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

Bousquet, Pierre

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

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\mathrm{div}a\left(\nabla u\right)+F\left[u\right]\left(x\right)=0,$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\partial \Omega .$ The vector field $a:{ℝ}^n\to {ℝ}^n$ satisfies an ellipticity condition and for a fixed $x,F\left[u\right]\left(x\right)$ 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 $\phi$ 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 $\phi :$ the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if $\phi$ is the restriction to $\partial \Omega$ 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 $\Omega .$

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

[1] P. Bousquet, The lower bounded slope condition. J. Convex Anal. 14 (2007) 119-136. | Zbl

[2] P. Bousquet and F. Clarke, Local Lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Eq. (to appear). | MR | Zbl

[3] F. Clarke, Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511-530. | Numdam | Zbl

[4] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl

[5] P. Hartman, On the bounded slope condition. Pacific J. Math. 18 (1966) 495-511. | Zbl

[6] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271-310. | Zbl

[7] G.M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations 6 (1981) 437-497. | Zbl

[8] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values. Arch. Rational Mech. Anal. 79 (1982) 305-323. | Zbl

[9] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data. Comm. Partial Differential Equations 11 (1986) 167-229. | Zbl

[10] M. Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in $n$ variabili. Ann. Scuola Norm. Sup. Pisa 19 (1965) 233-249. | Numdam | Zbl