Soluciones viscosas de problemas de Hamilton - Jacobi.

Abstract

In this work we study the viscosity solutions for completely nonlinear second-order parabolic equations. In general, it addresses the problem ∂tu + H(x, t, u, Du, D^2u) = 0, with (x, t) ∈ R^N × [0, +∞[ and initial condition u(x, 0) = u_0 where u_0 is continuous and bounded.We will use Perron’s method to obtain the existence of discontinuous viscosity solutions from a comparison principle. For the comparison principle we will take advantage of certain hypotheses about the Hamiltonian H and we will use the technique of separation of variables for its proof. Finally, we will work on finding the uniqueness of the solutions. All these results will be adapted from Cardaliaguet’s course notes in [2] and Topp and Yangari’s article in [9], detailing proofs of their work.