An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations

Abstract

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.