Mollification of $\mathcal{D}$-solutions to Fully Nonlinear PDE Systems

Abstract

In a recent paper (arXiv:1501.06164) the author has introduced a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows the interpretation of merely measurable maps as solutions. This approach is duality-free and builds on the probabilistic representation of limits of difference quotients via Young measures over certain compactifications of the "state space". Herein we establish a systematic regularisation scheme of this notion of solution which, by analogy, is the counterpart of the usual mollification by convolution of weak solutions and of the mollification by sup/inf convolutions of viscosity solutions.