ON THE NEUMANN ( p, q ) -EIGENVALUE PROBLEM IN HÖLDER SINGULAR DOMAINS

. In the article we study the Neumann ( p,q ) -eigenvalue problems in bounded Hölder γ -singular domains Ω γ ⊂ R n . In the case 1 < p < ∞ and 1 < q < p ∗ γ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of ( p,q ) -eigenvalues.

1 qualitative properties of the eigenfunctions of (1.3).For p = q in (1.1) under the hypothesis that Ω = B 1 ∪B 2 for two disjoint balls B 1 and B 2 , Croce-Henrot-Pisante [4] proved existence of a bounded eigenfunction of (1.1) in the space W 1,p per (Ω), which stands for the Sobolev space of functions in W 1,p (Ω) taking constant boundary values, see also Nazarov [25] for related results.
Estimates of Neumann eigenvalues of the p-Laplace operator in non-convex domains is a long-standing complicated problem [20,26,27].This problem was partially solved on the base of composition operators on Sobolev spaces, see for example, [13,15,16].In this article we prove that the first non-trivial Neumann (p, q)-eigenvalues can be characterized by the Min-Max Principle By using this characterization we give lower estimates of Neumann (p, q)-eigenvalues in bounded Hölder singular domains Ω γ ⊂ R n .In particular, we give the following lower estimate of the first nontrivial Neumann (p, q)-eigenvalue So, this example gives, in particular, in the frameworks the conjecture by V. Maz'ya [23], that exact Poincaré-Sobolev constants in anisotropic Hölder singular domains depends not on γ = γ 1 + ... + γ n−1 + 1 only.
In addition, in any bounded Hölder singular domain Ω γ , for q = 2, we prove existence of eigenfunction of (1.1) in W 1,p (Ω γ ) (see Theorems 4.1-4.2) with zero mean, but not necessarily takes the constant boundary values as in [4].Further, we study the associated minimizing problem of (1.1) (see Theorem 4.2) and prove the boundedness of the eigenfunctions among other qualitative properties (see Theorem 4.3).To this end, we follow the approach from Ercole [7].
This article is organized as follows: In Section 2, we prove Min-Max Principle for the first non-trivial Neumann (p, q)-eigenvalue.In Section 3, we give estimates of Neumann (p, q)-eigenvalues in Hölder singular domains.Finally, in Section 4, we mention the functional setting related to the problem (1.1) and further state and prove our regularity results.

Neumann eigenvalue problem
2.1.Sobolev spaces.Let Ω ⊂ R n be an open set.Then the Sobolev space W 1,p (Ω), 1 ≤ p ≤ ∞, is defined as a Banach space of locally integrable weakly differentiable functions u : Ω → R equipped with the following norm: where L p (Ω) is the Lebesgue space with the standard norm.In accordance with the non-linear potential theory [19,24] we consider elements of Sobolev spaces W 1,p (Ω) as equivalence classes up to a set of p-capacity zero [23].
The Sobolev space W 1,p loc (Ω) is defined as follows: u ∈ W 1,p loc (Ω) if and only if u ∈ W 1,p (U ) for every open and bounded set U ⊂ Ω such that U ⊂ Ω, where U is the closure of the set U .
Theorem 2.1.Let Ω γ ⊂ R n be a domain with anisotropic Hölder γ-singularities.Suppose 1 < p < γ.Then the embedding operator is compact for any 1 < r < p * γ , where p * γ = γp/(γ − p).Throughout Sections 2-3, we assume that 1 < p < γ and 1 < q < p * γ unless otherwise mentioned.Now for λ ∈ R, we consider the following Neumann (p, q)eigenvalue problem where Ω γ ⊂ R n and ν is the outward unit normal to ∂Ω γ in the weak formulation: We refer to λ as an eigenvalue and u as the eigenfunction corresponding to λ.Now we prove the Min-Max Principle for the first non-trivial Neumann (p, q)eigenvalue.We establish Theorem 2.3 following the proof of [4, Lemma 2].To this end, first we obtain the following auxiliary result.
Proof.By contradiction, suppose for every n ∈ N, there exists Without loss of generality, let us assume that v n L p (Ωγ ) = 1.If not, we define , 3) holds for u n and also ´Ωγ |u n | q−2 u n dx = 0.By (2.3), since ∇v n L p (Ωγ ) → 0 as n → ∞, we have that the sequence of functions and g ∈ L q (Ω γ ) such that Since ∇v n L p (Ωγ ) → 0 as n → ∞, we have ∇v n ⇀0 weakly in L p (Ω γ ), hence ∇v = 0 a.e. in Ω γ , which gives that v = constant a.e. in Ω γ .This combined with the fact that gives that v = 0 a.e. in Ω γ .This contradicts the hypothesis that v n L p (Ωγ ) = 1.This completes the proof.
Proof.Let n ∈ N and define the functionals and , where we denoted λ p,q (Ω γ ) by λ p,q .By the definition of infimum, for every n ∈ N, there exists u n ∈ W 1,p (Ω γ ) \ {0} such that ´Ωγ |u n | q−2 u n dx = 0 and H 1 n (u n ) < 0. Without loss of generality, let us assume that ∇u n L p (Ωγ ) = 1.By Lemma 2.2, the sequence {u n } is uniformly bounded in W 1,p (Ω γ ).Hence, by Theorem 2.1, because the embedding operator ) and u n → u strongly in L q (Ω γ ) and there exists g ∈ L q (Ω γ ) such that |u n | ≤ g a.e. in Ω γ and ∇u n ⇀∇u weakly in L q (Ω γ ).
Since |u n | ≤ g a.e. in Ω γ and u n → u a.e. in Ω γ .Then, So, by the Lebesgue Dominated Convergence Theorem (see, for example, [8]), it follows that Since ∇u n ⇀∇u weakly in L p (Ω γ ), by the weak lower semicontinuity of norm, we have So, by passing to the limit in (2.4), we get .
Therefore, by the definition of λ p,q , we obtain .

Estimates of Neumann (p, q)-eigenvalues
In this section we give estimates of Neumann (p, q)-eigenvalues in Hölder singular domains.The suggested method is based on Theorem 2.3 and on the composition operators theory on Sobolev spaces [29,32,33,34].
3.1.Composition operators on Sobolev spaces.The seminormed Sobolev space L 1,p (Ω) in a domain Ω ⊂ R n is the space of all locally integrable weakly differentiable functions with the following seminorm: Let Ω and Ω be domains in the Euclidean space ⊂ R n .Then a homeomorphism ϕ : Ω → Ω belongs to the Sobolev class W 1,p loc (Ω), 1 ≤ p ≤ ∞, if its coordinate functions ϕ j belong to W 1,p loc (Ω), j = 1, . . ., n.In this case the formal Jacobi matrix Dϕ(x) = ∂ϕi ∂xj (x) , i, j = 1, . . ., n, and its determinant (Jacobian) J(x, ϕ) = det Dϕ(x) are well defined at almost all points x ∈ Ω.The norm |Dϕ(x)| of the matrix Dϕ(x) is the norm of the corresponding linear operator Dϕ(x) : R n → R n defined by the matrix Dϕ(x).
holds.We denote by B r,s (Ω) the best constant in this inequality.
The weak quasiconformal mappings permits us to "transfer" the Sobolev-Poincaré inequalities from one domain to another.In the work [17], the authors obtained the following result.Theorem 3.2.Let a bounded domain Ω ⊂ R n be a (r, s)-Sobolev-Poincaré domain, 1 < s ≤ r < ∞, and there exists a weak (p, s)-quasiconformal homeomorphism ϕ : Ω → Ω of a domain Ω onto a bounded domain Ω, possesses the Luzin Nproperty (an image of a set of measure zero has measure zero) and such that M r,q (Ω) = ˆΩ |J(x, ϕ)| r r−q dx r−q rq < ∞ for some 1 ≤ q < r.Then in the domain Ω the (q, p)-Sobolev-Poincaré inequality holds and for 1 < s < p, we have Here B r,s (Ω) is the best constant in the (r, s)-Sobolev-Poincaré inequality in the domain Ω and K p,s (ϕ; Ω) is as defined in (3.1).
Let us check conditions of Theorem 3.2.Since ϕ a is a weak (p, s)-quasiconformal mapping, by Theorem 3.3, the constant K p,s (ϕ a ; Ω 1 ) is finite and satisfy the estimate The domain Ω 1 is a Lipschitz domain and so is a (r, s)-Sobolev-Poincaré domain, i.e.B r,s (Ω 1 ) < ∞.Let 1 < q < r < ∞.Next we estimate the quantity M r,q (Ω 1 ).
Remark 3.5.The estimate of the constant in the (r, s)-Sobolev-Poincaré inequality in the domain Ω 1 was obtained in [16]: The problem of exact value of constants in the (r, s)-Sobolev-Poincaré inequality in the case s = r is a complicated open problem even in the case of the unit disk D ⊂ R 2 .
Corollary 3.6.Let us consider an application of Theorem 3.4 to the following spectral problem with the Neumann boundary condition in an anisotropic Hölder γ-singular domains Ω γ , where Then the first non-trivial Neumann (3, 2)-eigenvalue of the spectral problem (3.4) satisfies the estimate: ≈ 12π.
If 3 < γ < 5 we can take a = 1 and so

Existence and regularity results
Throughout this section, we assume that 1 < p < γ and q = 2 unless otherwise stated.Let By Theorem 2.1, we endow the norm • X on X defined by (a) There exists a sequence {w n } n∈N ⊂ X ∩ Y such that w n Y = 1 and for every v ∈ X, we have where (b) Moreover, the sequences {µ n } n∈N and { w n+1 p X } n∈N given by (4.3) are nonincreasing and converge to the same limit µ, which is bounded below by λ.Further, there exists a subsequence {n j } j∈N such that both {w nj } j∈N and {w nj+1 } j∈N converges in X to the same limit w ∈ X ∩ Y with w Y = 1 and (µ, w) is an eigenpair of (2.1).Theorem 4.2.Let 1 < p < γ and q = 2. Suppose {u n } n∈N ⊂ X∩Y is a minimizing sequence for λ, that is u n Y = 1 and u n p X → λ.Then there exists a subsequence {u nj } j∈N which converges weakly in X to u ∈ X ∩Y such that λ = u p X .Moreover, u is an eigenfunction of (1.1) corresponding to λ and its associated eigenfunctions are precisely the scalar multiple of those vectors at which λ is reached.
Moreover, we have the following regularity results.Theorem 4.3.Let 1 < p < γ and q = 2. Assume that λ > 0 is an eigenvalue of the problem (1.1) and u ∈ X \ {0} is a corresponding eigenfunction.Then (i) u ∈ L ∞ (Ω γ ).(ii) Moreover, if u ∈ X \ {0} is nonnegative in Ω γ , then u > 0 in Ω γ .Further, for every ω ⋐ Ω γ there exists a positive constant c depending on ω such that u ≥ c > 0 in ω.First we state some useful results.The following result from [3, Theorem 9.14] will be useful for us.Theorem 4.5.Let V be a real separable reflexive Banach space and V * be the dual of V .Assume that A : V → V * is a bounded, continuous, coercive and monotone operator.Then A is surjective, i.e., given any f ∈ V * , there exists u ∈ V such that A(u) = f .If A is strictly monotone, then A is also injective.Next, we prove the following result.Proof.(i) Continuity: We only prove the continuity of A, since the continuity of B would follow similarly.To this end, suppose v n ∈ X such that v n → v in the norm of X.Thus, up to a subsequence ∇v n → ∇v in Ω γ .We observe that (4.7) for some constant c > 0, which is independent of n.Thus, up to a subsequence, we have Since, the weak limit is independent of the choice of the subsequence, as a consequence of (4.8), we have lim n→∞ Av n , w = Av, w for every w ∈ X.Thus A is continuous.
(ii) Boundedness: Using the estimate (4.9), we have Thus, A is bounded.

Coercivity: We observe that
Av, v = v p X .Since p > 1, we have A is coercive.Monotonicity: Using Lemma 4.6, it follows that there exists a constant C = C(p) > 0 such that for every v, w ∈ X, we have Thus, A is a monotone operator.holds for every v, w ∈ X.We claim that either v = 0 or w = 0 or v = tw for some constant t > 0. Indeed, if v = 0 or w = 0, this is trivial.Therefore, we assume v = 0 and w = 0 and prove that v = tw for some constant t > 0. By the estimate (4.9) if the equality (4.10) holds, then we have .

Lemma 4 . 7 .
(i) The operators A defined by (4.4) and B defined by (4.5) are continuous.(ii) Moreover, A is bounded, coercive and monotone.