Some stability results under domain variation for Neumann problems in metric spaces

Abstract

A famous result of Chenais (8) (1975) says that if ›n is a sequence of extension domains in R N that converges to › in the characteristic functions topology, then the weak solutions un for the problem (0.1) ( i¢un + un = f in ›n; @ un = 0 on @›n converge strongly to the solution u of the same problem in ›. It is also proved in (8) using the method of Calderon that an -cone condition is sucient to obtain uniform extension domains. In this paper we establish this result in a metric space framework, replacing the classical Sobolev space H 1 (›) by the Newtonian space N 1;2 (›). Moreover, using the latest results about extension domains contained in (2), and which rely on the techniques of Jones, we give weaker conditions on the domains for still getting stability of the Neumann problem. Finally we prove that the Neumann problem is stable for a sequence of quasiballs with uniform distortion constant that converge in a certain measure sense. The latter result gives a new existence theorem for some shape optimisation problems under quasiconformal variations.