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M. Lanza de Cristoforis

Publications and source records attributed to M. Lanza de Cristoforis.

13 recordsLinked to original sources
arXiv2026

A Helmholtz-Weyl decomposition for Hölder spaces. A nonvariational approach

We prove a form of the fundamental theorem of vector calculus for Hölder continuous vector fields with no assumption on the derivatives that exploits Schauder spaces with negative exponents. Then by developing an idea of von Wahl in the context of Lebesgue spaces, we prove a form of the Helmholtz Weyl decomposition Theorem for spaces of Hölder continuous functions, with explicit formulas for the projection operators.

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arXiv2026

Representation theorems for nonvariational solutions of the Helmholtz equation

We consider a possibly multiply connected bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},α}$ for some $m\in {\mathbb{N}}$, $α\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for the Helmholtz equation in $Ω$ and in the exterior of $Ω$ in terms of acoustic layer potentials. Then we turn to prove an integral representation theorem solutions of the Helmholtz equation in terms of an acoustic single layer potential. The main focus of the paper is on $α$-Hölder continuous solutions which may not have a classical normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$\, \textit{i.e.}, case $m=0$. Namely for solutions that do not belong to the classical variational setting.

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arXiv2026

A nonvariational form of the acoustic single layer potential

We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$ and the space $V^{-1,α}(\partialΩ)$ of (distributional) normal derivatives on the boundary of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,α}(\overlineΩ)$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $Ω$ with boundary data in $V^{-1,α}(\partialΩ)$ and solutions in the space of $α$-Hölder continuous functions in $Ω$ that have Laplace operator in $C^{-1,α}(\overlineΩ)$, \textit{i.e.}, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting.

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arXiv2025

A nonvariational Neumann problem for the Helmholtz equation

We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $Ω$ and in the exterior of $Ω$. We look for solutions in the space for $α$-Hölder continuous functions that may not have a classical normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. Namely for solutions that do not belong to the classical variational setting.

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arXiv2025

A uniqueness theorem for nonvariational solutions of the Helmholtz equation

We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$, and we define a distributional outward unit normal derivative for $α$-Hölder continuous solutions of the Helmholtz equation in the exterior of $Ω$ that may not have a classical outward unit normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for $α$-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for $α$-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting.

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arXiv2025

The volume potential for elliptic differential operators in Schauder spaces

The aim of this paper is to prove continuity results for the volume potential corresponding to the fundamental solution of a second order differential operator with constant coefficients in Schauder spaces of negative exponent and to generalize some classical results in Schauder spaces of positive exponents.

math.AP↗
arXiv2024

A nonvariational form of the Neumann problem for the Poisson equation

We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are Hölder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary for the solutions that extends that for harmonic functions that has been introduced in a previous paper and we solve the nonvariational Neumann problem for data in the interior with a negative Schauder exponent and for data on the boundary that belong to a certain space of distributions on the boundary.

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arXiv2024

A nonvariational form of the Neumann problem for Hölder continuous harmonic functions

We present a nonvariational setting for the Neumann problem for harmonic functions that are Hölder continuous and that may have infinite Dirichlet integral. Then we introduce a space of distributions on the boundary (a space of first order traces for Hölder continuous harmonic functions), we analyze the properties of the corresponding distributional single layer potential and we prove a representation theorem for harmonic Hölder continuous functions in terms of distributional single layer potentials. As an application, we solve the interior and exterior Neumann problem with distributional data in the space of first order traces that has been introduced.

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arXiv2024

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 Hölder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.

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arXiv2023

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 Hölder 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 Hölder 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\'ıa-Cuerva and Gatto in the frame of Hölder spaces and later by the author.

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arXiv2023

Classes of kernels and continuity properties of the double layer potential in Hölder 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 Hölder 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.

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arXiv2016

Integral potential method for a transmission problem with Lipschitz interface in ${\mathbb R}^3$ 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 ${\mathbb R}^3$, one of them is a bounded Lipschitz domain $Ω$ with connected boundary, and another one is the exterior Lipschitz domain ${\mathbb R}^3\setminus \overline{Ω}$. 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.

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