Optimal Regularity in Transmission Problems]{Optimal regularity for variational solutions of free transmission problems

In this article we study functionals of the type considered in \cite{HS21}, i.e. $$J(v):=\int_{B_1} A(x,u)|\nabla u|^2 +f(x,u)u+ Q(x)\lambda (u)\,dx$$ here $A(x,u)= A_+(x)\chi_{\{u>0\}}+A_-(x) \chi _{\{u<0\}}$, $f(x,u)= f_+(x)\chi_{\{u>0\}}+f_-(x) \chi _{\{u<0\}}$ and $\lambda(x,u) = \lambda_+(x) \chi_{\{u>0\}} + \lambda_-(x) \chi_{\{u\le 0\}}$. We prove the optimal $C^{0,1^-}$ regularity of minimizers of the functional indicated above (with precise H\"older estimates) when the coefficients $A_{\pm}$ are continuous functions and $\mu \le A_{\pm}\le \frac{1}{\mu}$ for some $0<\mu<1$, with $f \in L^N(B_1)$ and $Q$ bounded. We do this by presenting a new compactness argument and approximation theory similar to the one developed by L. Caffarelli in \cite{Ca89} to treat the regularity theory for solutions to fully nonlinear PDEs. Moreover, we introduce the $\mathcal{T}_{a,b}$ 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.