Some stability results under domain variation for Neumann problems in metric spaces . Estibalitz Durand - Cartagena and Antoine Lemenant

A famous result of Denise Chenais [8] (1975) says that if Ωn is a sequence of extension domains in R that converges to Ω for the characteristic functions topology, then the weak solutions un for the problem −∆un + un = f in Ωn ∂ ∂ν un = 0 on ∂Ωn (0.1) converge strongly to the solution u of the same problem in Ω. It is also proved in [8] using the method of Calderón that an ε-cone condition is sufficient to obtain uniform extension domains. In this paper we establish this result in a metric space framework, replacing the classical Sobolev space H(Ω) by the Newtonian space N(Ω). Moreover, using the latest results about extension domains contained in [2], and which rely on the technics of P. Jones, we give weaker conditions on the domains for still getting stability for 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. AMS classification. 58J99, 30L10, 49J99, 46E35