Optimal regularity for variational solutions of free transmission problems

In this article we study functionals of the type considered in [36], i.e. here , and . We prove the optimal regularity of minimizers of the functional indicated above (with precise estimates) when the coefficients are continuous functions and for some , with and Q bounded. We do this by presenting a new compactness argument and approximation theory similar to the one developed by L. Caffarelli in [9] to treat the regularity theory for solutions to fully nonlinear PDEs. Moreover, we introduce the operator that allows one to transfer minimizers from the transmission problems to the Alt-Caffarelli-Friedman type functionals, in small scales, allowing this way the study of the regularity theory of minimizers of Bernoulli type free transmission problems.