We consider a class of stationary viscous Hamilton-Jacobi equations as lambda u - div(A(x)del u) = H(x, del u) in Omega u = 0 on partial derivative Omega where lambda >= 0, A(x) is a bounded and uniformly elliptic matrix and H(x, xi) is convex in xi and grows at most like \xi\(q) + f (x), with 1 < q < 2 and f is an element of L-N/q'(Omega). Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate, i.e. (1 + \u\)((q) over bar -1) u is an element of H-0(1) (Omega), for a certain (optimal) exponent (q) over bar. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.

Barles, G., Porretta, A. (2006). Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 5(1), 107-136.

Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations

PORRETTA, ALESSIO
2006-01-01

Abstract

We consider a class of stationary viscous Hamilton-Jacobi equations as lambda u - div(A(x)del u) = H(x, del u) in Omega u = 0 on partial derivative Omega where lambda >= 0, A(x) is a bounded and uniformly elliptic matrix and H(x, xi) is convex in xi and grows at most like \xi\(q) + f (x), with 1 < q < 2 and f is an element of L-N/q'(Omega). Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate, i.e. (1 + \u\)((q) over bar -1) u is an element of H-0(1) (Omega), for a certain (optimal) exponent (q) over bar. This completes the recent results in [15], where the existence of at least one solution in this class has been proved.
2006
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
NONLINEAR ELLIPTIC-EQUATIONS; LOWER ORDER TERM; RIGHT-HAND SIDE; RENORMALIZED SOLUTIONS; EXISTENCE
Barles, G., Porretta, A. (2006). Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 5(1), 107-136.
Barles, G; Porretta, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/38896
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