A STABILITY RESULT FOR NONLINEAR NEUMANN PROBLEMS IN REIFENBERG FLAT DOMAINS IN R-N

作者：Lemenant Antoine^*; Milakis Emmanouil

来源：Publicacions Matematiques, 2011, 55(2): 413-432.

DOI：10.5565/PUBLMAT_55211_08

• Boundary value problems
• nonlinear elliptic equations
• Hausdorff distance
• Reifenberg-flat sets
• Mosco convergence

In this paper we prove that if Omega(k) is a sequence of Reifenberg-flat domains in R-N that converges to Omega for the complementary Hausdorff distance and if in addition the sequence Omega(k) has a "uniform size of holes", then the solutions u(k) of a Neumann problem of the form
(0.1) {-div a(x, del u(k)) + b(x, u(k)) = 0 in Omega(k)
a(x, del u(k)) nu = 0 on partial derivative Omega(k)
converge to the solution u of the same Neumann problem in Omega. The result is obtained by proving the Mosco convergence of some Sobolev spaces, that follows from the extension property of Reifenberg-flat domains.

• 出版日期2011