发表机构
School of Mathematics East China University of Science and Technology（华东理工大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究晶格图 $\mathbb Z^N$ 上 Born--Infeld 能量的尖锐 Sobolev 型不等式，确定最优常数的正性阈值，在临界指数处给出常数表达式并证明不可达到，在超临界区域证明极值函数的存在性。
AI 中文摘要
本文研究了与晶格图 $\mathbb Z^N$（$N\geq3$）上 Born--Infeld 能量相关的尖锐 Sobolev 型不等式：\\[ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,\alpha} \left(\sum_{x\in\mathbb Z^N}|u(x)|^\alpha\right)^{\frac{N}{N+\alpha}}, \\] 其中 $u\in D^{1,2}(\mathbb Z^N)\cap\ell^\alpha(\mathbb Z^N)$ 且对每个 $x\sim y$ 满足 $|\nabla_{xy}u|\leq1$，$C_{N,\alpha}$ 表示最优常数。我们确定了精确的正性阈值，并证明 $C_{N,\alpha}>0$ 当且仅当 $\alpha\geq2^*:=2N/(N-2)$。在 Sobolev 临界指数 $\alpha=2^*$ 处，我们确定了最优常数为 \\[ C_{N,2^*}=\frac12\mathcal S_2, \\] 其中 $\mathcal S_2$ 是最优离散 Sobolev 常数，并证明该常数不可达到。在超临界区域，存在 $\varepsilon_0=\varepsilon_0(N)>0$ 使得对每个 $2^*<\alpha<2^*+\varepsilon_0$，$C_{N,\alpha}$ 均可达到，且极值函数为非负 Schwarz 对称函数。主要的紧性困难源于 $\mathbb Z^N$ 上缺乏合适的缩放以及 Born--Infeld 能量的非齐次性。我们通过将离散 Schwarz 重排与 $Q_1$ 离散到连续的比较相结合，并建立当 $\ell^\alpha$ 范数趋于无穷时的严格分离来克服这些困难。
英文摘要
In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph \(\mathbb Z^N\), \(N\geq3\): \[ \frac12\sum_{x\in\mathbb Z^N}\sum_{y\sim x} \left(1-\sqrt{1-|\nabla_{xy}u|^2}\right) \geq C_{N,α} \left(\sum_{x\in\mathbb Z^N}|u(x)|^α\right)^{\frac{N}{N+α}}, \] for \(u\in D^{1,2}(\mathbb Z^N)\cap\ell^α(\mathbb Z^N) \) satisfying \(|\nabla_{xy}u|\leq1\) for every \(x\sim y\), where \(C_{N,α}\) denotes the optimal constant. We determine the exact positivity threshold and prove that \(C_{N,α}>0\) if and only if \(α\geq2^*:=2N/(N-2)\). At the Sobolev critical exponent \(α=2^*\), we identify the optimal constant as \[ C_{N,2^*}=\frac12\mathcal S_2, \] where \(\mathcal S_2\) is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists \(\varepsilon_0=\varepsilon_0(N)>0\) such that \(C_{N,α}\) is attained for every \(2^*<α<2^*+\varepsilon_0\), with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on \(\mathbb Z^N\) and from the nonhomogeneity of the Born--Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a \(Q_1\) discrete-to-continuum comparison and by establishing a strict separation as the \(\ell^α\)-norm tends to infinity.
Comments26 pages,2 figures.Comments and suggestions are most welcome
