论文标题
凸域中的两相无界边界问题
Two-phase free boundary problems in convex domains
论文作者
论文摘要
我们研究了两相自由边界问题的最小化器的规律性。对于一类N维凸域，我们在Neumann边界条件下建立了最小化器至固定边界的Lipschitz连续性。我们的证明使用限制在凸形域的Alt-Caffarelli-Friedman功能中使用了几乎单调的公式。这需要具有Neumann边界条件的球体凸的经典弗里兰 - 海曼不平等的变体。为了应用这种不等式，除了凸度外，我们还需要一个第二个条件，以控制固定边界在每个边界点收敛到其极限锥的速率。
We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the rate at which the fixed boundary converges to its limit cone at each boundary point.
