Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

Guy Barles; Alessio Porretta

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 1, page 107-136
  • ISSN: 0391-173X

Abstract

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We consider a class of stationary viscous Hamilton-Jacobi equations aswhere λ ≥ 0 , A ( x ) is a bounded and uniformly elliptic matrix and H ( x , ξ ) is convex in ξ and grows at most like | ξ | q + f ( x ) , with 1 < q < 2 and f ∈ L N / q ' ( Ω ) . Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,i.e. ( 1 + | u | ) q ¯ - 1 u ∈ H 0 1 ( Ω ) , for a certain (optimal) exponent q ¯ . This completes the recent results in [15], where the existence of at least one solution in this class has been proved.

How to cite

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Barles, Guy, and Porretta, Alessio. "Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.1 (2006): 107-136. <http://eudml.org/doc/239988>.

@article{Barles2006,
abstract = {We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $\lambda \ge 0$, $A(x)$ is a bounded and uniformly elliptic matrix and $H(x,\xi )$ is convex in $\xi $ and grows at most like $|\xi |^q+f(x)$, with $1&lt;q&lt;2$ and $f\in L^\{N/\{q^\{\prime \}\}\}(\Omega )$. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,i.e.$(1+|u|)^\{\bar\{q\}-1\}\,u\in H^1_0(\Omega )$, for a certain (optimal) exponent $\bar\{q\}$. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.},
author = {Barles, Guy, Porretta, Alessio},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {107-136},
publisher = {Scuola Normale Superiore, Pisa},
title = {Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations},
url = {http://eudml.org/doc/239988},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Barles, Guy
AU - Porretta, Alessio
TI - Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 1
SP - 107
EP - 136
AB - We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $\lambda \ge 0$, $A(x)$ is a bounded and uniformly elliptic matrix and $H(x,\xi )$ is convex in $\xi $ and grows at most like $|\xi |^q+f(x)$, with $1&lt;q&lt;2$ and $f\in L^{N/{q^{\prime }}}(\Omega )$. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,i.e.$(1+|u|)^{\bar{q}-1}\,u\in H^1_0(\Omega )$, for a certain (optimal) exponent $\bar{q}$. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.
LA - eng
UR - http://eudml.org/doc/239988
ER -

References

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  1. [1] N. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal. 24 (1993), 23–35. Zbl0809.35021MR1199524
  2. [2] G. Barles and F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Ration. Mech. Anal. 133 (1995), 77–101. Zbl0859.35031MR1367357
  3. [3] G. Barles, A-P. Blanc, C. Georgelin and M. Kobylanski, Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 381–404. Zbl0940.35078MR1736522
  4. [4] F. Betta, A. Mercaldo, F. Murat, M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in L 1 ( Ω ) . A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var. 8 (2002), 239–272. Zbl1092.35032MR1932952
  5. [5] F. Betta, A. Mercaldo, F. Murat and M. Porzio, Uniqueness results for nonlinear elliptic equations with a lower order term, Nonlinear Anal. 63 (2005), 153–170. Zbl1125.35343MR2165494
  6. [6] L. Boccardo, I. Diaz, D. Giachetti and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, J. Differential Equations 106 (1993), 215–237. Zbl0803.35046MR1251852
  7. [7] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655. Zbl0812.35043MR1163440
  8. [8] L. Boccardo, F. Murat and J.P. Puel, L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992), 326–333. Zbl0785.35033MR1147866
  9. [9] L. Boccardo, F. Murat and J.P. Puel, Existence de solutions faibles pour des équations elliptiques quasi-linèaires à croissance quadratique, In: “Nonlinear Partial Differential Equations and their Applications”. College de France Seminar, Vol. IV (Paris, 1981/1982), Res. Notes in Math. 84, Pitman, Boston, Mass. – London, 1983, 19–73. Zbl0588.35041MR716511
  10. [10] A. Dall’Aglio, D. Giachetti and J.P. Puel, Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl. (4) 181 (2002), 407–426. Zbl1097.35050MR1939689
  11. [11] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 741–808. Zbl0958.35045MR1760541
  12. [12] A. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), 321–366. Zbl0698.45010MR1014927
  13. [13] E. Ferone and F. Murat, Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, In: “Equations aux Dérivées Partielles et Applications”, Gauthier-Villars, Ed. Sci. Méd. Elsevier, Paris, 1998, 497–515. Zbl0917.35039MR1648236
  14. [14] N. Grenon, F. Murat and A. Porretta, Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 23–28. Zbl1149.35352MR2193390
  15. [15] N. Grenon, F. Murat and A. Porretta, Elliptic equations with superlinear gradient dependent terms, in preparation. Zbl1305.35058
  16. [16] K. Hansson, V. Maz’ja and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat. 37 (1999), 87–120. Zbl1087.35513MR1673427
  17. [17] P. L. Lions, Résolution de problèmes elliptiques quasilinèaires, Arch. Ration. Mech. Anal. 74 (1980), 335–353. Zbl0449.35036
  18. [18] P. L. Lions, Quelques remarques sur les problèmes elliptiques quasilinèaires du second ordre, J. Anal. Math. 45 (1985), 234–254. Zbl0614.35034MR833413
  19. [19] P. L. Lions and F. Murat, Solutions renormalisées d’équations elliptiques non linéaires, unpublished paper. 
  20. [20] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258. Zbl0151.15401MR192177

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