Osvaldo Méndez

The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that if 0<p<1, 0<q<1, ℝ, while if 0<p<1, 1≤q<+∞, ∝, and if 1≤p<+∞, 0<q<1, ℝ.Applications to questions regarding the global interior regularity of solutions to Poisson type problems for the three-dimensional Lamé system in Lipschitz domains are presented.

The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains

Given a bounded Lipschitz domain Ω ⊂ Rn, n ≥ 3, we prove that the Poisson’s problem for the Laplacian with right-hand side in Lp−t(Ω), Robin-type boundary datum in the Besov space B1−1/p−t,p p (∂Ω) and non-negative, non-everywhere vanishing Robin coefficient b ∈ Ln−1(∂Ω), is uniquely solvable in the class Lp2−t(Ω) for (t, 1 p) ∈ V , where V ( ≥ 0) is an open (Ω,b)-dependent plane region and V0 is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson’s problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.

Differential operators on spaces of variable integrability

The theory of Lebesgue and Sobolev spaces with variable integrability is experiencing a steady expansion, and is the subject of much vigorous research by functional analysts, function-space analysts and specialists in nonlinear analysis. These spaces have attracted attention not only because of their intrinsic mathematical importance as natural, interesting examples of non-rearrangement-invariant function spaces but also in view of their applications, which include the mathematical modeling of electrorheological fluids and image restoration.The main focus of this book is to provide a solid functional-analytic background for the study of differential operators on spaces with variable integrability. It includes some novel stability phenomena which the authors have recently discovered.At the present time, this is the only book which focuses systematically on differential operators on spaces with variable integrability. The authors present a concise, natural introduction to the basic material and steadily move toward differential operators on these spaces, leading the reader quickly to current research topics.

Complex Powers of the Neumann Laplacian in Lipschitz Domains

Let Lr(Ω) be the usual scale of Sobolev spaces and let ∆N be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does (−∆N )− r 2 +iδ map {f ∈ Lq(Ω); ∫ Ω f = 0} isomorphically onto Lr(Ω)/IR ?

Modular Eigenvalues of the Dirichlet p(·)-Laplacian and Their Stability

The concept of the modular first Dirichlet eigenvalue for the p(·)-Laplacian is introduced as a generalization of the constant case. An important property of the corresponding eigenfunctions is obtained. We prove a qualitative stability result for such eigenvalues in terms of the magnitude of the perturbation of the variable modular exponent p(·).

Recent Advances on Generalized Trigonometric Systems in Higher Dimensions

We present a survey of current research on the basis properties of several trigonometric systems in higher dimensions.

Solutions of nonlinear elliptic equations in unbounded Lipschitz domains

Abstract This work is devoted to the study of the elliptic equation Δu = f(x, u) in an exterior non-smooth domain. Applying the method of upper and lower solutions and a diagonal argument, we prove the existence of solutions under various boundary conditions.

Banach Envelopes of the Besov and Triebel-Lizorkin Spaces and Applications to PDE’s

If the “hat” denotes the Banach envelope, we show that for \(s \in \mathbb{R},\)then For 0<p, q<1, one has

NONLINEAR BOUNDARY CONDITIONS FOR ELLIPTIC EQUATIONS

\widehat{B_p^{s,q} (\mathbb{R}^n )} = \widehat{F_p^{s,q} (\mathbb{R}^n )} = B_1^{s - n(\frac{1} {p} - 1),1} (\mathbb{R}^n ),