In the 1950s, Lax and Malgrange proved that a solution v of a linear elliptic equation with analytic coefficients Pv=0 in a compact set can be approximated by a global solution u of Pu=0 provided that the complement of the set is connected. This theorem is key to showing the existence of minimal graphs on the unit ball whose transverse intersection with a horizontal hyperplane has arbitrarily large (n-1)-measure. The proof hinges on the construction of minimal graphs which are almost flat but have small oscillations of prescribed geometry. In addition, I will present an overview of our recent generalization of the Lax-Malgrange result to the case of parabolic equations and discuss applications to the study of hot spots. This is a joint work with A. Enciso and D. Peralta-Salas.