We prove local Lipschitz regularity and non-degeneracy estimates for viscosity solutions to one-phase free boundary problems governed by non-homogeneous fully non-linear equations with unbounded ...ingredients and quadratic growth in the gradient. We use the approach in De Silva (Interfaces Free Bound 13:223–238,
2011
) to show that flat or Lipschitz free boundaries are graphs
C
1
,
α
.
In this paper, we prove up to the boundary gradient estimates for viscosity solutions to inhomogeneous nonlinear Free Boundary Problems (FBP) governed by fully nonlinear and quasilinear elliptic ...equations with unbounded measurable ingredients. Here, we build upon our previous results in
9
to construct Inhomogeneous Pucci Barriers (IPB) for the Pucci extremal equations with unbounded coefficients. Using these barriers, we obtain a version of a boundary growth type lemma for inhomogeneous nonlinear equations that may be of independent interest. In a certain way, this lemma detects the expansion of the level sets of supersolutions from the boundary to the interior. The use of this boundary growth type lemma together with the geometry of IPB bridge the interchanging information between the free boundary condition and Dirichlet boundary data on the free and fixed boundary respectively. This produces an estimate on the trace of solutions to FBP along the fixed boundary. This way, control of such solutions (up to the boundary) by the distance to the negative phase is obtained. Finally, this distance control combined with the PDE boundary gradient estimates render our final result.
In this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients. More precisely, we deal with ...coefficients and right hand sides (RHS) in
L
q
with
q
≥
n
. This extends the result of Caffarelli on the
C
loc
1
,
1
regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded RHS. Moreover, we also provide a regularity result with estimates for
ω
-semiconvex functions that are supersolutions to the same type of equations with unbounded RHS (i.e, RHS in
L
q
,
q
≥
n
). By a completely different method, our results here extend the recent regularity results obtained by Braga et al. (Adv Math 334:184–242,
2018
) for
q
>
n
, as far as fully nonlinear PDEs are concerned. These results include, in particular, the apriori estimate obtained by Caffarelli et al. (Commun Pure Appl Math 38(2):209–252,
1985
) on the modulus of continuity of the gradient of
ω
-semiconvex supersolutions (for linear equations and bounded RHS) that have a Hölder modulus of semiconvexity.
In this paper we present a short proof of the following classification Theorem for
g
−harmonic functions in half-spaces. Assume that
u
is a nonnegative solution to Δ
g
u
= 0 in {
x
n
> 0} that ...continuously vanishes on the flat boundary {
x
n
= 0}. Then, modulo normalization,
u
(
x
) =
x
n
in {
x
n
≥ 0}. Our proof depends on a recent quantitative version of the Hopf-Oleı̆nik Lemma proven by the authors in Braga and Moreira (
Adv. Math.
334
, 184–242,
2018
). Moreover, in this paper, we show how to adapt the proofs in the literature to extend Carleson Estimate, Boundary Harnack Inequality and Schwartz Reflection Principle to the context of nonnegative
g
−harmonic functions. These results are also ingredients for the proof of the main result.
In this paper, we study classes of minimizers of inhomogeneous two-phase Alt–Caffarelli functionals of the type
J
G
(
u
,
Ω
)
=
∫
Ω
G
(
|
∇
u
|
)
+
f
1
(
x
)
H
1
(
u
+
)
+
f
2
(
x
)
H
2
(
u
-
)
+
Q
...(
h
1
,
h
2
)
(
u
)
(
x
)
d
x
,
on a bounded domain
Ω
⊂
R
n
, where
G
,
H
1
and
H
2
are power-like
N
-functions,
f
1
,
f
2
∈
L
q
(
Ω
)
for suitable
n
≤
q
≤
∞
, and
h
1
,
h
2
∈
L
∞
(
Ω
)
. Hölder and non-degeneracy estimates for minima are obtained and in the particular case where such minimizers are weak solutions of non-singular PDEs we provide log-Lipschitz type estimates. In the sequel, since the Alt–Caffarelli–Friedman monotonicity formula is missing in our context and there is a mistake in a proof of Lipschitz continuity for minimizers in Zheng et al. (Monatsh Math 172:441–475,
2013
), we extend the results of Braga et al. (Ann Inst H Poincaré Anal Non Linéaire 31(4):823–850,
2014
) establishing the Lipschitz regularity for more general class of minima under the additional condition of small Lebesgue density on one of the phases along the free boundary. We finish this paper with a result that establishes density estimates from below for the positive and negative phase on points inside the contact set between the free boundaries in the case where minimizers are not Lipschitz. Such estimates allow us to provide a preliminary full description of the free boundary for any minima even if the Lipschitz regularity (as optimal regularity) is unknown.
Our main result in this note can be stated as follows: Assume
E
⊂
B
1
and
0.1
F
(
D
2
u
(
x
)
,
∇
u
(
x
)
,
u
(
x
)
,
x
)
≤
ψ
(
x
)
in
B
1
\
E
holds in the
C
-
viscosity sense where
|
E
|
=
0
and
F
...is a degenerate elliptic operator. This way, (
0.1
) holds in the whole unit ball
B
1
(i.e,
E
is removable for (
0.1
)) provided
0.2
M
λ
,
Λ
-
(
D
2
u
)
-
γ
|
∇
u
|
≤
f
in
B
1
where
f
∈
L
n
(
B
1
)
. Zeroth order term can appear in (
0.2
) provided
u
is bounded in
B
1
. This extends a result due to Caffarelli et al. proven in (Commun Pure Appl Math 66(1):109–143,
2013
) where a second order linear uniformly elliptic PDE with bounded RHS appeared in place of (
0.2
).
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
.
In this paper, we provide an inhomogeneous version of the Carleson estimate for quasilinear elliptic equations with g-Laplace type growth and unbounded right-hand side. We use this result to extend ...exponential growth theorems in cylindrical unbounded domains proven by Berestycki et al. (Duke Math J 81:467–494, 1996). We finish this paper showing a boundary Harnack-type inequality.
In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under ...the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g-Laplace type growth. Finally, we prove (sharp) regularity for ω-semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in 13 in 1985 as well as a more recent result on the C1,1 regularity for convex supersolutions obtained by C. Imbert in 33 in 2006.
We prove uniform up to the boundary gradient estimates for one phase nonlinear inhomogeneous singular perturbation problems with unbounded measurable ingredients governed by fully nonlinear elliptic ...equations. We present similar results for quasilinear PDEs with bounded RHS. Our proof is based on the Lipschitz regularity up to the boundary for free boundary problems (FBP) found in Braga and Moreira (2022).
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