Elliptic Regularity on Manifolds

partial-differential-equations manifolds riemannian-geometry

Chapter 6 of Warner (Foundations of Differentiable Manifolds and Lie Groups) covers elliptic operators on manifolds. Specifically, you're probably interested in Theorem 6.5 and its proof in 6.32. The proof uses coordinate charts and elliptic regularity in $\mathbb{R}^n$ to establish smooth local solutions and then pieces the global solution together with a partition of unity.

You may also be interested in Booss-Bleeker (Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics). Additionally, here is a sketch of some of the ideas from a course at Indiana University several years ago.

Related

Uniform Convergence verification for Sequence of functions - NBHM
When the mathematical community consider the inclusion of a new axiom?.
Solving the quadratic equation for matrices
Explanation for the integral of differential forms
Set of elements of degree $2^n$ over a base field is itself a field
Continuity of the derivative
Is a morphism of reduced schemes over an algebraically closed field determined by its values on closed points?
Isomorphism in cohomology is an isomorphism in homology
Checking if $\langle 2 \rangle$ is a maximal ideal in $\mathbb{Z}[i]$
Area of the given triangle
If the Lie algebra is a direct sum, then the Lie group is a direct product?
How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$