Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, p. 107-136

We consider a class of stationary viscous Hamilton-Jacobi equations aswhere λ0, 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 fL 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 uH 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.

Classification:  35J60,  35R05,  35Dxx
@article{ASNSP_2006_5_5_1_107_0,
     author = {Barles, Guy and Porretta, Alessio},
     title = {Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     pages = {107-136},
     zbl = {1150.35030},
     mrnumber = {2240185},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_1_107_0}
}
Barles, Guy; Porretta, Alessio. Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, pp. 107-136. http://www.numdam.org/item/ASNSP_2006_5_5_1_107_0/

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