Annales Academiĉ Scientiarum Fennicĉ

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 *H*^{1}(\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.

Full document as PDF file

doi:10.5186/aasfm.2010.3533

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