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Results tagged with elliptic-pde
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user 37103
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
11
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2
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What's wrong with the Courant nodal domain theorem?
The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. How …
- 5,169
2
votes
Accepted
Solution to inhomogenous PDE
This kernel is called the Bessel potential. It is smooth away from $0$, and in your case, this kernel is locally integrable.
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2
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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
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Moser estimates?
Integrating by parts, using a cutoff function if necessary, we have
$$ \int_{B(r)} |\nabla u|^2\ll \int_{B(r)} a_{ij}\partial_iu\partial_ju=\int_{B(R)} uLu+O(u|\nabla u|)\ll \int_{B(R)} |u\nabla u|. …
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