This article develops a unified and intrinsic framework for the theory of Sobolev spaces on vector bundles over Riemannian manifolds. The analytical core of our approach is an explicit higher-order geometric integration by parts formula, which characterizes the formal adjoint of the covariant derivative as a global differential operator. This identity is established on arbitrary Riemannian manifolds with boundary, without assuming completeness or compactness.
While first-order integration by parts identities are classical, explicit higher-order formulas with precise boundary terms are rarely stated in the literature. As applications of this framework, we recover the classical Meyers--Serrin theorem on arbitrary manifolds and, in the compact case, the Sobolev embedding and Rellich--Kondrachov compactness theorems, providing direct and self-contained proofs. At the end of this work we also stablish a Green Formula and we use it to stablish norm equivalence in Sobolev Spaces on vector bundles with closed manifold as base space and the Bochner laplacian operator. As a corollary we show that, in case of trivial vector bundles this equivalence reduces to a well known (but non proved rigorously in literature) result for closed manifolds and the Laplace-Beltrami operator.
By emphasizing intrinsic global arguments and sharp local-to-global norm equivalence estimates, rather than ad hoc coordinate patching, this work offers a transparent and accessible foundation for the study of Sobolev spaces on vector bundles, suitable for researchers in global analysis, differential geometry, and partial differential equations.
We prove Sobolev embedding Theorems with weights for vector bundles in a complete riemannian manifold. We also get general Gaffney's inequality with weights. As a consequence, under a "weak bounded geometry" hypothesis, we improve classical Sobolev embedding Theorems for vector bundles in a complete riemannian manifold. We also improve known results on Gaffney's inequality in a complete riemannian manifold.
In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion $\nabla u\in K$ subject to suitable affine boundary conditions for $ u$ with $$ K:= SO(2)\left[\begin{array}{ ccc } 1 & δ\\ 0 & 1 \end{array}\right] \cup SO(2)\left[\begin{array}{ ccc } 1 & -δ\\ 0 & 1 \end{array}\right] $$ such that the associated deformation gradients $\nabla u$ enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape…
The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. Results of this type in published literature generally can be found only for integer order Sobolev s…
We study Sobolev estimates for the solutions of parabolic equations acting on a vector bundle, in a complete, compact or non compact, riemannian manifold $M.$ The idea is to introduce geometric weights on $M.$ We get global Sobolev estimates with these weights. As applications, we find and improve "classical results", i.e. results without weights, by use of a Theorem by Hebey and Herzlich. As an example we get Sobolev estimates for the solutions of the heat equation on $p$-forms when the manifold has "weak bounded geometry " of order $1$.
We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assumption on the injectivity radii can be replaced with the request of a controlled growth of the full curvature tensor at order $0$. The control in the assumptions occur via non-necessarily polynomial growth functions. This construction largely extend previously known results in vari…
We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of re…
This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagli…
We observe, utilize dualities in differential equations and differential inequalities, dualities between comparison theorems in differential equations, and obtain dualities in "swapping" comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II and lead to comparison theorems in Riemannian geometry with analytic, geometric, P.D.E.'s and physical applications. In particular, we prove Hessian comparison theorems and Laplacian comparison theorem under varied radial Ricci curvature or radial curvature assumptions, …
We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $\nabla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $\nabla$-Sobolev spaces and of the $\nabla$-differential operators on general manifolds. For …
We construct divergence-free Sobolev vector fields in C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. We then show that the vector fields we produce have at least as many integral curves starting from a.e. point of T^d as the number of distinct positive solutions to the continuity equation these vector fields admit. Our work uses convex integration techniques for the continuity equation introduced in [L. Szekelyhidi and S. Modena, Annals of PDE, 2018] and [E. Brue, M. Colombo, C. De Lellis, A…
We construct infinitely many incompressible Sobolev vector fields $u \in C_t W^{1,\tilde p}_x$ on the periodic domain $\mathbb{T}^d$ for which uniqueness of solutions to the transport equation fails in the class of densities $ρ\in C_t L^p_x$, provided $1/p + 1/\tilde p > 1 + 1/d$. The same result applies to the transport-diffusion equation, if, in addition $p'<d$.
The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^α(\mathbb{S}^{d-1}),$ with $α\in \mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $α>0$ this characterization involves a …
We prove a one-parameter family of diffusion hypercontractivity and present the associated Log-Sobolev, Poincare and Talagrand inequalities. A mean-field type Bakry-Emery iterative calculus and volume measure based integration formula (Yano's formula) are presented. Our results are based on the interpolation among divergence functional, generalized diffusion process, and generalized optimal transport metric. As a result, an inequality among Pearson divergence (P), negative Sobolev metric H^-1 and generalized Fisher information functional (I), named PH^{-1}I inequality, is derived.
