论文标题
具有单数项（非线性椭圆情况）的自由边界的凸度
Convexity for free boundaries with singular term (nonlinear elliptic case)
论文作者
论文摘要
我们考虑外部域中的一个自由边界问题\ begin {cases} \ begin {array} {cc} lu = g（u） \ end {array} \ end {case}其中$ k $是$ \ m马理{r}^n $（$ n \ ge2 $），$ω= \ {u> 0 \> 0 \} \ supset k $是一个unkey $ p $ $ p $ p $ p $ - 在$ K $和$ g $的合适假设下，我们证明存在非负准共晶解决方案来解决上述问题。我们还考虑$ \ {x_n = 0 \} $中包含的集合$ k $的情况，并获得类似的结果。
We consider a free boundary problem in an exterior domain \begin{cases}\begin{array}{cc} Lu=g(u) & \text{in }Ω\setminus K, \\ u=1 & \text{on }\partial K,\\ |\nabla u|=0 &\text{on }\partial Ω, \end{array}\end{cases} where $K$ is a (given) convex and compact set in $\mathbb{R}^n$ ($n\ge2$), $Ω=\{u>0\}\supset K$ is an unknown set, and $L$ is either a fully nonlinear or the $p$-Laplace operator. Under suitable assumptions on $K$ and $g$, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set $K$ is contained in $\{x_n=0\}$, and obtain similar results.
