What is it about?
The paper delas with a class of Hamilton-Jacobi equations with an Hamiltonian discontinuous with respect to the space variable. The aim is to prove the existence of solutions via an approximating procedure, in order to generalize the notion of viscosity solution.
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Why is it important?
My interest in this topic finds its motivation in the study of non (sequentially weakly lower) semicontinuous variational problems, which ar classically reduced to the solution of suitable equations of Hamilton-Jacobi type. This approach leads to investigate first order fully non linear partial differential equations of the considered kind and I have devoted my effort to this study, proving in subsequent works an interesting criterion of strong convergence in Sobolev spaces.
Perspectives
Classical viscosity solutions enjoy the property of stability, which in several cases takes the form of strong relative compactness in Sobolev spaces. In my papers on this matter I have deeply investigated this tool in order to apply the results to the minimization of non (s.w.l.) semicontinuos functionals. Roughly speaking my aim is to construct minimizing sequences of non semicontinuous functionals as solutions of suitable first order partial differential equations having the compactness property specified above. This fact would improve the comprehension of non semicontinuous variational problems.
Prof. Sandro Zagatti
Scuola Internazionale Superiore di Studi Avanzati
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This page is a summary of: Generalized solution for a class of Hamilton–Jacobi equations, Advances in Pure and Applied Mathematics, January 2016, De Gruyter,
DOI: 10.1515/apam-2015-0014.
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