No abstract

We study a free transmission problem driven by degenerate fully nonlinear operators. By framing the equation in the context of viscosity inequalities, we produce optimal regularity results for viscosity solutions and certain strong solutions to the problem. Our findings include regularity in C 1,α spaces, and an explicit characterization of α in terms of the degeneracy rates. As a consequence, we examine geometric properties of the associated free boundary. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation.

No abstract

In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the p(x)-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class C 1,κ for a universal constant 0 < κ < 1. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.

We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such non-linear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli's classification scheme: Flat and Lipschitz free boundaries are locally C 1,β for some 0 < β (universal) < 1. Particularly, our findings are new even for the toy modelWe also bring to light other interesting doubly degenerate settings where our results still work. Finally, we present some key tools in the theory of degenerate fully nonlinear PDEs, which may have their own mathematical importance and applicability.