Electron. J. Diff. Equ., Vol. 2015 (2015), No. 217, pp. 1-14.

Limit of nonlinear elliptic equations with concentrated terms and varying domains: the non uniformly Lipschitz case

Gleiciane da Silva Aragao, Simone Mazzini Bruschi

In this article, we analyze the limit of the solutions of nonlinear elliptic equations with Neumann boundary conditions, when nonlinear terms are concentrated in a region which neighbors the boundary of domain and this boundary presents a highly oscillatory behavior which is non uniformly Lipschitz. More precisely, if the Neumann boundary conditions are nonlinear and the nonlinearity in the boundary is dissipative, then we obtain a limit equation with homogeneous Dirichlet boundary conditions. Moreover, if the Neumann boundary conditions are homogeneous, then we obtain a limit equation with nonlinear Neumann boundary conditions, which captures the behavior of the concentration's region. We also prove the upper semicontinuity of the families of solutions for both cases.

Submitted February 28, 2015. Published August 19, 2015.
Math Subject Classifications: 35J60, 30E25, 35B20, 35B27.
Key Words: Nonlinear elliptic equation; boundary value problem; varying boundary; oscillatory behavior; concentrating term; upper semicontinuity.

Show me the PDF file (297 KB), TEX file, and other files for this article.

Gleiciane da Silva Aragão
Departamento de Ciências Exatas e da Terra
Universidade Federal de São Paulo
Rua Professor Artur Riedel, 275, Jardim Eldorado
Cep 09972-270, Diadema-SP, Brazil
email: gleiciane.aragao@unifesp.br, Phone (+55 11) 33193300
Simone Mazzini Bruschi
Departamento de Matemática
Universidade de Brasília
Campus Universitário Darcy Ribeiro
ICC Centro, Bloco A, Asa Norte
Cep 70910-900, Brasília-DF, Brazil
email: sbruschi@unb.br, Phone (+55 61) 31076389

Return to the EJDE web page