Maximizing mean exit-time of the Brownian motion on Riemannian manifolds

Abstract

We study the functional Ω↦E(Ω), where Ω runs in the set of all compact domains of fixed volume v in any Riemannian manifold (M,g) and where E(Ω) is the mean exit-time of the Brownian motion (also called torsional rigidity) of Ω. We first prove that, when (M,g) is strictly isoperimetric at one of its points, the maximum of this functional is realized by the geodesic ball centered at this point. When (M,g) is any Riemannian manifold, for every domain Ω in M, we prove that E(Ω)≤E(Ω∗), where Ω∗ is the corresponding symmetrized domain on a model-space (M∗,g∗). We also consider the functional Ω↦E(Ω), when Ω runs in the set of all compact domains, with smooth boundary in the class of all Riemannian manifolds with “bounded” geometry. We prove two results in this direction. In the first one (Theorem 1.9) we prove that for every complete, connected Riemannian manifold (M,g) whose Ricci curvature satisfies Ricg≥(n−1)g and for every compact domain with smooth boundary Ω in M one has E(Ω)≤E(Ω∗), where Ω∗ is a geodesic ball of the canonical sphere (Sn,g0) such that Vol(Ω∗,g0)Vol(Sn,g0)=Vol(Ω,g)Vol(M,g). Morever, if there exists some domain Ω⊂M such that E(Ω)=E(Ω∗) then (M,g) is isometric to (Sn,g0) and Ω is isometric to Ω∗. The second result (Theorem 1.10) shows that if (M,g) is any compact Riemannian manifold and Ω is any compact domain with smooth boundary in M such that Vol(Ω)≤12 Vol(M), then E(Ω)≤1H(M,g)2, where H(M,g) is Cheeger’s isoperimetric constant

On the heat content functional and its critical domains

“…As a consequence of Serrin's rigidity theorem (see [35]) we know that the only domains in R n which are critical for T 1 under volume preserving deformations are balls, and these are all maxima by the classical isoperimetric result by Polya ([26,27]). The same rigidity holds in the spaces H n and S n + : the only critical domains are geodesic balls (also in this case, these are absolute maxima, see for example [5] and [6]).…”

On the heat content functional and its critical domains

“…As a consequence of Serrin's rigidity theorem (see [33]) we know that the only domains in R n which are critical for T 1 under volume preserving deformations are balls, and these are all maxima by the classical isoperimetric result by Polya ([25], [26]). The same rigidity holds in the spaces H n and S n + : the only critical domains are geodesic balls (also in this case, these are absolute maxima, see for example [5] and [6]). On the sphere there are many critical domains which are not geodesic balls, for example, domains bounded by any connected isoparametric hypersurface, as proved in [34] (see also [31]).…”

“…1 − j)!j! and E k (x) is the k-th exit time function defined in(6). In particular, Ω is critical for the k-th exit time moment, for all k ≥ 1 and for all volume preserving deformations, if and only if ∂E k ∂N is constant on the boundary, for all k.…”

Talenti's comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature

“…The collection {T k (Ω)} ∞ k=1 and {J k (Ω)} ∞ k=1 are called the L 1 -moment spectrum and L ∞ -moment spectrum of Ω, respectively. For interpretation for L 1 and L ∞ -moment spectrum in probability theory, we refer the reader to [11,25,26]. For related results for torsional rigidity, L 1 and L ∞ -moment spectrum, one can consult [2,9,11,12,14,16,18,22,25,26] and references therein.…”

“…For interpretation for L 1 and L ∞ -moment spectrum in probability theory, we refer the reader to [11,25,26]. For related results for torsional rigidity, L 1 and L ∞ -moment spectrum, one can consult [2,9,11,12,14,16,18,22,25,26] and references therein.…”