Non-Linear Balayage and applications

Abstract

A theory of capacities has been extentively studied for Besov spaces [1]. However not much seems to have been done regarding non-linear potentials. We develop some of this here as consequences of the form of certain metric projections.

The non-linear potential theory is used to derive the form of tangent cones for a class of convex sets in Besov spaces. Tangent cones for obstacle problem arise when studying differentiability of metric projection. Characterising the tangent cones is the first step in these considerations. This has been done in some of the Sobolev spaces using Hilbert space methods. In this article we describe tangent cones for obstacle problems precisely, using non-linear potential theoretic ideas, for all Besov spaces $B^{p,q}_{\alpha}$, $1 \lt p \lt \infty$, $1 \lt q \lt \infty$, $\alpha \gt 0$.