I will describe a class of Hamilton-Jacobi equation in metric spaces, they correspond to variational problems defined on metric-space-valued curves. Such problems arise in a variety of contexts such as continuum mechanics, large deviations, etc. One can develop a well posedness theory for such problems.

It is common that some continuum mechanics problems have a condensation of mass property. In this context, the talk will review a previously open problem on well-posedness for a Hamilton-Jacobi equation in Wasserstein space. Metric space result developed in the first half of the talk will be applied here. In particular, the condensation of mass property is solved by using a concept of geometric tangent cone and its connection with probability coupling techniques.

This is a joint work with Luigi Ambrosio.