Nonlinear semigroups generated by $j$-elliptic functionals

Abstract

We generalise the theory of energy functionals used in the study of gradient systems to the case where the domain of definition of the functional cannot be embedded into the Hilbert space $H$ on which the associated operator acts, such as when $H$ is a trace space. We show that under weak conditions on the functional $\varphi$ and the map $j$ from the effective domain of $\varphi$ to $H$, which in opposition to the classical theory does not have to be injective or even continuous, the operator on $H$ naturally associated with the pair $(\varphi ,j)$ nevertheless generates a nonlinear semigroup of contractions on $H$. We show that this operator, which we call the $j$-subgradient of $\varphi$, is the (classical) subgradient of another functional on $H$, and give an extensive characterisation of this functional in terms of $\varphi$ and $j$. In the case where $H$ is an $L^2$-space, we also characterise the positivity, $L^\infty$-contractivity and existence of order-preserving extrapolations to $L^q$ of the semigroup in terms of $\varphi$ and $j$. This theory is illustrated through numerous examples, including the $p$-Dirichlet-to-Neumann operator, general Robin-type parabolic boundary value problems for the $p$-Laplacian on very rough domains, and certain coupled parabolic-elliptic systems.