An operator [A.sub.d] is called an
elliptic operator if
where [[phi].sub.j] is an eigenfunction of the
elliptic operatorSuppose that [??] be an
elliptic operator in the divergence form as in (1.1), where the coefficient matrix ([a.sub.ij]) satisfies the degenerate ellipticity condition
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a quasilinear
elliptic operator, 1 < q < p < N, 0 < [??] < + [infinity], 0 [less than or equal to] [mu] < [bar.[mu]] with [bar.[mu]] = [((N - p)/p).sup.p], [lambda] [greater than or equal to] 0, and [alpha], [beta] > 1 satisfy [alpha] + [beta] = [p.sup.*], [p.sup.*] [??] (Np/(N - p)) denotes the critical Sobolev exponent, and Q [member of] C([R.sup.N]) [intersection] [L.sup.[infinity]]([R.sup.N]) and [h.sub.i] [member of] [L.sup.[theta]]([R.sup.N]) (i = 1,2) with d = Np/(Np - q(N - p)) are G-symmetric functions (see Section 2 for details) with respect to a closed subgroup G of O(N).
Consider now the problem (NBVP) in [R.sup.n] (n > 1) for the nonlinear
elliptic operatorFor instance, such a structure arises when decomposing the domain of definition of an
elliptic operator using unidirectional stripes, or more generally, for a decomposition such that (in addition to a corresponding portion of the original boundary) each subdomain has a common boundary only with its previous and next neighbours in the sequence of subdomains.
Exploring recent results in spectral geometry and its links with shape optimization, contributors are interested with whether there exists a set that minimizes (or maximizes) the k-th eigenvalue of a given
elliptic operator with given boundary conditions, among sets of given volume, and if so what can be said about the regularity of the optimal set.
The boundary conditions are chosen to be homogeneous only for simplicity, [OMEGA] is a given bounded Lipschitz domain and L is an
elliptic operator, i.e.
Before presenting our main theorem, we introduce the second-order
elliptic operator L as follows: For [xi] = ([[xi].sub.1], ..., [[xi].sub.n]) [member of] [C.sup.n], denote its complex conjugate ([[bar.[xi]].sub.1], ..., [[bar.[xi]].sub.n]) by [bar.[xi]].