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 nonlinear 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 nonlinear 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 in the toy model

$${{\cal G}_{p,q}}\left[u \right]: = \left[{{{\left| {\nabla u} \right|}^p} + \mathfrak{a}\left(x \right){{\left| {\nabla u} \right|}^q}} \right]\Delta u,\,\,\,\,\,{\rm{for}}\,\,0 < p < q {{\cal G}_{p,q}}\left[u \right]: = \left[{{{\left| {\nabla u} \right|}^p} + \mathfrak{a}\left(x \right){{\left| {\nabla u} \right|}^q}} \right]\Delta u,\,\,\,\,\,{\rm{for}}\,\,0 < p < q < \infty \,\,{\rm{and}}\,\,0 \le \mathfrak{a} \in {C^0}\left(\Omega \right) \infty \,\,{\rm{and}}\,\,0 \le \mathfrak{a} \in {C^0}\left(\Omega \right)$$

We 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.