LpPolyharmonic Dirichlet problems in regular domains I: the unit disc

Limits of Nonlinear Dirichlet Problems in Varying Domains

for suitable constants 0 < c 1 < c 2 < +o% 1 < p < n. For every ge Lq(f2), 1/p + 1/q = 1, we denote by mh(g) and Mh(g) respectively the minimum value and the set of all minimum points of problem (0.1). We shall prove the following compactness theorem (Section 6): for every sequence (Eh) of closed subsets of f2 there exist a subsequence (Ec(h)) and a non-negative Borel measure g, vanishing on ev...

A Monotonicity Approach to Nonlinear Dirichlet Problems in Perforated Domains

Abstract. We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation,...

Half-Dirichlet problems for Dirac operators in Lipschitz domains

Recall that in the case of the Dirichlet problem for the Laplace operator ∂2 x +∂ 2 y in Ω ⊆ R2, one prescribes the whole trace of a harmonic function in, say, L2(∂Ω). On the other hand, for the Cauchy-Riemann operator ∂x + i∂y, natural boundary problems are obtained by prescribing “half” of the trace of the analytic function in L2(∂Ω). Such half-Dirichlet problems arise when, for example, one ...

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Ω ⊂ R be a ball or an annulus and f : R → R absolutely continuous, superlinear, subcritical, and such that f(0) = 0. We prove that the least energy nodal solution of −∆u = f(u), u ∈ H 0 (Ω), is not radial. We also prove that Fučik eigenfunctions, i. e. solutions u ∈ H 0 (Ω) of −∆u = λu − μu−, with eigenvalue (λ, μ) on the first nontrivial curve of the Fučik spectrum, are not radial. A relat...

The Dirichlet Space on the bi-disc