Annales Academić Scientiarum Fennicć
Mathematica
Volumen 35, 2010, 537-563

SOME STABILITY RESULTS UNDER DOMAIN VARIATION FOR NEUMANN PROBLEMS IN METRIC SPACES

Estibalitz Durand-Cartagena and Antoine Lemenant

Universidad Complutense de Madrid, Departamento de Análisis Matemático
28040 Madrid, Spain; estibalitzdurand 'at' mat.ucm.es

Scuola Normale Superiore, Centro E. De Giorgi
Piazza dei Cavalieri 3, I-56100 Pisa, Italy; antoine.lemenant 'at' sns.it

Abstract. A famous result of Chenais [8] (1975) says that if \Omega_n is a sequence of extension domains in R^N that converges to \Omega in the characteristic functions topology, then the weak solutions u_n for the problem

-\Delta u_n + u_n= f in \Omega_n,
\frac{\partial}{\partial \nu} u_n = 0 on \partial \Omega_n

converge strongly to the solution u of the same problem in \Omega. It is also proved in [8] using the method of Calderón that an \varepsilon-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 H¹(\Omega) by the Newtonian space N^1,2(\Omega). 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.

2000 Mathematics Subject Classification: Primary 58J99, 30L10, 49J99, 46E35.

Key words: Newtonian spaces, \gamma-convergence, Mosco convergence, Neumann problem, differentiability in metric spaces, quasiconformal mappings, shape optimisation.

Reference to this article: E. Durand-Cartagena and A. Lemenant: Some stability results under domain variation for Neumann problems in metric spaces. Ann. Acad. Sci. Fenn. Math. 35 (2010), 537-563.

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doi:10.5186/aasfm.2010.3533