Sharp isoperimetric inequalities via the ABP

Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in Rn satisfying a concavity condition in the cone. The condition is equivalent to a natural curvature-dimension bound and also to the nonnegativeness of a Bakry-Emery Ricci tensor. Even that our weights are nonradial, still balls are minimizers of the weighted isoperimetric problem. A particular important case is that of monomial weights. Our proof uses the ABP method applied to an appropriate linear Neumann problem. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and PacellaPeer ReviewedPostprint (published version

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions

Space-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space $H$ are developed. The well-posedness of these equations in the Hilbert space $L^{2}(H,\mu)$ of functions on $H$, which are square-integrable with respect to a Gaussian measure $\mu$ on $H$, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener-Itô decomposition of $L^{2}(H,\mu)$, are introduced and are shown to converge quasioptimally with respect to the nonlinear, best $N$-term approximation benchmark. As a consequence, the proposed adaptive Galerkin solution algorithms perform quasioptimally with respect to the best $N$-term approximation in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of "active" coordinates identified by the proposed adaptive Galerkin approximation algorithms