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On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent

Published online by Cambridge University Press:  15 August 2002

Françoise Demengel
Françoise Demengel*
Affiliation:
Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvain, 95302 Cergy-Pontoise Cedex, France; Francoise.DEMENGEL@math.u-cergy.fr.
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Abstract

Let Ω be a smooth bounded domain in ${\bf R}^n$, n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star = {n\over n-1}$. We are concerned here with the existence of solution in $BV(\Omega)$, positive or not, to the problem:


$$ \left\{ \begin{array}{rl} -{\rm div}\ \sigma+a(x) sign\ u &= f|u|^{1^\star-2} u\cr \sigma.\nabla u &= |\nabla u|\ {\rm in}\ \Omega\cr u\ {\rm is\ not \ identically\ zero}, &-\sigma.n (u) = |u|\ {\rm on }\ \partial \Omega.\end{array}\right.$$

This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$.

Keywords

BV functionsbest constant for Sobolev embeddings.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 667 - 686
Copyright
© EDP Sciences, SMAI, 1999

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References

Aubin, T., Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geom. 11 (1976) 573-598. CrossRef
T. Aubin, Nonlinear Analysis on manifolds-Monge-Ampère Equations. Grundlehern der Mathematischen Wissenschaften (1982) 252.
Bahri, A. and Coron, J.M., On a non linear elliptic equation involving the critical Sobolev exponent: The effet of the topology of the domain. Comm. Pure Appl. Math. 41 (1988) 253-294. CrossRef
Bobkov and Ch. Houdré, Some connections between isoperimetric and Sobolev type Inequalities. Mem. Amer. Math. Soc. 616 (1997).
Demengel, F., Some compactness result for some spaces of functions with bounded derivatives. Arch. Rational Mech. Anal. 105 (1989) 123-161.
Demengel, F. and Hebey, E., On some nonlinear equations involving the p-Laplacian with critical Sobolev growth. I. Adv. Partial Differential Equations 3 (1998) 533-574.
I. Ekeland and R. Temam, Convex Analysis and variational problems. North-Holland (1976).
E. Giusti, Minimal surfaces and functions of bounded variation, notes de cours rédigés par G.H. Williams. Department of Mathematics Australian National University, Canberra (1977), et Birkhaüser (1984).
Hebey, E., La méthode d'isométrie concentration dans le cas d'un problème non linéaire sur les variétés compactes à bord avec exposant critique de Sobolev. Bull. Sci. Math. 116 (1992) 35-51.
Hebey, E. and Vaugon, M., Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev Growth. J. Funct. Anal. 119 (1994) 298-318. CrossRef
Lions, P.L., La méthode de compacité concentration, I et II. Revista Ibero Americana 1 (1985) 145.
Kohn, R.V. and Temam, R., Dual spaces of stress and strains with applications to Hencky plasticity. Appl. Math. Optim. 10 (1983) 1-35. CrossRef
B. Nazaret, Stability results for some nonlinear elliptic equations involving the p-Laplacian with critical Sobolev growth, COCV, accepted Version française : Prepublication de l'Université de Cergy-Pontoise N 5/98, Avril 1998.
Talenti, Best, constants in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110 (1976) 353-372.
G. Strang and R. Temam, Functions with bounded variations. Arch. Rational Mech. Anal. (1980) 493-527.
Suquet, P., Sur les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. (6) 1 (1979) 77-87. CrossRef
Ziemmer, Weakly Differentiable functions. Springer Verlag, Lectures Notes in Math. 120 (1989).