Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

By Guy Barles, Alessio Porretta. 2006

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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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12c3f57fed0b7cf64939508024744d12b3d6e5c4

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Partial Differential Equations Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms (2006, background): “In contrast, uniqueness results for the solutions of (1) which satisfy the regularity requirement (7) have been recently proved in [ 2 ].”
Solvability of Uniformly Elliptic Fully Nonlinear PDE (2010, unknown): “Uniqueness in the natural Sobolev spaces was proved in [BM], [BBGK], [ BP ].”
UNIQUENESS OF SOLUTIONS FOR SOME ELLIPTIC EQUATIONS WITH A QUADRATIC GRADIENT TERM (2010, background): “Other uniqueness results for equations with a gradient term can be found in [31], where the assumption β(s) ≤ 0 is considered, [26] for uniqueness of the zero solution for a sign-changing nonsingular function β and f(x, s)s ≤ 0, [8] for the case of subquadratic terms in ∇u, [11–13] for equations wit”