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

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 $m\ge 1$ and every $1\le p<\mathrm{\infty }$, 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.