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Uniqueness for unbounded solutions to stationary viscous Hamilton--Jacobi equations

Abstract : We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la\,u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{ in }\Omega ,\\ u=0\mbox{ on }\partial\Omega\end{array} \right. $$ where $\la\geq 0$, $A(x)$ 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 $1 < q < 2$ and $f \in \elle {\frac N{q'}}$. 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. $(1+|u|)^{\bar q-1}\,u\in \acca$, for a certain (optimal) exponent $\bar q$. This completes the recent results in \cite{GMP}, where the existence of at least one solution in this class has been proved.
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Submitted on : Thursday, January 26, 2006 - 9:23:11 AM
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Guy Barles, Alessio Porretta. Uniqueness for unbounded solutions to stationary viscous Hamilton--Jacobi equations. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2006, 5 (1), pp.107--136. ⟨hal-00017876⟩

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