关于黎曼 $p$-拉普拉斯算子的正交曲面自由边界问题

中文摘要 AI 辅助

我们研究了一个由光滑紧致有限维黎曼流形上的黎曼 $p$-拉普拉斯算子驱动的正交曲面自由边界问题。我们将该问题表述为一个形状优化问题，并基于一致黎曼 $RC$-$GNP$ 条件~\cite{DS3} 发展了一个内在的容许类。我们建立了容许域的紧性以及在强 $RC$-$GNP$ 收敛下相关 Dirichlet 问题的稳定性。特别地，状态在 $W^{1,p}$ 中强收敛，且相关的 $p$-扭转能量是连续的。在额外的边界正则性下，我们推导了形状泛函的第一变分，并得到了自由边界 Euler--Lagrange 条件 \\[ |\nabla_g u_\Omega|_g^p=\sigma H_{\partial\Omega}+k^p. \\] 我们还公式化了接触最优性不等式，证明了在切向接触处第二基本形式的黎曼比较，并获得了参考域上的一个充分条件，该条件排除了接触并产生了自由边界问题的解。文中包含了球面测地球上的显式径向例子。

英文摘要

We study a quadrature-surface free boundary problem driven by the Riemannian $p$-Laplacian on a smooth compact finite-dimensional Riemannian manifold. We formulate the problem as a shape optimization problem and develop an intrinsic admissible class based on a uniform Riemannian $RC$-$GNP$ condition~\cite{DS3}. We establish compactness of admissible domains and stability of the associated Dirichlet problems under strong $RC$-$GNP$ convergence. In particular, the states converge strongly in $W^{1,p}$ and the associated $p$-torsional energy is continuous. Under additional boundary regularity, we derive the first variation of the shape functional and obtain the free-boundary Euler--Lagrange condition \[ |\nabla_g u_Ω|_g^p=σH_{\partialΩ}+k^p. \] We also formulate the contact optimality inequality, prove the Riemannian comparison of second fundamental forms at tangential contact, and obtain a sufficient condition on a reference domain which rules out contact and yields a solution of the free-boundary problem. Explicit radial examples on geodesic balls of the round sphere are included.

发表机构

• Ecole Doctorale de Mathématiques et Informatique U.C.A.D.（数学与信息博士学院）

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