AI 中文总结
研究与分数阶p-拉普拉斯算子相关的一相Alt-Caffarelli问题极小值正则性，通过多种方法证明极小值存在及性质，包括非负性等，还得到局部Hölder连续性、正性集内解齐次方程及最优自由边界增长估计。
AI 中文摘要
我们研究了在p≥2范围内与分数阶p-拉普拉斯算子相关的一相Alt-Caffarelli问题的极小值的正则性性质。我们考虑由正性集的测度惩罚的分数阶p-能量的极小值，具有规定的非负外部数据。我们证明了极小值的存在性并推导了它们的基本变分性质：极小值是非负的并且是齐次分数阶p-拉普拉斯方程的弱次解。主要的正则性论证结合了分数阶p-调和替换、能隙估计、非局部尾部界和Campanato型迭代。这产生了极小值的局部Hölder连续性，并意味着它们在其正性集中解齐次方程。最后，我们证明了最优自由边界增长估计，表明线性分数阶伯努利问题中已知的尖锐阶在分数阶p-拉普拉斯算子设置中持续存在。
英文摘要
We study regularity properties of minimizers for the one-phase Alt--Caffarelli problem associated with the fractional $p$-Laplacian in the range $p\geq2$. We consider minimizers of the fractional $p$-energy penalized by the measure of the positivity set, with prescribed nonnegative exterior datum. We prove existence of minimizers and derive their basic variational properties: minimizers are nonnegative and are weak subsolutions of the homogeneous fractional $p$-Laplace equation. The main regularity argument combines fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, and a Campanato-type iteration. This yields local Hölder continuity of minimizers and implies that they solve the homogeneous equation in their positivity set. Finally, we prove the optimal free boundary growth estimate, showing that the sharp order known in the linear fractional Bernoulli problem persists in the fractional $p$-Laplacian setting.
Comments24 pages
