7
votes
3answers
209 views

Spectrum of Dirichlet Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and ...
1
vote
1answer
219 views

Reference request: Spectral analysis of advection diffusion PDE

As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and NOT on existence/uniqueness etc. which is usually the ...
1
vote
0answers
88 views

null controllability of linear wave equation

Consider the linear wave equation : $$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$ Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
1
vote
0answers
54 views

strong stability for the wave equation

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
1
vote
0answers
132 views

Weyl quantization and convexity

Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf ...
11
votes
1answer
938 views

Relationship between Green's function and geodesic distance?

I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and ...
2
votes
2answers
548 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
1
vote
2answers
293 views

Showing a solution of elliptic PDe is non-degenerate

Dear Mathoverflowers: I am interested in radial positive solutions of $-\Delta u(r) = r^\alpha u(r)^p$ in the unit ball in $ R^N$ with $ u=0$ on the boundary. Here $p>1$ and $ \alpha >0$. ...
14
votes
2answers
438 views

Is there a spectral theory approach to non-explicit Plancherel-type theorems?

Teaching graduate analysis has inspired me to think about the completeness theorem for Fourier series and the more difficult Plancherel theorem for the Fourier transform on $\mathbb{R}$. There are ...
1
vote
0answers
309 views

Definition of spectral gradient

Consider this differential operator $$ \mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x})) $$ where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow ...
2
votes
0answers
448 views

Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background Consider the one dimensional second order elliptic PDE, $$ \left\{\!\! \begin{aligned} & -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\ & u(0)=u(1)=0 \end{aligned} ...
3
votes
2answers
439 views

Localization of Laplacian eigenfunction on the unit square?

Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k ...
3
votes
0answers
243 views

Controlling the Second Eigenvalue of a Schrödinger Operator

Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$. Let $L$ be the operator $$ L=\Delta+V $$ where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
3
votes
0answers
163 views

Generalizations of group algebras for arbitrary manifolds?

In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take ...
2
votes
3answers
703 views

Sobolev norms of eigenfunctions

Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything ...
5
votes
2answers
1k views

Eigenvalues of Laplacian

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = ...