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Results tagged with regularity
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user 37103
regularity of solutions of PDEs.
11
votes
2
answers
3k
views
What's wrong with the Courant nodal domain theorem?
However, Chavel in Eigenvalues in Riemannian Geometry seems to point out (P23) that Divergence theorem is used, so the regularity of the nodal set matters, whose proof by Cheng in dimension $\ge 3$ is …
- 5,169
2
votes
Elliptic theory on compact manifolds
For the specific question of getting rid of the low-order term $\|u\|_p$, what you really need is the compact embedding of $W^{2,p}$ in $L^p$ (Rellich–Kondrachov theorem). Indeed, suppose there is no …
- 5,169
2
votes
Moser estimates?
. $$
(If $u$ does not have $W^{1,2}$ regularity to start with, you can approximate the left hand side by difference quotients.)
Now do the same for $\nabla u$, $\nabla^2 u$, etc. …
- 5,169