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.

Classification : 35J60, 35R05, 35Dxx

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