AI 中文总结
研究紧致度量图上\(p -\)拉普拉斯算子的\(p -\)扭转刚度，通过变分法确立相关性质，推导出\(p -\)圣维南和\(p -\)科勒 - 乔宾两个精确等周不等式，扩展了\(p = 2\)时的线性理论到\(1 < p < \infty\)范围。
AI 中文摘要
我们研究在配备狄利克雷条件（在一度顶点的非空集合\(\mathcal{V}^D\)上）和非线性基尔霍夫条件（在所有其余顶点处）的紧致连通度量图上，\(1 < p < \infty\)时\(p -\)拉普拉斯算子的\(p -\)扭转刚度。我们确立了\(p -\)扭转函数的存在性、唯一性和正性，以及\(p -\)扭转刚度的变分特征。主要贡献是推导了两个精确的等周不等式。首先证明了一个\(p -\)圣维南不等式，表明在所有规定总长度的紧致度量图中，\(p -\)扭转刚度恰好由具有单个狄利克雷端点的区间最大化。然后推导了一个精确的\(p -\)科勒 - 乔宾不等式，根据\(p -\)扭转刚度为\(p -\)拉普拉斯算子的第一特征值提供了一个尺度不变的下界。这些结果在紧致度量图的背景下产生了经典圣维南和科勒 - 乔宾不等式的非线性对应物，并将由穆格诺洛和普吕默发展的\(p = 2\)时的线性理论扩展到了\(1 < p < \infty\)的全范围。
英文摘要
We investigate the $p$-torsional rigidity for the $p$-Laplacian, $1<p<\infty$, on compact connected metric graphs equipped with Dirichlet conditions on a nonempty set $\mathcal{V}^D$ of degree-one vertices and nonlinear Kirchhoff conditions at all remaining vertices. We establish the existence, uniqueness, and positivity of the $p$-torsion function, together with a variational characterization of the $p$-torsional rigidity. Our main contribution is the derivation of two sharp isoperimetric inequalities. We first prove a $p$-Saint-Venant inequality, showing that, among all compact metric graphs of prescribed total length, the $p$-torsional rigidity is maximized precisely by the interval with a single Dirichlet endpoint. We then derive a sharp $p$-Kohler--Jobin inequality, providing a scale-invariant lower bound for the first eigenvalue of the $p$-Laplacian in terms of the $p$-torsional rigidity. These results yield nonlinear counterparts, in the setting of compact metric graphs, of the classical Saint-Venant and Kohler--Jobin inequalities, and extend the linear theory, where $p=2$, developed by Mugnolo and Plümer to the full range $1<p<\infty$.
