NUMERICAL IMPLEMENTATIONS OF THEORETICAL RESULTS IN PARTIAL DIFFERENTIAL EQUATIONS

In this work, we present two theoretical results in nonlinear partial differential equations and we exploit both of them to produce novel visualizations of their solutions. First, we show a proof of the existence and uniqueness of viscosity solutions to the p-Laplace equation in the setting of the double obstacle problem. These solutions are built by adopting the framework provided by so-called random tug-of-war games. Using the theoretical result, in this context we employ a finite elements method to obtain visualizations of various approximate solutions. Second, we develop a proof of the density of C 1,α solutions to the Monge-Ampère equation in the set of continuous functions. This proof was obtained in the framework provided by the technique of convex integration. The proof is written with all due details, which allowus to give explicit bounds for every involved quantity. By means of numerical computations, we construct approximations of anomalous solutions to the Monge-Ampère equation, suitably guided by the bounds obtained by our theoretical results. Finally, we use these approximations to give a graphical description of anomalous solutions. The visualizations and the numerical results provide insight into the approximating constructions