Kavli Affiliate: Yi Zhou
| First 5 Authors: Lili Du, Yi Zhou, , ,
| Summary:
In the previous work [Interfaces Free Bound., 19, 351-369, 2017], de Queiroz
and Shahgholian investigated the regularity of the solution to the obstacle
problem with singular logarithmic forcing term
begin{equation*}
-Delta u = log u , chi_{{u>0}} quad text{in} quad Omega,
end{equation*}
where $chi_{{u>0}}$ denotes the characteristic function of the set
${u>0}$ and $Omega subset mathbb{R}^n$ ($n geq 2$) is a smooth bounded
domain. The solution solves the minimum problem for the following functional,
begin{equation*}
mathscr{J}(u):=int_{Omega}left(frac{|nabla u|^2}{2}-u^+ (log
u-1)right) , dx,
end{equation*}
where $u^+=max{{0,u}}$. In this paper, based on the regularity of the
solution, we establish the $C^{1,alpha}$ regularity of the free boundary
$Omega cap partial{u>0}$ near the regular points for some $alphain
(0,1)$.
The logarithmic forcing term becomes singular near the free boundary
$Omegacappartial{u>0}$ and lacks the scaling properties, which are very
crucial in studying the regularity of the free boundary. Despite these
challenges, we draw inspiration for our overall strategy from the
"epiperimetric inequality" method introduced by Weiss in 1999 [Invent. Math.,
138, 23-50, 1999]. Central to our approach is the introduction of a new type of
energy contraction. This allows us to achieve energy decay, which in turn
ensures the uniqueness of the blow-up limit, and subsequently leads to the
regularity of the free boundary.
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