On the tangential gradient of the kernel of the double layer potential

In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in H\"{o}lder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.Comment: arXiv admin note: text overlap with arXiv:2305.1967

A survey on the boundary behavior of the double layer potential in Schauder spaces in the frame of an abstract approach

We provide a summary of the continuity properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder and Schauder spaces on the boundary of a bounded open subset of ${\mathbb{R}}^n$. The purpose is two-fold. On one hand we try present in a single paper all the known continuity results on the topic with the best known exponents in a H\"{o}lder and Schauder space setting and on the other hand we show that many of the properties we present can be deduced by applying results that hold in an abstract setting of metric spaces with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by Garc\'{\i}a-Cuerva and Gatto in the frame of H\"{o}lder spaces and later by the author.Comment: arXiv admin note: text overlap with arXiv:2305.19672, arXiv:2307.0477

Classes of kernels and continuity properties of the double layer potential in H\"{o}lder spaces

We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of integral operators and a generalization of a result for integral operators on differentiable manifolds.Comment: arXiv admin note: substantial text overlap with arXiv:2305.19672, arXiv:2103.06971, arXiv:2307.0415

Analytic dependence of a periodic analog of a fundamental solution upon the periodicity paramaters

We prove an analyticity result in Sobolev-Bessel potential spaces for the periodic analog of the fundamental solution of a general elliptic partial differential operator upon the parameters which determine the periodicity cell. Then we show concrete applications to the Helmholtz and the Laplace operators. In particular, we show that the periodic analogs of the fundamental solution of the Helmholtz and of the Laplace operator are jointly analytic in the spatial variable and in the parameters which determine the size of the periodicity cell. The analysis of the present paper is motivated by the application of the potential theoretic method to periodic anisotropic boundary value problems in which the `degree of anisotropy' is a parameter of the problem

Mapping properties of weakly singular periodic volume potentials in Roumieu classes

The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper, we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. This result extends to the periodic case of some previous results obtained by the authors for nonperiodic potentials, and it is motivated by the study of perturbation problems for the solutions of boundary value problems for elliptic differential equations in periodic domains

Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

An Inequality for H\uf6lder Continuous Functions Generalizing a Result of Carlo Miranda

We prove an inequality for the H"{o}lder continuity of a continuously differentiable function in a bounded open Lipschitz set, which generalizes an inequality of Carlo Miranda in a bounded open hypograph of a function of class $C^{1,alpha}$ for some $alphain ]0,1]$ and which enables to simplify a proof of a result of Carlo Miranda for layer potentials with moment in a Schauder space

A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach

Let $\Omega$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in \Omega$ and that $\mathbb{R}^n \setminus \mathrm{cl}\Omega$ is connected. Then we take $q_{11},..., q_{nn}\in ]0,+\infty[$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $\epsilon$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[\Omega_\epsilon]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+\epsilon \Omega +\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $\epsilon$ small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set $\mathbb{S}[\Omega_\epsilon]^{-}$. Namely, we consider a Dirichlet condition on the boundary of the set $p+\epsilon \Omega$, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of $\epsilon$ and of the Dirichlet datum on $p+\epsilon \partial \Omega$, around a degenerate pair with $\epsilon=0$

Composition Operators in Grand Lebesgue Spaces

Let $\Omega$ be an open subset of $\mathbb{R}^n$ of finite measure. Let $f$ be a Borel measurable function from $\mathbb{R}$ to $\mathbb{R}$. We prove necessary and sufficient conditions on $f$ in order that the composite function $T_f[g]=f\circ g$ belongs to the Grand Lebesgue space $L_{p),\theta}(\Omega)$ whenever $g$ belongs to $L_{p),\theta}(\Omega)$. We also study continuity, uniform continuity, H\"older and Lipschitz continuity of the composition operator $T_f [\cdot]$ in $L_{p),\theta}(\Omega)$