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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Generalized Fuchsian-type PDE?

Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
Math2024's user avatar
0 votes
1 answer
129 views

About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain

I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$: Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
Stack_Underflow's user avatar
7 votes
2 answers
452 views

Intuition for Agmon-Douglis-Nirenberg ellipticity

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently. I am trying to understand the definition of ellipticity of systems due to ...
G. Blaickner's user avatar
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1 vote
1 answer
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Green's function for a linear PDE initial value problem

For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem $$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
Abhishek Halder's user avatar
4 votes
1 answer
169 views

Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$

I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given: \begin{align*} \partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\ ...
Paul Joh's user avatar
  • 161
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0 answers
80 views

Reversing heat transfer with respect to time

Fact: One can easily compute heat dispersion in a plane using the heat equation. Question: Has any research been done on computing the process in the reverse time direction? That is, given a heat map $...
Snared's user avatar
  • 109
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0 answers
159 views

Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?

I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
53Demonslayer's user avatar
2 votes
0 answers
53 views

Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
Paruru's user avatar
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1 vote
0 answers
111 views

Regularity of elliptic equation with Neumann boundary conditions

In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
Cathelion's user avatar
1 vote
0 answers
31 views

Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation

I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO). Consider the following initial boundary value problem for the linear ...
user111's user avatar
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1 vote
1 answer
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Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data

I already asked the question on MSE, and have tried to figure it out myself. But the problem seems trickier than expected, so I guess MO is a better place to ask.. For the sake of completeness, I ...
Isaac's user avatar
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1 vote
0 answers
63 views

Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
SebastianP's user avatar
2 votes
0 answers
120 views

Uniqueness of the solution to systems of first-order linear PDEs

Context: Let $\Omega \subset \mathbb{R}^p$ be an domain. For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
Paruru's user avatar
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0 answers
29 views

Quantitative global Schauder estimate for solution operator to second order elliptic equation via extension

Let $\Omega\to\mathbb R^n$ be a bounded Lipschitz domain. We can choose an extension operator $E$ such that $E:W^{k,p}(\Omega)\to W^{k,p}(\mathbb R^n)$ is bounded for all $k\ge0$ and $1<p<\infty$...
Liding Yao's user avatar
4 votes
2 answers
255 views

Regularity of solution of $(-\Delta + w)f = 0$

I am studying the following Schrödinger equation: $$(-\Delta + w)f = 0$$ which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
MathMath's user avatar
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5 votes
1 answer
167 views

$L^2$ regularity theory for elliptic equations: Is there another method other that the difference quotient method? Reference request

So i'm interested in the following classical theorem or similar variants. Consider the following elliptic PDE $$ -D_\alpha(a^{ij}D_\beta u) = f. $$ If we assume that the coefficients $a^{ij}$ are ...
Franlezana's user avatar
1 vote
1 answer
236 views

Estimates for linear elliptic PDE in the whole of $\mathbb{R}^n$

I am struggling to track down any literature on the topic of elliptic regularity when the domain in question is the whole space $\mathbb{R}^n$. Consider the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^n)$,...
Benjamin's user avatar
4 votes
2 answers
755 views

Is there any bilinear Poincaré/Sobolev inequality?

Is the following, I call it bilinear Poincaré inequality, true? Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
Hao Yu's user avatar
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3 votes
0 answers
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Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field

Setting Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
MySheperd's user avatar
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0 answers
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Studying the evolution of laplacian in NS equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by: \begin{equation}\label{Eq1} \dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
MrPie 's user avatar
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0 answers
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Obtaining metric and compatible differential equations on codimension one foliation of $n$-cube

Essentially the same content as this post on Math stack exchange: https://math.stackexchange.com/q/4741835/460999. I don't expect an answer there and after waiting a few days I've decided to post here....
geocalc33's user avatar
1 vote
0 answers
37 views

Mixed boundary condition of parabolic equations

Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that $$ \partial\Omega=\partial\Omega_D\cup\partial\Omega_N, $$ where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
Luis Yanka Annalisc's user avatar
0 votes
0 answers
47 views

Convergence of wave equation with friction

I asked this in MSE, but I didnt get any hint or answer. I’m studying the following 1-D wave equation with friction $$\begin{cases} u_{tt}+2\epsilon u_t-u_{xx}=0,\,x\in(0,\pi),\,t>0,\\ u_x(0,t)=0=...
Guillermo García Sáez's user avatar
4 votes
1 answer
138 views

Continuous up to the boundary without boundary smoothness

Let $\Omega$ be a bounded domain, $f\in L^{\infty}(\Omega),$ and $0\leq u\in H_0^1(\Omega)$ is a non-negative solution of $\Delta u=f$. My question is as follows: Can we conclude that $u\in C^0(\bar{\...
sorrymaker's user avatar
2 votes
1 answer
136 views

Method of characteristics with 2 dependent variables in 3 dimensions

I'm interested in solving a first-order linear PDE with 2 dependent variables in 3 dimensions by the method of characteristics. Something of this general form: $$ A \frac{\partial u}{\partial x} + B \...
Computerish's user avatar
2 votes
0 answers
60 views

Dirichlet's problem for Laplace's equation in the model domain

Let $$\Omega_\alpha=\left\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\right\},$$ $a>0$. we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{ w}_0^...
sidi mohamd deval's user avatar
2 votes
0 answers
140 views

Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply. When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
monotone operator's user avatar
5 votes
0 answers
304 views

All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)

I have asked this question on MSE, but this is a better place. The heat equation and the heat kernel. Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
Lorenzo Pompili's user avatar
4 votes
0 answers
104 views

Continuity of solutions of Elliptic PDE with respect to parameters

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy $$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$ where $f$ is a fast decaying smooth function. I would like to know how the ...
Matchmaticians's user avatar
3 votes
0 answers
92 views

Comparison principle for Elliptic PDE with exponential nonlinearity

Suppose $\varphi$ is a radial (and radially decreasing) solution of $$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$ with $ R>0$, and $\psi$ is a decreasing radial function ...
Matchmaticians's user avatar
2 votes
1 answer
153 views

Status of the Bressan conjecture

Let me first recall what is the Bressan conjecture. Take a $BV\cap L^\infty$ vector field $X$ on some open subset of $\mathbb R^n$ such that there exists an $L^\infty$ function $\alpha\ge 1$ so that $ ...
Bazin's user avatar
  • 15k
2 votes
0 answers
101 views

A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define $$ H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N. $$ By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
ThiagoGM's user avatar
  • 121
1 vote
1 answer
102 views

Solving a particular delay PDE $\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s)$

I recently encountered a particular delay PDE in my work, the solution of which corresponds to the Laplace transform of some probability distribution. I'm having trouble to solve this equation. The ...
ely's user avatar
  • 13
1 vote
0 answers
229 views

Maximal regularity heat equation

Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate \begin{align*} \forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
Ayman Moussa's user avatar
  • 2,540
3 votes
1 answer
328 views

Existence of solution to linear inhomogeneous first order PDEs systems

Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response. For $i=1,\ldots, r$, ...
A. J. Pan-Collantes's user avatar
4 votes
0 answers
286 views

Pohozaev identity for linear equations

For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says $$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$ Here $u_T$ are the tangential ...
Adi's user avatar
  • 483
2 votes
1 answer
238 views

Generalizing a formula with distributions — Distributional Radon transform

I will try to describe the problem, it will necessarily be incomplete, so please if you have questions or remarks to make it more clear do not hesitate to leave them in comments. The problem Let $a$ ...
metaUser's user avatar
1 vote
0 answers
118 views

Surjectivity of perturbed linear operators

Consider two Banach spaces $X$ and $Y$ and two linear bounded operators $A,B:Y\rightarrow Y$. Suppose the following: (1) Y is reflexive (or even uniformly convex); (2) $X\cap Y$ is dense in $X$ and $Y$...
Eddy's user avatar
  • 11
3 votes
1 answer
372 views

Linear PDE, analytic continuation, Green's function and boundary conditions

I'm looking at the linear PDE in 3+1 dimensions, $$ \left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1} $$ Where $\xi$ is generally a ...
Fetchinson0234's user avatar
1 vote
0 answers
193 views

Fundamental Solution to Biharmonic Equation in 3D

(This is a repost of a question posed in StackExchange that didn't get any replies.) Is anything known about the fundamental solution to the equation: $$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$ for ...
Jap88's user avatar
  • 431
2 votes
0 answers
120 views

On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases} -\Delta ...
student's user avatar
  • 1,320
1 vote
0 answers
63 views

Behavior of Green's function $G(x)$ for $x\to 0$ for general second order PDE

Let's have a generic elliptic second order PDE in $n$-dimensions with a Dirac delta on the right hand side $$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...
Fetchinson0234's user avatar
3 votes
1 answer
199 views

Schauder estimates with boundary conditions

For the elliptic equation with non-divergence form $$ \sum_{i,j=1}^na_{ij}(x)\partial_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1), $$ where $ \{a_{ij}(x)\} $ is a ...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
220 views

Find an integral kernel for the solution of a partial differential equation: an initial value problem

Consider the following partial differential equation with an initial condition $u(x,0)=f(x)$: \begin{equation} \frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\...
Mirar's user avatar
  • 350
1 vote
0 answers
237 views

Has anyone studied the PDE generalization of Teichmüller Space?

We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize). Given a surface $S$ we can define Teichmüller space $T(S)$ to ...
Sidharth Ghoshal's user avatar
3 votes
0 answers
92 views

On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
Pelle Steffens's user avatar
0 votes
1 answer
174 views

Ergodicity question

Consider a dynamical system given by the system of ODE. $$\frac{d x_i}{d t} = F_i(\mathbf{x}).$$ It seems to be a well-known fact that this system is ergodic if and only if the kernel of the Koopman ...
Igor Rivin's user avatar
  • 95.5k
3 votes
1 answer
152 views

Solvability of general linear PDE with constant coefficients

Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$. Is it true that the equation $$Du=f$$ is solvable in any ...
asv's user avatar
  • 21.1k
2 votes
0 answers
42 views

Polynomial solutions of differential equations vs smooth ones

Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words $$Im(D_1)\subset Ker(...
asv's user avatar
  • 21.1k
2 votes
0 answers
99 views

Question about the second order linear elliptic PDE on closed manifold

Recently I see a question linear second order PDE in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE $$ \Delta u +B^{i}(x)u_{i}+C(x)u=f $$ is ...
Elio Li's user avatar
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