Negative Energy Solutions for a New Fractional p x -Kirchhoff Problem without the (AR) Condition

In this paper, we investigate the following Kirchhoff type problem involving the fractional p x -Laplacian operator. a − b ∫ Ω × Ω u x − u y p x , y / p x , y x − y N + s p x , y d x d y L u = λ u q x − 2 u + f x , u x ∈ Ω u = 0 x ∈ ∂ Ω , , where Ω is a bounded domain in ℝ N with Lipschitz boundary, a ≥ b > 0 are constants, p x , y is a function defined on Ω ¯ × Ω ¯ , s ∈ 0 , 1 , and q x > 1 , L u is the fractional p x -Laplacian operator, N > s p x , y , for any x , y ∈ Ω ¯ × Ω ¯ , p x ∗ = p x , x N / N − s p x , x , λ is a given positive parameter, and f is a continuous function. By using Ekeland’s variational principle and dual fountain theorem, we obtain some new existence and multiplicity of negative energy solutions for the above problem without the Ambrosetti-Rabinowitz ((AR) for short) condition.

The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d with Lipschitz boundary $$\Gamma $$ Γ . More precisely, using form methods, we show that the associated operator on the ground space $$L^2(\Omega )\times L^2(\Gamma )$$ L 2 ( Ω ) × L 2 ( Γ ) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY

We consider the linearized and nonlinear systems describing the motion of incompressible flow around a rotating and translating rigid body Ɗ in the exterior domain Ω = ℝ3 \ Ɗ, where Ɗ ⊂ ℝ3 is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity essentially behaves near the infinity as a constant times the first column of the fundamental solution of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare this solution with the solution in the exterior domain Ω to get the truncation error estimate.

On the Identication of Coecient and Source Parameters in Elliptic Systems Modelled with Many Boundary Values Problems

The inverse problem for determination of parameters related to the support and/or functions describing the intensity of coefficient and sources in models based strongly elliptic second order systems is posed with Cauchy data over specification at boundary. This stablish a set of various boundary value problems associated with the same group of unknown parameters. A Lipschitz boundary dissection is used for decomposing each Cauchy data into pairs of complementary mixed boundary values problems. The concept of Calderon projector is introduced as a tool to check the consistency of the Cauchy data and to demonstrate the equivalence of these two problems. This lets you define a discrepancy function to measure the distance between the solutions of problems obtained by dissecting Lipschitz Cauchy data. This discrepancy appears as a consequence of inadequate parameters values in the constitutive relations. For Cauchy noisy data, the difference between these solutions would be small if the parameters used in the solution are correct. The methodology we propose explores concepts as Lipschitz Boundary Dissection, Complementary Mixed Problems with trial parameters and Internal Discrepancy fields. Differentiable and non-differentiable optimizations algorithms can then be used in the reconstruction of these parameters simultaneously. Numerical experiments are presented.

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

Homogenization of Perforated Elastic Structures

The paper is dedicated to the asymptotic behavior of $\varepsilon$ ε -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ ε → 0 . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ 3 D to $2D$ 2 D or $1D$ 1 D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as $\varepsilon \to 0$ ε → 0 we use the periodic unfolding method.

Morrey spaces and generalized Cheeger sets

We maximize the functional\frac{\int_{E}h(x)\,dx}{P(E)},where {E\subset\overline{\Omega}} is a set of finite perimeter, Ω is an open bounded set with Lipschitz boundary and h is nonnegative. Solutions to this problem are called generalized Cheeger sets in Ω. We show that the Morrey spaces {L^{1,\lambda}(\Omega)}, {\lambda\geq n-1}, are natural spaces to study this problem. We prove that if {h\in L^{1,\lambda}(\Omega)}, {\lambda>n-1}, then generalized Cheeger sets exist. We also study the embedding of Morrey spaces into {L^{p}} spaces. We show that, for any {0<\lambda<n}, the Morrey space {L^{1,\lambda}(\Omega)} is not contained in any {L^{q}(\Omega)}, {1<q<p=\frac{n}{n-\lambda}}. We also show that if {h\in L^{1,\lambda}(\Omega)}, {\lambda>n-1}, then the reduced boundary in Ω of a generalized Cheeger set is {C^{1,\alpha}} and the singular set has Hausdorff dimension at most {n-8} (empty if {n\leq 7}). For the critical case {h\in L^{1,n-1}(\Omega)}, we demonstrate that this strong regularity fails. We prove that a bounded generalized Cheeger set E in {\mathbb{R}^{n}} with {h\in L^{1}(\mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h.