In this paper we consider a viscosity solution u of the Hamilton-Jacobi equation partial derivative(t)u + H(D(x)u) = 0 in Omega subset of [0,T] x R-n. where H is smooth and convex. We prove that when d(t,center dot) := H-p(D(x)u(t,center dot)), H-p := del H is BV for all t epsilon [0, T] and suitable hypotheses on the Lagrangian L hold, the Radon measure divd(t,center dot) can have Cantor part only for a countable number of t's in [0,T]. This result extends a result of Robyr for genuinely nonlinear scalar balance laws and a result of Bianchini, De Lellis and Robyr for uniformly convex Hamiltonians.

SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian / Bianchini, S.; Tonon, D.. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 391:1(2012), pp. 190-208. [10.1016/j.jmaa.2012.02.017]

SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian

Bianchini, S.;Tonon, D.
2012-01-01

Abstract

In this paper we consider a viscosity solution u of the Hamilton-Jacobi equation partial derivative(t)u + H(D(x)u) = 0 in Omega subset of [0,T] x R-n. where H is smooth and convex. We prove that when d(t,center dot) := H-p(D(x)u(t,center dot)), H-p := del H is BV for all t epsilon [0, T] and suitable hypotheses on the Lagrangian L hold, the Radon measure divd(t,center dot) can have Cantor part only for a countable number of t's in [0,T]. This result extends a result of Robyr for genuinely nonlinear scalar balance laws and a result of Bianchini, De Lellis and Robyr for uniformly convex Hamiltonians.
2012
391
1
190
208
https://doi.org/10.1016/j.jmaa.2012.02.017
Bianchini, S.; Tonon, D.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/13909
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