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hal-00017876, version 1

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

Guy Barles () 1, Alessio Porretta 2

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

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.

• 1:  Laboratoire de Mathématiques et Physique Théorique (LMPT)
• CNRS : UMR6083 – Université François Rabelais - Tours
• 2:  Dipartimento di Matematica [Roma II] (DIPMAT)
• Universita degli studi di Roma Tor Vergata
• Domain : Mathematics/Analysis of PDEs
• Keywords : viscous Hamilton-Jacobi Equations – uniqueness results

• hal-00017876, version 1
• oai:hal.archives-ouvertes.fr:hal-00017876
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• Submitted on: Thursday, 26 January 2006 09:23:11
• Updated on: Thursday, 30 August 2007 09:38:25