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

Gilles Francfort; François Murat; Luc Tartar

Bollettino dell'Unione Matematica Italiana (2004)

- Volume: 7-B, Issue: 1, page 23-59
- ISSN: 0392-4041

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topFrancfort, 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 -

## References

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