Existence and optimal regularity theory for weak solutions of free transmission problems of quasilinear type via Leray-Lions method

We study existence and regularity of weak solutions for the following PDE -\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;in $B_1$. where $A(x,s)=A_{+}(x){\chi }_{\{s>0\}}+A_{-}(x){\chi }_{\{s\le 0\}}$ and $f(x,s)=f_{+}(x){\chi }_{\{s>0\}}+f_{-}(x){\chi }_{\{s\le 0\}}$. Under the ellipticity assumption that $\frac{1}{\mu }\le A_{\pm }\le \mu$, $A_{\pm }\in C(\text{Ø})$ and $f_{\pm }\in L^N(\text{Ø})$, we prove that under appropriate conditions the PDE above admits a weak solution in $W^{1,p}(B_1)$ which is also $C_{loc}^{0,\alpha }$ for every $\alpha \in (0,1)$ with precise estimates. Our methods relies on similar techniques as those developed by Caffarelli to treat viscosity solutions for fully non-linear PDEs (c.f. \cite{C89}). Other key ingredients in our proofs are the ${\text{TT}}_{a,b}$ operator (which was introduced in \cite{MS22}) and Leray-Lions method (c.f. \cite{BM92}, \cite{MT03}).