论文标题
关于伯努利问题，无限跳跃
On the Bernoulli problem with unbounded jumps
论文作者
论文摘要
我们调查了开处方无限跳跃条件的Bernoulli自由边界问题。数学设置会导致对形式的非不同最小化问题的分析（\ nabla u \ cdot（a（x）\ nabla u） +φ（x） +φ（x）1 _ {\ {\ {u> 0 \ 0 \}}}}} \ right）具有有界，可测量系数和$φ$的椭圆矩阵不一定是局部界限的。我们证明了最小化和两相问题的最小化器的普遍连续性。还获得了沿自由边界的尖锐规律性估计。此外，我们对无限跳跃点$ξ$的几何形状进行彻底分析，$之一，$ξ\ inφ^{ - 1}（\ infty）$。我们表明，它取决于$φ$接近$ξ$的爆炸率，并且我们获得了这种牙牙几何形状的分析描述。
We investigate Bernoulli free boundary problems prescribing infinite jump conditions. The mathematical set-up leads to the analysis of non-differentiable minimization problems of the form $\int \left(\nabla u\cdot (A(x)\nabla u) + φ(x) 1_{\{u>0\}}\right) \,\mathrm{d}x \to \text{min}$, where $A(x)$ is an elliptic matrix with bounded, measurable coefficients and $φ$ is not necessarily locally bounded. We prove universal Hölder continuity of minimizers for the one- and two-phase problems. Sharp regularity estimates along the free boundary are also obtained. Furthermore, we perform a thorough analysis of the geometry of the free boundary around a point $ξ$ of infinite jump, $ξ\in φ^{-1}(\infty)$. We show that it is determined by the blow-up rate of $φ$ near $ξ$ and we obtain an analytical description of such cusp geometries.
