Abstract
We prove pointwise and L^{p}-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an {{,mathrm{RCD},}}(K,N) metric measure space, with K>0 and Nin (1,infty )). The obtained Talenti-type comparison is sharp, rigid and stable with respect to L^{2}/measured-Gromov–Hausdorff topology; moreover, several aspects seem new even for smooth Riemannian manifolds. As applications of such Talenti-type comparison, we prove a series of improved Sobolev-type inequalities, and an {{,mathrm{RCD},}} version of the St. Venant-Pólya torsional rigidity comparison theorem (with associated rigidity and stability statements). Finally, we give a probabilistic interpretation (in the setting of smooth Riemannian manifolds) of the aforementioned comparison results, in terms of exit time from an open subset for the Brownian motion.
Highlights
The classical notion of Schwarz symmetrization of a function plays a notable role in proving results such as the Rayleigh–Faber–Krahn Inequality, as well as several variational inequalities for differential boundary problems
Estimates on solutions to differential boundary value problems via Schwarz symmetrization have been obtained by several mathematicians, let us mention Weinberger [51], Bandle [11], Talenti [49], P
The main result of the paper is a Talenti-type comparison theorem where we compare the weak solution to a Poisson problem as in (1) defined on an open set of an RCD(K, N ) space (K > 0, N ∈ (1, ∞)) with the solution of an analogous Poisson problem defined on the model space (3)
Summary
In the study of geometric and variational problems in Euclidean spaces, a tool which often proves useful is the technique of symmetrization: one can frequently simplify a complex problem by reducing it to the study of spherically symmetric objects. Let us stress that several results of the paper seems new even for smooth Riemannian manifolds with Ricci curvature bounded below by a positive constant. The main result of the paper is a Talenti-type comparison theorem where we compare the weak solution to a Poisson problem as in (1) defined on an open set of an RCD(K , N ) space (K > 0, N ∈ (1, ∞)) with the solution of an analogous Poisson problem defined on the model space (3) (see Theorem 3.10). A rigidity result (Theorem 4.4) roughly stating that if equality in the Talenti-type comparison Theorem 3.10 is achieved, the space is a spherical suspension;. A stability result (Theorem 4.15) roughly stating that equality in the Talenti-type comparison Theorem 3.10 is almost achieved (in L2-sense) if and only if the space is mGH-close to a spherical suspension. For the reader’s convenience, the Appendix gives a self-contained presentation of the statements of the main results for a smooth Riemannian manifold with positive Ricci curvature (as several aspects seem to be new even in this setting)
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