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.

我们研究两相自由边界问题的极小化子的正则性。对于一类n维凸域，我们建立了在Neumann边界条件下直至固定边界的最小化子的Lipschitz连续性。我们的证明对限制在凸域上的Alt–Caffarelli–Friedman函数使用几乎单调的公式。对于具有Neumann边界条件的球的测地线凸子集，这需要经典Friedland-Hayman不等式的变体。为了应用此不等式，除了凸度外，我们还需要一个Dini条件，该条件控制固定边界在每个边界点收敛到其极限锥的速率。