Regularizing effects of homogeneous evolution equations: the case of homogeneity order zero

Daniel Hauer and José Mazón

Abstract

In this paper, we develop a functional analytical theory for establishing that mild solutions of first-order Cauchy problems involving homogeneous operators of order zero are strong solutions; in particular, the first-order time derivative satisfies a global regularity estimate depending only on the initial value and the positive time. We apply those results to the Cauchy problem associated with the total variational flow operator and the nonlocal fractional $1$-Laplace operator.

Keywords: Nonlinear semigroups, local and nonlocal operators, $1$-Laplace operator, regularity, homogenous operators.

AMS Subject Classification: Primary 47H20; secondary 47H06, 47J35.