We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set.Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren's frequency formula, first established by Caffarelli, Salsa, and Silvestre.2000 Mathematics Subject Classification. 35R35.
In this paper we prove C 1,α regularity (near flat points) of the free boundary ∂{u > 0} ∩ Ω in the Alt-Caffarelli type minimum problem for the p-Laplace operator: (2000): 35R35, 35J60
Mathematics Subject Classification
We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.2000 Mathematics Subject Classification. Primary 35R35, 35K85.
We study the regularity of the free boundary in a Stefan-type problem
\[
Δ
u
−
∂
t
u
=
χ
Ω
in
D
⊂
R
n
×
R
,
u
=
|
∇
u
|
=
0
on
D
∖
Ω
\Delta u - \partial _t u = \chi _\Omega \quad \text {in $D\subset \mathbb {R}^n\times \mathbb {R}$}, \qquad u = |\nabla u| = 0 \quad \text {on $D\setminus \Omega $}
\]
with no sign assumptions on
u
u
and the time derivative
∂
t
u
\partial _t u
.
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