TY - UNPB

T1 - A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

AU - Karakhanyan, Aram

AU - Dipierro, Serena

PY - 2015/9/1

Y1 - 2015/9/1

N2 - We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional
ϕp(r,u,x0)=1r4∫Br(x0)|∇u+(x)|p|x−x0|N−2dx∫Br(x0)|∇u−(x)|p|x−x0|N−2dx
where x0 is a free boundary point, i.e. x0∈∂{u>0} and u is a minimizer of the functional
J(u):=∫Ω|∇u|p+λp+χ{u>0}+λp−χ{u≤0},
for some bounded smooth domain Ω⊂RN and positive constants λ± with Λ:=λp+−λp−>0.
Here we show the discrete monotonicity of ϕp(r,u,x0) in two spatial dimensions at non-flat points, when p is sufficiently close to 2, and then establish the linear growth. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

AB - We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional
ϕp(r,u,x0)=1r4∫Br(x0)|∇u+(x)|p|x−x0|N−2dx∫Br(x0)|∇u−(x)|p|x−x0|N−2dx
where x0 is a free boundary point, i.e. x0∈∂{u>0} and u is a minimizer of the functional
J(u):=∫Ω|∇u|p+λp+χ{u>0}+λp−χ{u≤0},
for some bounded smooth domain Ω⊂RN and positive constants λ± with Λ:=λp+−λp−>0.
Here we show the discrete monotonicity of ϕp(r,u,x0) in two spatial dimensions at non-flat points, when p is sufficiently close to 2, and then establish the linear growth. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

M3 - Working paper

BT - A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

PB - ArXiv

ER -