Density of the domain of doubly nonlinear operators in $L^1$

Abstract

The aim of this paper is to provide sufficient conditions implying that the effective domain $D(A\varphi )$ of an $m$-accretive operator $A\varphi$ in $L^1$ is dense in $L^1$. Here, $A\varphi$ refers to the composition $A\circ \varphi$ in $L^1$ of the part $A=(\partial \mathcal{E}{)}_{|L^{1\cap \infty }}$ in $L^{1\cap \infty }\times L^{1\cap \infty }$ of the subgradient $\partial \mathcal{E}$ in $L^2$ of a convex, proper, lower semicontinuous functional $\mathcal{E}$ on $L^2$ and a continuous, strictly increasing function $\varphi$ on the real line $\mathbb{R}$. To illustrate the role of the sufficient conditions, we apply our main result to the class of doubly nonlinear operators $A\varphi$, where $A$ is a classical Leray-Lions operator.