Michel Chipot

This paper is devoted to some nonlocal, nonlinear elliptic problems with an emphasis on the computation of the solution of such problems, reducing it in particular to a fixed point argument in [ ]. Error estimates and numerical experiments are provided.

This note is devoted to some nonlocal, nonlinear elliptic problems with an emphasis on the computation of the solution of such problems, reducing it in particular to a fixed point argument in R. Errors estimates and numerical experiments are provided.

The goal of this note is to present different uniqueness results for problems associated with pseudo-monotone operators having the structure of the p -Laplace operator. The results are quite different when the equations at hand have a strictly monotone lower order term. The topic uses different test functions depending on the assumptions of the coefficients of the operators. Some of them are used in the references mentioned below but most of the time we simplify them and compare their strength. At the end of the note, we introduce some new type of such operators for which uniqueness can be proved roughly speaking the same way. MSC2020-Mathematics Subject Classification: 35A16, 35J60, 35J66, 35J92.

The goal of this note is to present regularity results regarding problems of the type [ ] in [ ], [ ] on [ ] which depend only on [ ] and not on the monotone function [ ]. This uses nice techniques developed by H. Brezis and collaborators.

The goal of this note is to consider Liouville type theorem for p-Laplacian type operators. In particular guided by the Laplacian case one establishes analogous results for the p-Laplacian and operators of this type.

The goal of this paper is to study the asymptotic behaviour of a linear combination of operators of the $ p $-Laplacian type in cylinder like domains having some directions going to infinity. The case of a combination of two such operators has been the topic of numerous studies in the recent years.

In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

The purpose of this note is to study the existence of a nontrivial solution for an elliptic system which comes from a newly introduced mathematical problem so called Field-Road model. Specifically, it consists of coupled equations set in domains of different dimensions together with some interaction of non classical type. We consider a truncated problem by imposing Dirichlet boundary conditions and an unbounded setting as well.