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.

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 $H\left(x,\xi \right)$ is convex in $\xi$ and grows at most like ${|\xi |}^{q}+f\left(x\right)$, with $1 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. ${\left(1+|u|\right)}^{\overline{q}-1}\phantom{\rule{0.166667em}{0ex}}u\in {H}_{0}^{1}\left(\Omega \right)$, 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.

Classification: 35J60,  35R05,  35Dxx
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title = {Uniqueness for unbounded solutions to stationary viscous {Hamilton-Jacobi} equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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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://archive.numdam.org/item/ASNSP_2006_5_5_1_107_0/

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