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Results tagged with ap.analysis-of-pdes
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user 13093
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
8
votes
First eigenvalue of the Laplacian on a regular polygon
For $N \geq 5$ it is still not known if the $N$-gon which minimizes the first eigenvalue under area constraint (which exists), is the regular one. I have done some numerical computations which suggest …
- 2,222
7
votes
A long-lasting conjecture of Pólya & Szegő
Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.
Our paper can be found here: On the poly …
- 2,222
2
votes
First eigenvalue of the Laplacian on a regular polygon
Some recent activity on the problem:
Results regarding the Hessian matrix with respect to vertex coordinates, numerical local minimality estimates and analytical bounds on the diameter of the optimal …
- 2,222
6
votes
1
answer
636
views
Eikonal equation - Snell's law
I am interested in equations of the form $|\nabla d|= F(x)$, where $F(x)$ is piecewise constant and $d(x) = 0$ on $\Gamma_D$, a subset of the boundary. In particular, like in the figure, one can consi …
- 12.9k
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
5
votes
1
answer
507
views
$C_0$-semigroups applications
My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is som …
CommunityBot
- 1
1
vote
Spectrum of Dirichlet Problem for Laplacian on a Parallelogram
This may not be what you wish for, but here is a list of the first $10$ eigenvalues calculated numerically for the rhombus with sidelength $1$:
$$\begin{array}{c}24.8982\\
52.6379\\
71.7085\\
…
- 2,222
2
votes
1
answer
878
views
Weak divergence implies weak differentiability of components?
Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.
We say that $\sigma$ has weak divergence if there exists …
- 16.2k
2
votes
0
answers
175
views
A limit involving a regularizing kernel
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
Some …
- 2,222