Global approximation theorems for PDEs and applications

A global approximation theorem for a differential operator P is a result ensuring that a solution v of P[v] =0 in a closed set S satisfying some topological assumptions can be approximated by a global solution u of the equation P[u]=0. The theory for elliptic equations has been widely developed. In this talk I will show global approximation theorems for parabolic equations. In addition, I will apply these results to prove the existence of solutions of the heat equation with local hot spots with prescribed behavior as well as minimal graphs with micro-oscillations. This is a joint work with A. Enciso and D. Peralta-Salas.