Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

Barles, Guy; Porretta, Alessio

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 107-136.

We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $\lambda \ge 0$, $A\left(x\right)$ is a bounded and uniformly elliptic matrix and $H(x,\xi )$ is convex in $\xi$ and grows at most like ${\left|\xi \right|}^q+f\left(x\right)$, with $1<q<2$ and $f\in L^{N/q^{\text{'}}}\left(\Omega \right)$. 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. ${(1+|u\left|\right)}^{\overline{q}-1}u\in H_0^1\left(\Omega \right)$, for a certain (optimal) exponent $\overline{q}$. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.

[1] N. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal. 24 (1993), 23-35. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[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 $L^1\left(\Omega \right)$. A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var. 8 (2002), 239-272. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[7] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655. | MR | Zbl

[8] L. Boccardo, F. Murat and J.P. Puel, $L^{\infty }$ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992), 326-333. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[15] N. Grenon, F. Murat and A. Porretta, Elliptic equations with superlinear gradient dependent terms, in preparation.

[16] K. Hansson, V. Maz'Ja and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat. 37 (1999), 87-120. | MR | Zbl

[17] P. L. Lions, Résolution de problèmes elliptiques quasilinèaires, Arch. Ration. Mech. Anal. 74 (1980), 335-353. | Zbl

[18] P. L. Lions, Quelques remarques sur les problèmes elliptiques quasilinèaires du second ordre, J. Anal. Math. 45 (1985), 234-254. | MR | Zbl

[19] P. L. Lions and F. Murat, Solutions renormalisées d'équations elliptiques non linéaires, unpublished paper.

[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. | Numdam | MR | Zbl