We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the -energy and the parameter . These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.
We investigate the solvability of the quasilinear Neumann problem (1.1) with sub- and supercritical exponents in an unbounded domain Ω. Under some integrability conditions on the coefficients we establish embedding theorems of weighted Sobolev spaces into weighted Lebesgue spaces. This is used to obtain solutions through a global minimization of a variational functional.
Inequalities of Korn's type involve a positive constant, which depends on the domain, in general. A question arises, whether the constants possess a positive infimum, if a class of bounded two-dimensional domains with Lipschitz boundary is considered. The proof of a positive answer to this question is shown for several types of boundary conditions and for two classes of domains.
We introduce a method to treat a semilinear elliptic equation in (see equation (1) below). This method is of a perturbative nature. It permits us to skip the problem of lack of compactness of but requires an oscillatory behavior of the potential b.
Under a suitable oscillatory behavior either at infinity or at zero of the nonlinear term, the existence of infinitely many weak solutions for a non-homogeneous Neumann problem, in an appropriate Orlicz--Sobolev setting, is proved. The technical approach is based on variational methods.
First we recall a Faber-Krahn type inequality and an estimate for in terms of the so-called Cheeger constant. Then we prove that the eigenvalue converges to the Cheeger constant as . The associated eigenfunction converges to the characteristic function of the Cheeger set, i.e. a subset of which minimizes the ratio among all simply connected . As a byproduct we prove that for convex the Cheeger set is also convex.