Estimates and asymptotic expansions for condenser p-capacities. the anisotropic case of segments

Abstract

We provide estimates and asymptotic expansions of condenser p-capacities and focus on the anisotropic case of (line) segments.

After preliminary results, we study p-capacities of points with respect to asymptotic approximations, positivity cases and convergence speed of descending continuity.

We introduce equidistant condensers to point out that the anisotropy caused by a segment in the p-Laplace equation is such that the Pólya–Szegő rearrangement inequality for Dirichlet type integrals yields a trivial lower bound. Moreover, when p > N, one cannot build an admissible solution for a segment, however small its length may be, by extending the case of a punctual obstacle.

Our main contribution is to provide a lower bound to the N-dimensional condenser p- capacity of a segment, by means of the N-dimensional and of the (N - 1)-dimensional condenser p-capacities of a point. The positivity cases follow for p-capacities of segments. Our method could be extended to obstacles with codimensions ≥2 in higher dimensions, such as surfaces in R⁴.

Introducing elliptical condensers, we obtain an estimate and the asymptotic expansion for the condenser 2-capacity of a segment in the plane. The topological gradient of the 2-capacity is not an appropriate tool to separate curves and obstacles with non-empty interior in 2D. In the case p ≠ 2, elliptical condensers should prove useful to obtain further estimates of p-capacities of segments.

Mathematics Subject Classification (2010): 31C15, 31C45, 35J92, 41A60, 42B37.

Keywords: Asymptotic analysis, topological gradient, topological sensitivity, capacity, condenser p-capacity, equidistant condenser, elliptical condenser, p-Laplace equation, quasilinear elliptic equations, nonlinear potential theory, line segment