An operator [A.sub.d] is called an elliptic operator
where [[phi].sub.j] is an eigenfunction of the elliptic operator
Let [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 operator
For 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
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]].