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Viscosity solutions of Hamilton-Jacobi equations and iterated minmax

  • Wei, Qiaoling
Publication Date
May 30, 2013
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In this thesis, we study the solutions of Hamilton-Jacobi equations. We will compare the viscosity solution and the minmax solution, with the latter defined by a geometric method. In the literature, there are well-known cases where these two solutions coincide: if the Hamiltonian is convex or concave with respect to the momentum variable, the 8 minmax can be reduced to min or max. The minmax and viscosity solutions are different in general. We will construct "iterated minmax" by iterating the minmax step by step and prove that, as the size of steps go to zero, the iterated minmax converge to the viscosity solution. In particular, we study the equations of conservation laws in dimension one, where, by the "front tracking" method, we shall see that in the case where the initial function is convex, the viscosity solution and the minmax are equal. And as an application, we use the limiting iterated process to describe the singularities of the viscosity solution. In the end, we show that the notion of minmax is not so obvious.

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