A nonlinear elliptic problem with terms concentrating in the boundary

Abstract

Abstract. In this paper we investigate the behavior of a family of steady state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a -neighborhood of a portion Γ of the boundary. We assume that this -neighborhood shrinks to Γ as the small parameter goes to zero. Also, we suppose the upper boundary of this -strip presents a highly oscillatory behavior. Our main goal here is to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood.

Concentrated reaction terms on the boundary of rough domains for a quasilinear equation

“…Also, χ O ε is the characteristic function of O ε and (H f ) f : R → R is a bounded function with bounded derivatives. Notice we are in agreement with previous works as [2,3,5] using the characteristic function χ O ε and term 1/ε γ+1 to express concentration on O ε . Our main goal here is to improve [1,4,6,8] dealing with a p-Laplacian equation in a perturbed n + 1-dimensional domain presenting roughness on boundary.…”

Concentrated terms and varying domains in elliptic equations: Lipschitz case

“…In order to obtain the upper semicontinuity of the family of solutions of (1) and (2), we study the behavior of the nonlinearities h , 0 Ä Ä 0 , defined by (4) and (5). Here, we also proved a more general result, which the behavior of potential V , 0 Ä Ä 0 , defined in is analyzed when !…”

“…In , some results of were adapted to a nonlinear elliptic problem posed on an open square Ω in , considering ω ε ⊂Ω and with highly oscillatory behavior in the boundary inside of Ω. Recently, the dynamics of the flow generated by a nonlinear parabolic problem posed on a C 2 domain Ω in , when some reaction and potential terms are concentrated in a neighborhood of the boundary and the ‘inner boundary’ of this neighborhood presents a highly oscillatory behavior, was studied in where the continuity of the family of attractors was proved.…”

“…In [5], some results of [3] were adapted to a nonlinear elliptic problem posed on an open square in R 2 , considering ! and with highly oscillatory behavior in the boundary inside of .…”

Dynamics of thermoelastic plate system with terms concentrated in the boundary

“…In this work we analyze the asymptotic behavior of the global compact attractors of autonomous thermoelastic plate systems with Neumann boundary conditions when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter ε goes to zero. There has been numerous studies to investigate the dynamics, in the sense of attractors, of systems when reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter ε goes to zero, see for instance [2,3,4,5,6,7,8,9,13,14] and references therein.…”