论文标题
具有高度奇异术语的完全非线性边界问题的对称性
Symmetry for a fully nonlinear free boundary problem with highly singular term
论文作者
论文摘要
在本文中，我们证明了在椭圆形和抛物线机制中具有单数右侧的自由边界问题解决方案的径向对称性。更确切地说，在单位球$ b_1 $中，我们考虑解决完全非线性椭圆问题的解决方案 $ \ begin {case} f（d^2u）= f（u）＆\ text {in} b_1 \ cap \ { $ u = 0 $，表现为$ u^a $，$ a \ in（-1,0）$的负值。由于缺乏$ u $的$ c^2 $ -SmoothNess和免费的边界$ \ partial \ {u> 0 \} $，我们无法应用众所周知的Serrin-type边界点引理。我们通过一阶扩展和第二阶的衰减以及临时比较原理的确切假设来绕过这一点。我们同样处理问题的抛物线情况，并陈述相应的结果。
In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear elliptic problem $$ \begin{cases} F(D^2u)=f(u)&\text{in }B_1 \cap \{u >0 \},\\ u=M&\text{on }\partial B_1,\\ 0\le u<M&\text{in }B_1,\end{cases}$$ where the right hand side $f(u) $, near $u=0$, behaves like $u^a$ with negative values for $a \in (-1,0)$. Due to lack of $C^2$-smoothness of both $u$ and the free boundary $\partial\{u>0\}$, we cannot apply the well-known Serrin-type boundary point lemma. We circumvent this by an exact assumption on a first order expansion and the decay on the second order, along with an ad-hoc comparison principle. We treat equally the parabolic case of the problem, and state a corresponding result.
