Related papers: Solving elliptic PDEs in unbounded domains
This paper aims to establish counterparts of fundamental regularity statements for solutions to elliptic equations in the setting of low-dimensional structures such as, for instance, glued manifolds or CW-complexes. The main result proves…
We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…
In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…
This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential…
An adaptive finite element method is presented for the elastic scattering of a time-harmonic plane wave by a periodic surface. First, the unbounded physical domain is truncated into a bounded computational domain by introducing the…
In this paper, we demonstrate the existence of positive solutions for certain weakly coupled elliptic systems of sublinear growth under homogeneous Dirichlet boundary conditions. Our findings generalize existing results related to sublinear…
In this paper, we propose a novel machine learning method based on an adaptive tensor neural network subspace for solving quasiperiodic elliptic problems. To this end, we first provide a theoretical analysis of the associated quasiperiodic…
This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated…
We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
This paper investigates the geometric constraints imposed on a domain by overdetermined problems for partial differential equations. Serrin's symmetry results are extended to overdetermined problems with potentially degenerate ellipticity…
We study iterative finite element approximations for the numerical approximation of semilinear elliptic boundary value problems with monotone nonlinear reactions of subcritical growth. The focus of our contribution is on an optimal a priori…
This paper is concerned with the mathematical analysis of the time-domain electromagnetic scattering problem in an infinite rectangular waveguide. A transparent boundary condition is developed to reformulate the problem into an equivalent…
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…
The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified…
Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion…
We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of…
Elliptic estimates in Hardy classes are proved on domains with minimally smooth boundary. The methodology is different from the original methods of Chang/Krantz/Stein.
In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…
In this paper, we present how high-order accurate solutions to elliptic partial differential equations can be achieved in arbitrary spatial domains using radial basis function-generated finite differences (RBF-FD) on unfitted node sets…