AI 中文总结
本文为黎曼向量丛上的Sobolev截面建立内蕴迹理论，构造连续迹算子并刻画Dirichlet边界条件，进而证明广义(p,q)-增长拟线性椭圆方程基态解的存在性，统一了Bochner p-拉普拉斯等几何算子。
AI 中文摘要
设 \\(\mathbf{E}\to M\\) 为紧致带非空光滑边界的黎曼流形上的有限秩黎曼向量丛，配备相容联络。我们发展了通过弱协变导数定义的Sobolev截面的内蕴迹理论。对每个整数 \\(m\ge1\\) 和每个 \\(1\le p<\infty\\)，我们构造了直到 \\(m-1\\) 阶协变导数的连续迹算子，并将具有齐次Dirichlet边界条件的Sobolev空间刻画为相应迹算子的核。作为应用，我们建立了具有广义 \\((p,q)\\)-增长的一类黎曼向量丛上拟线性椭圆方程基态解的存在性。所提出的框架统一了若干重要的几何算子，包括Bochner \\(p\\)-拉普拉斯算子、\\((p,q)\\)-Bochner拉普拉斯算子以及预定平均曲率型算子。
英文摘要
Let \(\mathbf{E}\to M\) be a finite-rank Riemannian vector bundle over a compact Riemannian manifold with nonempty smooth boundary, equipped with a compatible connection. We develop an intrinsic trace theory for Sobolev sections defined through weak covariant derivatives. For every integer \(m\ge1\) and every \(1\le p<\infty\), we construct continuous trace operators for the covariant derivatives up to order \(m-1\) and characterize the Sobolev space with homogeneous Dirichlet boundary conditions as the kernel of the corresponding trace operator. As an application, we establish the existence of ground state solutions for a class of quasilinear elliptic equations on Riemannian vector bundles with generalized \((p,q)\)-growth. The proposed framework unifies several important geometric operators, including the Bochner \(p\)-Laplacian, the \((p,q)\)-Bochner Laplacian, and prescribed mean curvature-type operators.
Comments30 pages
