Monotone operators in divergence form with -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

Abstract

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We prove the existence of solutions to , together with appropriate boundary conditions, whenever is a maximal monotone graph in , for every fixed . 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 rotation, for every fixed ; in other words, the graph is defined through , where is a Carathéodory contraction in . This definition is shown to be equivalent to the fact that is pointwise monotone and that, for any and any , the equation has a solution with . Under additional coercivity and growth assumptions, the existence of solutions to is then established.

How to cite

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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 -

References

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  6. CHIADÒ PIAT, V.- DAL MASO, G.- DEFRANCESCHI, A., G-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (3) (1990), 123-160. Zbl0731.35033
  7. FEDERER, H., Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York1969. xiv+676 pp. Zbl0176.00801MR257325
  8. LERAY, J., LIONS, J.-L., Quelques resultats de Višik sur les problemes elliptiques nonlineaires par les methodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. Zbl0132.10502
  9. LIONS, J-.L., Quelques methodes de resolution des problemes aux limites non lineaires, Dunod; Gauthier-Villars, Paris1969. xx+554 pp. Zbl0189.40603
  10. MINTY, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. Zbl0111.31202MR169064

Citations in EuDML Documents

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  1. François Murat, Existence of a solution to with a maximal monotone graph in for every given
  2. Dalibor Pražák, Kumbakonam R. Rajagopal, Mechanical oscillators with dampers defined by implicit constitutive relations

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