Analysis Seminar

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Regularity of Viscosity Solutions for Hamilton-Jacobi Equations

Speaker: Daniela Tonon, ICERM and SISSA

Location: Warren Weaver Hall 1302

Date: Thursday, December 15, 2011, 11 a.m.

Synopsis:

We present two results on the regularity of viscosity solutions of Hamilton-Jacobi equations obtained in collaboration with Professor Stefano Bianchini. When the Hamiltonian is strictly convex viscosity solutions are semiconcave, hence their gradient is BV. First we prove the SBV regularity of the gradient of a viscosity solution of the Hamilton-Jacobi equation \(u_t+ H(t,x,D_x u)=0\) in an open set of \(R^{n+1}\), under the hypothesis of uniform convexity of the Hamiltonian \(H\) in the last variable. Secondly we remove the uniform convexity hypothesis on the Hamiltonian, considering a viscosity solution \(u\) of the Hamilton-Jacobi equation \(u_t+ H(D_x u)=0\) in an open set of \(R^{n+1}\) where \(H\) is smooth and convex. In this case the viscosity solution is only locally Lipschitz. However when the vector field \(d(t,x):=H_p(D_xu(t,x))\), here \(H_p\) is the gradient of \(H\), is BV for all \(t\) in \([0,T]\) and suitable hypotheses on the Lagrangian \(L\) hold, the divergence of \(d(t, )\) can have Cantor part only for a countable number of \(t\)'s in \([0,T]\). These results extend a result of Bianchini, De Lellis and Robyr for a uniformly convex Hamiltonian which depends only on the spatial gradient of the solution.