由重合集中边界距离驱动的自由边界问题
A free boundary problem driven by boundary distance in the coincidence set
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中文总结 AI 辅助
研究由边界距离驱动的自由边界问题，通过解决泛函定义问题证明极小值存在，建立弱下半连续性结果，推导包括变分不等式、正性集上PDE及自由边界条件等极小值的平稳性条件。
中文摘要 AI 辅助
我们研究一个自由边界问题，即最小化一个包含非局部项的泛函，该非局部项奖励进入零相位的深度。对于有界开集Ω⊂Rd上u≥0的情况，我们最小化泛函J(u)。这种泛函例如源于具有粘附接触能的双膜问题。我们首先解决由泛函的非局部、边界敏感性质引起的定义良好性问题，并证明极小值的存在性，在此过程中建立非局部项的弱下半连续性结果。然后我们推导极小值的平稳性条件。
英文摘要
We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for $u\geqslant 0$ on a bounded, open set $Ω\subset\mathbb R^d$, we minimise $$J(u) = \int_Ω\left(\frac12|\nabla u|^2 - fu\right) \;-\; \int_{\{u=0\}} F\big(\mathrm{dist}(x,\partial \{u=0\})\big)\;\mathrm{d}x. $$ This kind of functional arises, for example, from a two-membranes problem with an adhesive contact energy. We first address a well-definedness issue caused by the non-local, boundary-sensitive nature of the functional and prove existence of minimisers, establishing along the way a weak lower semicontinuity result for the non-local term. We then derive stationarity conditions for minimisers, including a variational (Euler--Lagrange type) inequality, a PDE on the positivity set, and, under a mild non-degeneracy assumption, a free boundary condition obtained via inner variations and a Danskin-type differentiation of the distance function.
