5
votes
1answer
163 views

Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...
0
votes
1answer
146 views

Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary [closed]

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form $$|u(t)|_{L^p} \leq ...
2
votes
1answer
144 views

If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?

Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function. When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...
6
votes
1answer
481 views

Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
3
votes
2answers
1k views

Sobolev imbedding on Riemannian manifolds

Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy. Let ...