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

Francfort, Gilles, Murat, François, and Tartar, Luc. "Monotone operators in divergence form with $x$-dependent multivalued graphs." Bollettino dell'Unione Matematica Italiana 7-B.1 (2004): 23-59. <http://eudml.org/doc/195327>.

@article{Francfort2004,
abstract = {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.},
author = {Francfort, Gilles, Murat, François, Tartar, Luc},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {23-59},
publisher = {Unione Matematica Italiana},
title = {Monotone operators in divergence form with $x$-dependent multivalued graphs},
url = {http://eudml.org/doc/195327},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Francfort, Gilles
AU - Murat, François
AU - Tartar, Luc
TI - Monotone operators in divergence form with $x$-dependent multivalued graphs
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/2//
PB - Unione Matematica Italiana
VL - 7-B
IS - 1
SP - 23
EP - 59
AB - 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.
LA - eng
UR - http://eudml.org/doc/195327
ER -