Estimates for the first eigenfunction of linear eigenvalue problems via Steiner symmetrization

Abstract

By means of Steiner symmetrization we get some estimates for the first eigenfunction of a class of linear problems, having as prototype the Laplacian with Dirichlet boundary conditions.

Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem

“…Theorem 1 extends previous results in the literature on the comparison of Steiner rearrangements which until now were merely related to linear problems (see [1], [13], [7], [8], [9] and their references). The case in which g is convex is considered in [12].…”

Steiner symmetrization for anisotropic quasilinear equations via partial discretization

“…in y ∈ Ω 2 , which easily imply the a priori estimate on u in L p or Orlicz norms. Similar results for m > 0 have also been proven in a more recent paper [14] by using a simpler approach; Neumann boundary value problems have been studied in [24] (see also [16]). The implications of these kinds of mass comparison are deep.…”

New Pólya–Szegö-type inequalities and an alternative approach to comparison results for PDE's

“…Proof of Theorem 1.3. By a standard approximation argument it is enough to prove our result when the datum f is analytic (see, for example, [21], [16]). This implies that the solution u is analytic too.…”