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Results tagged with elliptic-pde
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user 13093
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
4
votes
0
answers
50
views
Computing quantities related to the Dirichlet Laplace eigenfunctions
Consider the Dirichlet-Laplace eigenvalue problem
$$\left\{ \begin{array}{rccl}
-\Delta u & = & \lambda u & \text{ in }\Omega\\
u& = & 0 & \text{ on }\Omega
\end{array}\right.$$
Suppose $\lambda$ is …
- 2,222
3
votes
An inequality for eigenvalues of the Dirichlet problem
The inequalities you state cannot hold. It is known that $\lambda(tA) = t^{-2}\lambda(A)$ so choosing $A = sB$ you would get
$$ \frac{1}{(ts+1-t)^2}\lambda(B) \leq \geq (t/s^2+1-t)\lambda(B)$$
Neither …
- 2,222
2
votes
0
answers
114
views
Reference request – a priori estimate – mixed boundary condition
I am interested in finding references regarding estimates of the form
$$ \| D^2 u\|_{L^2(\Omega)} \leq C(\|f\|_{L^2(\Omega)}+\|g\|_{S} )$$
where $\|D^2 u\|_{L^2(\Omega)}^2 = \sum\limits_{i,j \in \{1,2 …
- 2,222
1
vote
0
answers
73
views
Eigenvalue overdetermined problem
Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$:
$$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\
u = 0 …
- 2,222