Regularizing effect of homogeneous evolution equations with perturbation

Daniel Hauer

Abstract

Since the pioneering work [C. R. Acad. Sci. Paris Sér., 1979] by Aronson & Bénilan and [Johns Hopkins Univ. Press, 1981] by Bénilan & Crandall, it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous for every positive time and if the underlying Banach space has the Radon-Nikodým property, then the mild solution is a.e. differentiable and the time-derivative satisfies global and point-wise bounds.

In this paper, we show that these results remain true if the homogeneous operator is perturbed by a Lipschitz continuous mapping. More precisely, we establish point-wise Aronson–Bénilan type estimates and global \(L^1\) Bénilan-Crandall type estimates. We apply our theory to derive global \(L^q\)-\(L^{\infty}\)-estimates on the time-derivative of the evolution problem governed by the Dirichlet-to-Neumann operator associated with the \(p\)-Laplace-Beltrami operator on a compact Riemannian manifold with Lipschitz boundary perturbed by a Lipschitz nonlinearity.

Keywords: Nonlinear semigroups, Aronson-Bénilan estimates, regularity of time-derivative, homogenous operators, \(p\)-Laplace Beltrami operator, Dirichlet-to-Neumann operator on manifolds.

AMS Subject Classification: Primary 47H20; secondary 47h06, 47H14, 47J35, 35B65.