Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
371
questions
2
votes
0
answers
168
views
Maslov canonical operator
Suppose $\Lambda$ is a Lagrangian submainfold of $M=T^*\mathbb{R}^n$. Let $x_i$ be the standard coordinate on the base manifold $\mathbb{R}^n$ and $\eta_j$ the coordinate on the dual. According to a ...
0
votes
1
answer
143
views
Solution of Poisson equation vanishing at the boundary of any order
Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and
$\Delta u=f$ in $\Omega$
such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \...
1
vote
0
answers
67
views
Smoothing in linear hyperbolic equations
This is a bit fuzzy, but I've somewhere read or heard something like:
"For linear hyperbolic equations smoothing in time leads to smoothing in space"
Is this in any sense true?
References, ...
1
vote
0
answers
109
views
Eigenvalues of elliptic operator analytic with respect to a parameter
I am interested when one can say the eigenvalues of an elliptic operator
are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
1
vote
1
answer
94
views
Decomposition of spectrum of a (unbounded, non-self-adjoint) linear operator in two spatial dimensions
Assume $L$ is unbounded, non-self adjoint operator for functions over two space dimensions $(x,y)\in \mathbb{R}^2$, such that upon fourier transforming w.r.t $y$, one can reduce the operator to (for ...
4
votes
1
answer
318
views
Fundamental gap for Schrödinger operator
Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$.
I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi_1>0$ and $ \phi_2$ ...
4
votes
0
answers
73
views
The sum of linear partial differential operators of equal strength
If $P$ and $P'$ are linear partial differential operators with constant complex coefficients on $U = \mathring U \subseteq \Bbb R^m$, we say that $P \sim P'$ if and only if $\dfrac {\tilde P} {\tilde {...
3
votes
1
answer
388
views
Harnack Inequality
In the Han and Lin book there is a Harnak inequality for elliptic operators of the kind:
$$
L u = D_i \big(a^{ij}\, D_ju\big),
$$
and the constant $C$ in the Harnack inequality does not depend on the ...
3
votes
1
answer
370
views
Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
3
votes
0
answers
553
views
2D laplace equation with Robin boundary condition (Green function)
Let's say that I know a fundamental solution for the Laplace equation in the whole plane:
$$\nabla^2u=\delta\quad \text{in the sense of distributions,}$$
and I need a solution for the laplace equation ...
1
vote
0
answers
71
views
Poisson boundary estimates
I asked this question Poisson equation estimates near boundary a few days ago but haven't gotten any response. So I will ask a related question. Suppose $-\Delta u(x)=f(x)$ in $B_1^+$ in the (...
1
vote
0
answers
115
views
Poisson equation estimates near boundary
Fix $ \Omega$ a bounded smooth domain in $\mathbb R^N$ (take $N$ big) and let $ \frac{N+1}{2}<p<N$. We now consider nonnegative smooth functions $f$ such that $-\Delta u(x)=f(x) $ in $ \Omega$ ...
0
votes
0
answers
47
views
$u$ satisfies Schrödinger equation implies $\mathcal{F}^{-1} \left(\chi_{2}(\xi) \hat{u} \right) $ also?
Consider the Schrödinger equation (SE):
$i \frac{\partial }{\partial t}u (x,t )+ \Delta u(x,t) =0, (x, t)\in \mathbb R^{N}\times \mathbb R.$
$u(0,x)=\phi(x).$
Then, formally,
the solution of (SE) ...
7
votes
1
answer
457
views
How can I construct D-modules over projective space using explicit differential equations?
Over $\mathbb{A}^n$, it is easy to construct D-modules by writing down an explicit linear system of PDE's and then writing a presentation of the associated D-module
$$
\mathcal{D}^n \xrightarrow{} \...
3
votes
1
answer
159
views
Solving a system of equations involving smooth functions
I have asked the following question in math.stackexchange, but I could not receive the answer. See here.
Suppose $h_{i\overline{j}}$, where $1\leq i, j\leq n$, are functions defined on $\mathbb{C}^n$ ...
1
vote
0
answers
220
views
Cauchy–Kowalevski Theorem for PDEs
I hope to extend the PDEs across the boundary using Cauchy–Kowalevski Theorem.
Given a real analytic manifold $M$ with boundary $\partial M$, the solution $u$ to the PDE $$\triangle u+b(x)\nabla u+c(...
1
vote
0
answers
105
views
Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
7
votes
1
answer
388
views
Does $(\nabla \times F) \cdot F= (\nabla \times F) \cdot \nabla f$ have a solution?
A grad student asked me this question during office hours, and I couldn't for the life of me come up with a proof or counterexample:
For a given $F:\mathbb{R}^3 \to \mathbb{R}^3$, does $(\nabla \...
0
votes
0
answers
145
views
Singular elliptic PDE: $-h\,\Delta u +\langle \nabla h,\nabla u\rangle =h$
Let $U\subset\mathbb{R}^n$ with $0\in U$. Fix $h\in L^2(U)\cap C^\infty(U)$ with $h(0)=0$. Is there some $C^1$-function $u\neq 0$ in such that $u$ is solution of
$$-h\,\Delta u +\langle \nabla h,\...
3
votes
0
answers
76
views
Smoothing inside the null space of a partial differential operator
Let $L$ be a linear partial differential operator with smooth coefficients in $U\subset\Bbb R^n$ and let $u\in W^{k,p}_{loc}(U)$ with $k\in\Bbb N$ and $p\in[1,\infty[$ satisfy $Lu=0$ in the ...
0
votes
0
answers
68
views
What can be said about moments of probability distribution if it satisifies hemholtz equation?
From physical considerations I have observed, that probability density in region of interest satisfies
$$
\Delta u(x) + \phi(x)u(x) = f(x),
$$
where $\phi(x)$ and $f(x)$ are both given functions and $...
3
votes
0
answers
198
views
Analytic solution to two component, first order, linear PDE system
I would like to obtain analytic solutions to the following PDE system:
\begin{equation}
\rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1}
\end{equation}
with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
3
votes
1
answer
591
views
When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)
Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator
$$
Lu:=-\Delta u+V(x)u
$$
where $V\geq0$ is the electric potential, ...
0
votes
1
answer
198
views
Differential Riccati-type equation
Setup
I have recently come across an ODE of the form
$$
0 = \dot{A}(t)^TG(t) + A(t)^T\dot{G}(t) + C(t) + \lambda B(t)^TA(t)G(t) + \frac{\lambda}{2} (D(t)A(t)G(t))(D(t)A(t)G(t))^T\bar{1},
$$
where $\...
2
votes
0
answers
101
views
Elliptic equation with Laplace-Beltrami boundary condition
For my research, I've come across the following type of equation (under variational form).
Assume $\Omega\subset\mathbb{R}^d$ is a Lipschitz domain, $\phi \in L^2(\partial \Omega)$ and $\nabla_{\...
2
votes
0
answers
175
views
are these norms equivalent?
If it is known that $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\geq \alpha^2|\xi|^2$, where $\xi=(\xi_1,\xi_2,...,\xi_n)\in\mathbb{R}^n$ then can it be said that $\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\...
0
votes
1
answer
236
views
Gradient bounds on Newtonian potentials
Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \...
3
votes
0
answers
331
views
Method of characteristic for a system of first order PDEs
I am working with this system of first order PDEs:
\begin{equation}
\left\{
\begin{aligned}
%Suscettibili
&\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
0
votes
0
answers
139
views
Fundamental solution matrix of a linear PDE
I've asked a very similar question also at math.stackexchange, but I've not received any answer.
A vectorial function $\boldsymbol{x}:\mathbb{R}^D \rightarrow \mathbb{R}^N$ satisfies the following ...
2
votes
0
answers
141
views
Fractional derivative of the Wright function
It is mentioned in some papers (Appendix in this paper, for example) that the (formal) solution of the fractional drift (or transport) equation
$$
\partial_{t}^{\alpha}u(t,x)+\partial_{x}u(t,x)=0\quad\...
8
votes
1
answer
476
views
Properties of connection Laplacian on vector fields
Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...
1
vote
0
answers
743
views
$C^{1,2}$ regularity of (weak) solutions to the heat equation
Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation
$$u_t - \Delta u = 0$$
$$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$
$$u(0) = u_0$$...
5
votes
1
answer
1k
views
$L^\infty$ estimate on heat equation with a lower order term
Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of
$$u_t - \Delta u = f$$
$$u(0) = u_0$$
$$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$
...
1
vote
0
answers
80
views
About the "method of lines": when are such solutions good approximations for **all** future time?
This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
0
votes
1
answer
362
views
Harmonic/Subharmonic lifting of functions on an annulus
Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...
1
vote
1
answer
231
views
How to find the eigenvalues equation of this PDE problem
Given the problem:
$$(\kappa(x)X^{'})^{'}+\lambda\rho(x)X=0$$
for $0<x<l$ with $X(0)=X(l)=0$
where $\kappa(x)=\kappa_{1}^{2}$ for $x<a$, $\kappa(x)=\kappa_{2}^{2}$ for $\kappa>a$. $\rho(x)=...
1
vote
1
answer
201
views
Why are the tangential derivatives in this diffraction problem zero? [closed]
I'm considering the diffraction problem described in section 3.16 of "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Uraltseva (1968). Let $\Omega$ be an open bounded subset in $\...
3
votes
0
answers
106
views
Constant in a trace Sobolev theorem for concave domains
I wonder is the following inequality is true/known:
Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then
$$
\int_{\partial\Omega} |u|^2 ds
\...
2
votes
0
answers
105
views
Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions
I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...
1
vote
2
answers
901
views
Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs
I have the following system of PDEs that I want to solve as "analytically" as possible:
$$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$
where $A$ and $B$ are constant, ...
11
votes
2
answers
687
views
Poincaré lemma for distributions
Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...
1
vote
0
answers
78
views
Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions
I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...
3
votes
2
answers
402
views
Airy's equation on $\mathbb R_-$
I am interested in Airy's equation
$$\frac{\partial u}{\partial t}(t,x)=-\frac{\partial^3 u}{\partial x^3}(t,x)$$
on a bounded or semi-bounded domain, e.g. on $(-\infty,0)$. In order to obtain a group ...
0
votes
0
answers
171
views
For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?
Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that $...
3
votes
2
answers
264
views
A Global Estimates for Linear Elliptic PDE
Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
1
vote
0
answers
74
views
Fundamental gap for Neumann BVP with potential
I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem
\begin{cases}...
1
vote
0
answers
190
views
Regularity on Neumann problem on polygonal domain
I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).
Let $ \Omega$ denote a cube in $ R^n$ and consider ...
3
votes
0
answers
379
views
Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]
I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi (x,y)...
1
vote
0
answers
37
views
Regularity of a flux induced by a potential
Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a scalar ...