Estimates and asymptotic expansions for condenser p-capacities. The anisotropic case of segments
Résumé
We provide estimates and asymptotic expansions of condenser p-capacities and focus on the anisotropic case of 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-Szegö rearrangement inequality for Dirichlet type integrals yields a trivial lower bound. Moreover, when p > N, one cannot build an admissible solution for the 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 codimension ≥ 2 in higher dimensions, such as surfaces in R^4. 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.
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