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

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

2006-01-26

arxiv.org/abs/math/0601635v1

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 5, 1 (2006) 107--136

Mathematics

Analysis of PDEs

Scientific paper

We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. $$ 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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