# 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

3
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### Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$

Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...

1
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0
answers

33
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### Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?

Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...

4
votes

3
answers

439
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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 $t$...

0
votes

1
answer

164
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### 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 $...

7
votes

2
answers

477
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 ...

1
vote

1
answer

170
views

### 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{...

4
votes

1
answer

175
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### 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\\
...

0
votes

0
answers

87
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 $...

1
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0
answers

205
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### Understanding the effect of PDE solution on critical strip?

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 ...

3
votes

0
answers

58
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
$$...

1
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0
answers

118
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 ...

1
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0
answers

31
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### 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 ...

1
vote

1
answer

111
views

### 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 ...

1
vote

0
answers

73
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 ...

2
votes

0
answers

124
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### 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 ...

0
votes

0
answers

30
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 $1<p<\infty$...

4
votes

2
answers

257
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 ...

5
votes

1
answer

185
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 ...

1
vote

1
answer

266
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 $\dot W^{1,p}(\mathbb{R}^n)$,...

4
votes

2
answers

764
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### 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 ...

3
votes

0
answers

59
views

### 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 ...

1
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0
answers

35
views

### 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 \...

1
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0
answers

38
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 ...

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=...

4
votes

1
answer

148
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{\...

2
votes

1
answer

146
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 \...

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^...

2
votes

0
answers

155
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 ...

5
votes

0
answers

346
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)&...

4
votes

0
answers

107
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 ...

3
votes

0
answers

95
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 ...

2
votes

1
answer

164
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
$
...

2
votes

0
answers

104
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_\...

1
vote

1
answer

103
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 ...

1
vote

0
answers

250
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 \...

3
votes

1
answer

350
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 $i=1,\ldots, r$, ...

4
votes

0
answers

295
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 ...

2
votes

1
answer

242
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 $a$ ...

1
vote

0
answers

122
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### 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 $Y$...

3
votes

1
answer

378
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 ...

1
vote

0
answers

215
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 ...

2
votes

0
answers

131
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 ...

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
$$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...

3
votes

1
answer

211
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 ...

1
vote

1
answer

226
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{\...

1
vote

0
answers

243
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 ...

3
votes

0
answers

97
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 ...

0
votes

1
answer

175
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 ...

3
votes

1
answer

163
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 ...

2
votes

0
answers

43
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(...