Related papers: Double-layer potentials, configuration constants a…

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…

Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $\Omega$ and obtain a compactness result for Hankel operators on the…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

Let $\cl{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where $\Omega \subseteq \bb{C}^m$ is a bounded domain. Let $\cl{M}_0\subseteq \cl{M}$ be the submodule of functions vanishing…

In this paper, we establish $C^{1, \alpha}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the…

We prove that for every $n \ge 2$, there exists a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ such that $\mathfrak{c}^0(\Omega) \subsetneq \mathfrak{c}^1(\Omega)$, where $\mathfrak{c}^k(\Omega)$ denotes the core of $\Omega$ with…

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincar\'e operator into a product of two self-adjoint transforms. Resurrecting some…

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…

Let $\Omega\subset\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $b\Omega$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $\Omega$.…

For smooth bounded pseudoconvex domains in $mathbb{C}^{2}$, we provide geometric conditions on (the points of infinite type in) the boundary which imply compactness of the $\bar{\partial}$-Neumann operator. It is noteworthy that the proof…

We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $\Omega$ with smooth boundary $b\Omega$ in…

Let $\Gamma(\cdot,\lambda)$ be smooth, i.e.\, $\mathcal C^\infty$, embeddings from $\bar{\Omega}$ onto $\bar{\Omega^{\lambda}}$, where $\Omega$ and $\Omega^\lambda$ are bounded domains with smooth boundary in the complex plane and $\lambda$…

In this thesis we solve the coboundary equation $\delta c=d$ with bounds for cochains with values in a coherent subsheaf of some $\mathcal{O}^p_{\Omega}$, where $\Omega$ is a Stein manifold. In particular the existence of a finite set of…

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) -…

Let $C_1(H)$ denote the space of all trace class operators on an arbitrary complex Hilbert space $H$. We prove that $C_1(H)$ satisfies the $\lambda$-property, and we determine the form of the $\lambda$-function of Aron and Lohman on the…

In this paper we establish a multivariable non-commutative generalization of L\"owner's classical theorem from 1934 characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex…

The homogenization of elliptic divergence-type fourth-order operators with periodic coefficients is studied in a (periodic) domain. The aim is to find an operator with constant coefficients and represent the equation through a perturbation…

In this work, we give a characterization of Lipschitz operators on spaces of $C^2(M)$ functions (also $C^{1,1}$, $C^{1,\gamma}$, $C^1$, $C^\gamma$) that obey the global comparison property-- i.e. those that preserve the global ordering of…

We study a family of non-local isoperimetric energies $E_{\gamma,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2…

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is shown that this operator has a…