By using probabilistic approaches, some uniform gradient estimates are obtained for Dirichlet heat semigroups on a Riemannian manifold with boundary. As an application, lower bound estimates of isoperimetric constants are presented in terms of functional inequalities.
"Gradient estimates of Dirichlet heat semigroups and application to isoperimetric inequalities." Ann. Probab. 32 (1A) 424 - 440, January 2004. https://doi.org/10.1214/aop/1078415841