Uniform Lipschitz regularity for two-phase singularly perturbed fully nonlinear elliptic equations

Abstract

We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$F(D^2{u}_{\epsilon })=\frac{\alpha }{\epsilon }\beta \left(\frac{u_{\epsilon }}{\epsilon }\right)\text{in }{B}_1\subset {\mathbb{R}}^n,$$ where $F$ is uniformly elliptic and $\beta \in C_c(-1,1)$ is nonnegative. We prove the scale-sharp estimate $$\Vert \nabla u_{\epsilon }{\Vert }_{L^{\infty }(B_{1/2})}\le C\left(\Vert u_{\epsilon }{\Vert }_{L^{\infty }(B_1)}+\sqrt{\alpha }\right),$$ with $C$ depending only on the dimension, the ellipticity constants, and $\beta$, and independent of $\epsilon$ and $\alpha$. This removes a longstanding compactness obstruction in the analysis of fully nonlinear two-phase singular perturbations. The difficulty is structural: at positive $\epsilon$ there is neither a free boundary nor a prescribed transmission law, while the general fully nonlinear setting provides no monotonicity formula capable of controlling the interaction of the two phases. The proof develops a diffuse counterpart of the De Silva--Savin decay-versus-Lipschitz alternative. Exact planar transitions furnish the local comparison geometry, and curved-test compactness carries this geometry across collapsing reaction layers. An intrinsic transition-region estimate reduces the problem to linear growth from buffered level boundaries. The resulting dyadic continuation is closed by a large-slope stopping argument: bounded accumulated slopes yield the desired growth directly, whereas unbounded slopes force the effective reaction strength to vanish after normalization and lead to a contradiction. The estimate is quantitatively optimal and supplies the scale-invariant compactness framework required for the subsequent sharp-interface analysis.

32 pages