发表机构
Weierstrass Institute; Bocconi University; University of Coimbra（韦伊尔斯拉斯研究所; 博科尼大学; 科英布拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带非齐次Neumann边界条件的非线性Poisson系统的Wasserstein提升，证明极小元存在性并推导最优性条件，发现高维中会出现测度值极小元，且提升问题与经典问题最小值不同，属非线性椭圆理论新现象。
AI 中文摘要
我们研究了具有非齐次Neumann边界条件的非线性Poisson系统的Wasserstein提升。我们利用涉及测度值迹的p-矩的新估计，建立了极小元的存在性。我们还推导了最优性条件，这些条件被证明是无压力Euler方程的矩阵值推广。虽然在适当的凸性假设和一维源域下存在确定性极小元，但我们证明在更高维中会出现真正的测度值极小元。特别地，我们构造了提升变分问题与经典变分问题具有不同最小值的例子，从而排除了提升问题的解是确定性的可能性，尽管没有任何先验机制强制测度值行为。据我们所知，这一现象在非线性椭圆理论中是新的。
英文摘要
We study the Wasserstein lift of nonlinear Poisson systems with inhomogeneous Neumann boundary conditions. We establish the existence of minimizers using new estimates involving the $p$-moments of measure-valued traces. We also derive the optimality conditions, which turn out to be matrix-valued generalizations of pressureless Euler equations. While deterministic minimizers exist under suitable convexity assumptions and for one-dimensional source domains, we show that genuinely measure-valued minimizers arise in higher dimensions. In particular, we construct examples in which the lifted and classical variational problems have different minimum values, thereby precluding the solutions to the lifted problem from being deterministic, despite the absence of any a priori mechanism enforcing measure-valued behavior. To the best of our knowledge, this phenomenon is new in the nonlinear elliptic theory.
