Related papers: About Three Dimensional Double-Sided Dirichlet and…
In this paper operator pencils $A(x,D,\lambda)$ are studied which act on a manifold with boundary and satisfy the condition of $N$-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by…
We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.
We establish the solvability of the $L^p$-Dirichlet and $L^{p^\prime}$-Neumann problems for the Laplacian for $p\in (\frac{n}{n-1}-\varepsilon,\frac{2n}{n-1}]$ for some $\varepsilon>0$ in $2$-sided chord-arc domains with unbounded boundary…
This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the…
In this paper, we study existence, uniqueness and asymptotic behavior of the Laplace equation with dynamical boundary conditions on regular non-cylindrical domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator and…
In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where $\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is…
New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…
The purpose of this paper is to investigate the existence of three different weak solutions to a nonlinear elliptic problem that is governed by the weighted {\varphi}-Laplacian operator and subjected to Dirichlet boundary conditions. We…
In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a…
In this paper we study nonlinear second-order differential inclusions involving the ordinary vector $p$-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying…
In our previous work [SIAM J. Sci. Comput. 43(3) (2021) B784-B810], an accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions has been developed. This work is devoted to studying the more…
This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet…
The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the…
We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that…
We study the $\bar\partial$ equation subject to various boundary value conditions on bounded simply connected Lipschitz domains $D\subset\mathbb C$: for the Dirichlet problem with datum in $L^p(bD, \sigma)$, this is simply a restatement of…
We prove the validity of a regularizing property on the boundary of the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in…
We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D} $ is an open bounded domain in…
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…
For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with $\overline\partial$-Neumann boundary conditions, which is a self-adjoint operator on the space of square-integrable differential…
Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed…