Related papers: A nonvariational form of the acoustic single layer…

We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $\Omega$ and in the exterior of $\Omega$. We look for…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$, and we define a distributional outward unit normal derivative for $\alpha$-H\"{o}lder continuous solutions of the Helmholtz…

We present a nonvariational setting for the Neumann problem for harmonic functions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. Then we introduce a space of distributions on the boundary (a space of first…

Given a compact convex planar domain $\Omega$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(\Omega)$ is the norm of the Neumann-Poincar\'e operator $K_\Omega$ acting on the space of continuous…

We generalize earlier studies on the Laplacian for a bounded open domain $\Omega\in \real^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this…

We establish the Hopf boundary point lemma for the Schr\"odinger operator $-\Delta + V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(\Omega)$. More precisely, we prove that among all supersolutions $u$ of $-\Delta +…

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the…

Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set…

We prove the existence of a family of compact subdomains $\Omega$ of the flat cylinder $\mathbb{R}^N\times \mathbb{R}/2\pi\mathbb{Z}$ for which the Neumann eigenvalue problem for the Laplacian on $\Omega$ admits eigenfunctions with constant…

In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $\Omega$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We perturb it to a domain $\Omega^\varepsilon$ attaching a family of small protuberances with "room-and-passage"-like geometry ($\varepsilon>0$ is a small parameter). Peculiar spectral…

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown…

We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the non-compactness of the boundary,…

Let $\Omega$ be an unbounded, pseudoconvex domain in $\Bbb C^n$ and let $\varphi$ be a $\mathcal C^2$-weight function plurisubharmonic on $\Omega$. We show both necessary and sufficient conditions for existence and compactness of a weighted…

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative…

We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that…

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

We show that the Neumann problem for Laplace's equation in a convex domain $\Omega$ with boundary data in $L^p(\partial\Omega)$ is uniquely solvable for $1<p<\infty$. As a consequence, we obtain the Helmholtz decomposition of vector fields…