {u

}

to u, it follows that lim

φ

(u

) = φ

(u), for all n ∈ N . Since u

∈ F

, we have that

φ

(u

) = 0, for every n ≤ k − 1,

thus φ

(u) = 0 for all n ∈ N . This means that u = 0, contradicting the fact that u ∈ M.

Remark 4.3. If Ω has m connected components Ω

, . . . , Ω

, equation (4.9) still defines an infinite sequence of eigenvalues, diverging at ∞, but in this case we have

σ

(Ω) = · · · = σ

(Ω) = 0,

corresponding to the (normalized) piecewise constant eigenfunctions c

=

Z

∂Ωi

%(x) dH

(x)

· 1

(x).

5. The first nontrivial eigenvalue

We are going to show that σ

(Ω) is actually the first nontrivial Stekloff eigenvalue of

− ∆ e

. In other words, the first eigenvalue σ = 0 is always isolated in the spectrum and any other eigenvalue has to be greater than σ

(Ω). Then the quantity σ

(Ω) can also be seen as the fundamental gap of the pseudo p−Laplacian, with Stekloff boundary conditions.

Theorem 5.1. Let u ∈ W

(Ω) be a Stekloff eigenfunction, with eigenvalue σ > 0. Then we have σ ≥ σ

(Ω).

2We recall that the Borsuk-Ulam states the following:

“for every continuous map f:S^k→R^k, there existsx0∈S^k such thatf(x0) =f(−x0)”.

Since our functionPk−1◦f is odd, this would give that 0∈Im(Pk−1◦f), that is a contradiction.