Monotone operators in divergence form with $x$-dependent multivalued graphs

Francfort, Gilles ; Murat, François ; Tartar, Luc

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 1, pp. 23-59

Résumé

We prove the existence of solutions to $-\text{div}\, a(x, \text{grad}\, u)=f$, together with appropriate boundary conditions, whenever $a(x, e)$ is a maximal monotone graph in $e$, for every fixed $x$. We propose an adequate setting for this problem, in particular as far as measurability is concerned. It consists in looking at the graph after a $45^{\circ}$ rotation, for every fixed $x$; in other words, the graph $d\in a(x, e)$ is defined through $d-e=\varphi (x, d+e)$, where $\varphi$ is a Carathéodory contraction in $\mathbb{R}^{N}$. This definition is shown to be equivalent to the fact that $a(x, \cdot)$ is pointwise monotone and that, for any $g\in [L^{p'} (\Omega)]^{N}$ and any $\delta > 0$, the equation $d + \delta |e|^{p-2}e= g$ has a solution $(e, d)$ with $d\in a(x, e)$. Under additional coercivity and growth assumptions, the existence of solutions to $- \text{div}\, a(x, \text{grad}\, u)= f$ is then established.