Existencia, unicidad y regularidad Hölder de soluciones de la ecuación de Hamilton-Jacobi con derivada temporal de tipo Caputo.

Abstract

We are interested in the following Cauchy's problem:

D^α u(t,x)+H(t,x,u(t,x),Du(t,x))=0, (t,x)∈Q≔(0,+∞)×T^N

subject to the initial condition u(0,x)=u_0 (x), ∀ x∈T^N,

where u_0∈"Lip" (T^N ) and H∈"C" (T^N×R×R^N ) are given functions. Moreover T^N denotes the N-dimensional Torus and "Lip" (T^N ) denotes the set of the Lipschitz continuos functions on T^N, u is the unknown function, Du is the spatial gradient of u, i.e., Du=(D_(x_1 ) u,…,D_(x_N ) u) and D^α is the Caputo's fractional derivative of order α∈(0,1).

The concept of a viscosity solution and discontinuous viscosity solution for this type of problem are defined. We will show the existence of discontinuous viscosity solutions using Perron's Method. We will show the Comparison Principle, which together with Perron's Method allows us to prove the existence and uniqueness of a viscosity solution to the problem. Finally, we analyze the regularity of the solution both in time and space.