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. $$J(v):={\int }_{B_1}A(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+f(x,u)u+Q(x)\lambda (u)dx$$ 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$\lambda(x,u) = \lambda+(x) \chi_{{u>0}} + \lambda_-(x) \chi_{{u\le 0}}$\mathrm{.}Weprovetheoptimal$C^{0,1^-}$ 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.