Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations

Bousquet, Pierre

doi:10.1051/cocv:2007035

Bousquet, Pierre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 707-716.

Résumé

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $div a (\nabla u) + F [u] (x) = 0,$ over the functions $u \in W^{1, 1} (Ω)$ that assume given boundary values $φ$ on $\partial Ω .$ The vector field $a : ℝ^{n} \to ℝ^{n}$ satisfies an ellipticity condition and for a fixed $x, F [u] (x)$ denotes a non-linear functional of $u .$ In considering the same problem, Hartman and Stampacchia [Acta Math. 115 (1966) 271-310] have obtained existence results in the space of uniformly Lipschitz continuous functions when $φ$ satisfies the classical bounded slope condition. In a variational context, Clarke [Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511-530] has introduced a new type of hypothesis on the boundary condition $φ :$ the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if $φ$ is the restriction to $\partial Ω$ of a convex function. We show that if $a$ and $F$ satisfy hypotheses similar to those of Hartman and Stampacchia, the lower bounded slope condition implies the existence of solutions in the space of locally Lipschitz continuous functions on $Ω .$

MR | 1 citation dans Numdam

DOI : 10.1051/cocv:2007035

Classification : 35J25, 35J60
Mots-clés : non-linear elliptic PDE's, Lipschitz continuous solutions, lower bounded slope condition

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     title = {Local {Lipschitz} continuity of solutions of non-linear elliptic differential-functional equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {707--716},
     publisher = {EDP-Sciences},
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     url = {https://www.numdam.org/articles/10.1051/cocv:2007035/}
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Bousquet, Pierre. Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 707-716. doi : 10.1051/cocv:2007035. https://www.numdam.org/articles/10.1051/cocv:2007035/

Bibliographie
Cité par

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