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On constructing the canonical boundary operator for a given differential operator

Given an $n\times n$ matrix $$X=\begin{pmatrix} x_{11} & x_{12} & \cdots & x_{1n} \\ x_{21} & x_{22} & \cdots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1}...
Ryan Hendricks's user avatar
7 votes
2 answers
2k views

Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
Puzzled's user avatar
  • 8,998
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0 answers
137 views

Elliptic partial differential equations with Robin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here. When I read the paper "On the attractor for a semilinear wave equation with critical ...
monotone operator's user avatar
2 votes
0 answers
267 views

Compactness of a nonlinear operator

Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$ ...
Mathlover's user avatar
0 votes
0 answers
76 views

Linear dependence of the derivatives of a vector valued function

Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function $$ g:\mathbb{R}^5\rightarrow\mathbb{R}^5 $$ given by $$ g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
Puzzled's user avatar
  • 8,998
0 votes
0 answers
40 views

Time regularity of traces

I have a question about the time regularity of the traces in one dimension. Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(...
TOT's user avatar
  • 11
1 vote
0 answers
236 views

Fredholmness of elliptic operator on Hölder spaces

Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
user158773's user avatar
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 ...
Elliott's user avatar
  • 325
4 votes
2 answers
410 views

Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$

The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function). $$(x^2y')'-x^2y=\lambda \;y$$ Now for a higher-degree ...
Bertrand's user avatar
  • 1,199
5 votes
2 answers
977 views

Symbol of the Laplace-Beltrami on $\mathbb{S}^2$

This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e. A differential operator $P=\sum_{|\...
BaoLing's user avatar
  • 329
3 votes
1 answer
467 views

Ground state for a double well potential (Schrödinger)

Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e. $V(x)=V(-x)$ for all $x\in\mathbb{R}$ $\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$ $V(x)\in \mathbb{...
M. Veruete's user avatar
3 votes
1 answer
247 views

Are there fundamental solutions of the laplacian that decay rapidly?

The question I consider the Laplacian $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function $$ \...
ClemensB's user avatar
5 votes
1 answer
187 views

Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, $$Y(t_1)...
tobias's user avatar
  • 749
0 votes
1 answer
568 views

How to solve this differential equation with an infinite sum?

I would like to find solutions of the following differential equation: $ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$ For example in space of function from $\mathbb R^*$ to $\mathbb ...
Bertrand's user avatar
  • 1,199
3 votes
1 answer
264 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
Jan Rozendaal's user avatar
2 votes
1 answer
4k views

Precise versions of "differential operators are unbounded but closed linear operators"

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign. Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...
Mark Kim-Mulgrew's user avatar
3 votes
0 answers
498 views

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases). Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
user16007's user avatar
  • 800