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

Françoise Demengel

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 667-686
  • ISSN: 1292-8119

Abstract

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Let Ω be a smooth bounded domain in 𝐑 n , n > 1, let a and f be continuous functions on Ω ¯ , 1 = n n - 1 . We are concerned here with the existence of solution in B V ( Ω ) , positive or not, to the problem:
 - div σ + a ( x ) s i g n u a m p ; = f | u | 1 - 2 u σ . u a m p ; = | u | in Ω u is not identically zero , a m p ; - σ . n ( u ) = | u | on Ω . This problem is closely related to the extremal functions for the problem of the best constant of W 1 , 1 ( Ω ) into L N N - 1 ( Ω ) .

How to cite

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Demengel, Françoise. "On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 667-686. <http://eudml.org/doc/197312>.

@article{Demengel2010,
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:
$$ \left\\{ \begin\{array\}\{rl\} -\{\rm div\}\ \sigma+a(x) sign\ u &amp;= f|u|^\{1^\star-2\} u\cr \sigma.\nabla u &amp;= |\nabla u|\ \{\rm in\}\ \Omega\cr u\ \{\rm is\ not \ identically\ zero\}, &amp;-\sigma.n (u) = |u|\ \{\rm on \}\ \partial \Omega.\end\{array\}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^\{1,1\}(\Omega)$ into $L^\{N\over N-1\}(\Omega)$. },
author = {Demengel, Françoise},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {BV functions; best constant for Sobolev embeddings.; 1-Laplacian; critical Sobolev exponent; existence},
language = {eng},
month = {3},
pages = {667-686},
publisher = {EDP Sciences},
title = {On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent},
url = {http://eudml.org/doc/197312},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Demengel, Françoise
TI - On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 667
EP - 686
AB - 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:
$$ \left\{ \begin{array}{rl} -{\rm div}\ \sigma+a(x) sign\ u &amp;= f|u|^{1^\star-2} u\cr \sigma.\nabla u &amp;= |\nabla u|\ {\rm in}\ \Omega\cr u\ {\rm is\ not \ identically\ zero}, &amp;-\sigma.n (u) = |u|\ {\rm on }\ \partial \Omega.\end{array}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$.
LA - eng
KW - BV functions; best constant for Sobolev embeddings.; 1-Laplacian; critical Sobolev exponent; existence
UR - http://eudml.org/doc/197312
ER -

References

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  1. T. Aubin, Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geom.11 (1976) 573-598.  Zbl0371.46011
  2. T. Aubin, Nonlinear Analysis on manifolds-Monge-Ampère Equations. Grundlehern der Mathematischen Wissenschaften (1982) 252.  Zbl0512.53044
  3. A. Bahri and J.M. Coron, 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.  Zbl0649.35033
  4. Bobkov and Ch. Houdré, Some connections between isoperimetric and Sobolev type Inequalities. Mem. Amer. Math. Soc. 616 (1997).  Zbl0886.49033
  5. F. Demengel, Some compactness result for some spaces of functions with bounded derivatives. Arch. Rational Mech. Anal.105 (1989) 123-161.  
  6. F. Demengel and E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth. I. Adv. Partial Differential Equations3 (1998) 533-574.  Zbl0955.35031
  7. I. Ekeland and R. Temam, Convex Analysis and variational problems. North-Holland (1976).  Zbl0322.90046
  8. 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).  
  9. E. Hebey, 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.  Zbl0756.35028
  10. E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev Growth. J. Funct. Anal.119 (1994) 298-318.  Zbl0798.35052
  11. P.L. Lions, La méthode de compacité concentration, I et II. Revista Ibero Americana1 (1985) 145.  
  12. R.V. Kohn and R. Temam, Dual spaces of stress and strains with applications to Hencky plasticity. Appl. Math. Optim.10 (1983) 1-35.  Zbl0532.73039
  13. 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.  
  14. Talenti, Best constants in Sobolev inequality. Ann. Mat. Pura Appl. (4)110 (1976) 353-372.  Zbl0353.46018
  15. G. Strang and R. Temam, Functions with bounded variations. Arch. Rational Mech. Anal. (1980) 493-527.  
  16. P. Suquet, Sur les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. (6)1 (1979) 77-87.  Zbl0405.46027
  17. Ziemmer, Weakly Differentiable functions. Springer Verlag, Lectures Notes in Math. 120 (1989).  

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