Calc. Var. (2020) 59:86
https://doi.org/10.1007/s00526-020-01733-5
Calculus of Variations
Free boundary theory for singular/degenerate nonlinear
equations with right hand side: a non-variational approach
J. Ederson M. Braga
1
· Raimundo A. Leitão
1
· J. Erivamberto L. Oliveira
1
Received: 5 November 2018 / Accepted: 28 February 2020 / Published online: 16 April 2020
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
We use non-variational type arguments to prove optimal regularity and smoothness of the
free boundary for one-phase solutions to inhomogeneous nonlinear free boundary problems
(FBP) governed by singular/degenerate elliptic PDEs with a nonzero right hand side (RHS).
In a precise way, we show that viscosity solutions to FBP, as previously mentioned, are
locally Lipschitz continuous and under certain conditions, flat or Lipschitz free boundaries,
are C
1,α
.
Mathematics Subject Classification 35B25 · 35B65 · 35D40 · 35J15 · 35J60 · 35J75 · 35R35
1 Introduction
In this paper, we are interested in the study of the local Lipschitz regularity and C
1,α
regularity
of the free boundary for viscosity solutions to the following model of FBP
⎧
⎨
⎩
g
u f (x ), in B
+
1
(u ):B
1
∩{u > 0},
|∇u | Q(x ), on F (u ):B
1
∩ ∂ {u > 0},
(1.1)
where
g
u :di v
g(|∇u |)
|∇u |
∇u
. (1.2)
Communicated by O.Savin.
B J. Ederson M. Braga
edersonbraga@mat.ufc.br
Raimundo A. Leitão
rleitao@mat.ufc.br
J. Erivamberto L. Oliveira
bertoliveira7@gmail.com
1
Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici - Bloco 914, Fortaleza,
Ceará CEP 60455-760, Brazil
123