Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

Abstract

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.

Fully nonlinear free transmission problems with nonhomogeneous degeneracies

“…This new model received lots of attention recently in the setting of free boundary problems, nonhomogeneous 1-laplacian equations or obstacle problems, cf. [20][21][22], while in [32] the authors carefully combine the approaches of [12,26] to derive local C 1;˛0 -regularity for the viscosity solution of the fully nonlinear equation with variable exponents and nonhomogeneous degeneracy OEjDuj p.x/ C a.x/jDuj q.x/ F .D 2 u/ D f .x/ in : (1.5)…”