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On a Riemannian Manifold $M^n$, the Hodge Laplacian is defined on k-forms by $\Delta\omega=\operatorname{d}\operatorname{d}^*\omega+\operatorname{d}^*\operatorname{d}\omega$. For 0-forms, e.i. smooth ...

Let $M$ be a simply-connected $d$-dimensional Riemannian manifold with boundary (for simplicity assume a ball). Consider the boundary value problem for $\omega\in\Omega^k(M)$, $$ d\omega = \alpha \...

(I previously asked this question on Math.SE but got no responses after two weeks.) Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...