Abstract
AbstractWe prove uniform parabolic Hölder estimates of De Giorgi–Nash–Moser type for sequences of minimizers of the functionals $$\begin{aligned} {\mathcal {E}}_\varepsilon (W) = \int _0^\infty \frac{e^{- t/\varepsilon }}{\varepsilon } \bigg \{ \int _{\mathbb {R}_+^{N+1}} y^a \left( \varepsilon |\partial _t W|^2 + |\nabla W|^2 \right) \textrm{d}X + \int _{\mathbb {R}^N \times \{0\}} \Phi (w) \,\textrm{d}x\bigg \}\,\textrm{d}t, \qquad \varepsilon \in (0,1) \end{aligned}$$
E
ε
(
W
)
=
∫
0
∞
e
-
t
/
ε
ε
{
∫
R
+
N
+
1
y
a
ε
|
∂
t
W
|
2
+
|
∇
W
|
2
d
X
+
∫
R
N
×
{
0
}
Φ
(
w
)
d
x
}
d
t
,
ε
∈
(
0
,
1
)
where $$a \in (-1,1)$$
a
∈
(
-
1
,
1
)
is a fixed parameter, $$\mathbb {R}_+^{N+1}$$
R
+
N
+
1
is the upper half-space and $$\textrm{d}X = \textrm{d}x \textrm{d}y$$
d
X
=
d
x
d
y
. As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted nonlinear Cauchy–Neumann problems arising in combustion theory and fractional diffusion.
Funder
HORIZON EUROPE European Research Council
Publisher
Springer Science and Business Media LLC
Subject
Mathematics (miscellaneous)
Reference46 articles.
1. G. Akagi, U. Stefanelli. A variational principle for gradient flows of nonconvex energies, J. Convex Anal. 23 (2016), 53–75.
2. I. Athanasopoulos, L. Caffarelli, E. Milakis. On the regularity of the Non-dynamic Parabolic Fractional Obstacle Problem, J. Differential Equations 265 (2018), 2614–2647.
3. A. Audrito, E. Serra, P. Tilli. A minimization procedure to the existence of segregated solutions to parabolic reaction-diffusion systems, Comm. Partial Differential Equations 46 (2021), 2268–2287.
4. A. Audrito, S. Terracini. On the nodal set of solutions to a class of nonlocal parabolic equations, to appear in Mem. Amer. Math. Soc., arxiv:1807.10135v2 (2020).
5. A. Banerjee, N. Garofalo. Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations, Adv. Math., 336 (2018), 149–241.