*Accepted Paper*

**Inserted:** 16 mar 2022

**Last Updated:** 2 jun 2022

**Journal:** JMPA

**Year:** 2021

**Abstract:**

n this article we study functionals of the type considered in \cite{HS21}, i.e.
J(v):=∫B1A(x,u)

∇u

2+f(x,u)u+Q(x)λ(u)dx
here A(x,u)=A+(x)χ{u>0}+A−(x)χ{u<0}, f(x,u)=f+(x)χ{u>0}+f−(x)χ{u<0} and λ(x,u)=λ+(x)χ{u>0}+λ−(x)χ{u≤0}. We prove the optimal C0,1− regularity of minimizers of the functional indicated above (with precise Hölder estimates) when the coefficients A± are continuous functions and μ≤A±≤1μ for some 0<μ<1, with f∈LN(B1) 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 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.