All Questions
Tagged with sp.spectral-theory elliptic-pde
21 questions with no upvoted or accepted answers
4
votes
0
answers
137
views
Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
- 2,095
4
votes
0
answers
82
views
On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 4,135
4
votes
0
answers
410
views
Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
- 325
3
votes
0
answers
67
views
Eigenvalues of an elliptic operator on shrinking domains
This was probably done somewhere 100 times, but I can't find a reference.
Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general ...
- 445
3
votes
0
answers
167
views
Asymptotic behaviour of principal eigenfunctions and large deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
- 5,380
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
2
votes
1
answer
196
views
Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
2
votes
0
answers
94
views
Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics
Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
- 2,095
2
votes
0
answers
67
views
A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
- 4,135
2
votes
0
answers
85
views
Proving an eigenvalue bound without resorting to Weyl's law
Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
- 4,135
2
votes
0
answers
77
views
Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator
I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
- 4,135
2
votes
0
answers
123
views
Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds
Consider the elliptic operator $Lu = - \Delta u + \langle \nabla u , X \rangle + c \, u $ acting on functions on a closed Riemannian manifold $M$. Here $\Delta$ denotes the Laplace-Beltrami operator, $...
- 386
2
votes
0
answers
59
views
Zero in the spectrum of an elliptic second order operator
This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator
but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...
- 21.8k
2
votes
0
answers
162
views
Spectrum of a linear elliptic operator
In the paper in quantum fields theory by
Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19;
in Section 3 the author makes the following claim from PDE and ...
- 21.8k
2
votes
0
answers
218
views
Weyl's law for minimal surfaces
I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...
- 1,616
2
votes
0
answers
71
views
Localized eigenfunctions of drift Laplacians
I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
- 21
1
vote
0
answers
148
views
Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
- 309
1
vote
0
answers
52
views
Laplacian eigenvalue problem for systems coupled along the boundary
I am looking for references on eigenvalue problems for systems of the following type:
Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...
1
vote
0
answers
259
views
An estimate for the solution of an elliptic PDE depending on a parameter
Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$.
We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator
$$
-\Delta:\ H^...
- 375
1
vote
0
answers
89
views
Eignfunctions of an elliptic operator
I am looking for a reference which makes it possible to say that one can devellop $f$ in the form of the sentence underlined by the yellow.
Thank you in advance.
- 135
0
votes
0
answers
161
views
When linear strongly elliptic operators are invertible?
I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like:
A linear differential operator, $$A : W^{...
- 1,774