Questions tagged [linear-pde]

Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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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 ...
Qijun Tan's user avatar
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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| \...
Matchmaticians's user avatar
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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, ...
F.M.R.'s user avatar
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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 ...
Math604's user avatar
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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 ...
mystupid_acct's user avatar
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$ ...
Math604's user avatar
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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 {...
Alex M.'s user avatar
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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 ...
Onil90's user avatar
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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 ...
Alex M.'s user avatar
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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 ...
user106481's user avatar
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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 ...
Manuel Pena's user avatar
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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 (...
Math604's user avatar
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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$ ...
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$u$ 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) ...
abcd's user avatar
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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{} \...
54321user's user avatar
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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 $\mathbb{C}^n$ ...
Paul's user avatar
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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(...
mathpde's user avatar
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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^...
Math604's user avatar
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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 \...
Kanye's user avatar
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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,\...
melomm's user avatar
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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 ...
Lapasotka's user avatar
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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 $...
Moonwalker's user avatar
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 $\rho = (\rho_0,\rho_1)^T$, $D$ ...
Frits Veerman's user avatar
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, ...
Lentes's user avatar
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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 $\...
ABIM's user avatar
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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_{\...
Florian Omnes's user avatar
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}{\...
Alexander's user avatar
  • 157
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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 $ \...
Math604's user avatar
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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,...
CrishaD's user avatar
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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 ...
Jommy's user avatar
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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\...
user's user avatar
  • 201
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 $\Delta_g$ ...
Anton Quelle's user avatar
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 $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$$...
ChristopherSail's user avatar
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 $$u_t - \Delta u = f$$ $$u(0) = u_0$$ $$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$ ...
TLE's user avatar
  • 53
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 $n$ ...
DC47's user avatar
  • 111
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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(\...
Matchmaticians's user avatar
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)=...
Galor's user avatar
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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 $\...
dh16's user avatar
  • 133
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 \...
poupy's user avatar
  • 175
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 ...
Hosein Rahnama's user avatar
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, ...
Josh Burkart's user avatar
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 ...
Bazin's user avatar
  • 15.1k
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 \,...
Adhvaitha's user avatar
  • 151
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 ...
Delio Mugnolo's user avatar
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 $...
Lao-tzu's user avatar
  • 1,856
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 ...
Matchmaticians's user avatar
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}...
Math604's user avatar
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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 ...
Math604's user avatar
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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)...
Hosein Rahnama's user avatar
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 ...
enrico's user avatar
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