An operator [A.sub.d] is called an

elliptic operator if

where [[phi].sub.j] is an eigenfunction of the

elliptic operatorLet [OMEGA] [subset] [R.sup.2] be a bounded domain and let L be a linear

elliptic operator of the form L = [[summation].sub.[absolute value of [alpha]][less than or equal to]m] [a.sub.[alpha]](x, y)[[partial derivative].sup.[alpha]], where [alpha] is a two-dimensional multi-index.

Suppose 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]].