All Questions
Tagged with ap.analysis-of-pdes linear-pde
303 questions
0
votes
0
answers
35
views
Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 455
0
votes
0
answers
17
views
Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 455
11
votes
0
answers
317
views
+50
Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 13k
0
votes
0
answers
87
views
Curl-Div equation with singular matrix
I want to solve the equation:
$$
\begin{cases}
\nabla \times (A \mathbf v)=f, \quad x\in \Omega \\
\operatorname{div}(\mathbf v)=0,
\end{cases}
$$
where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
6
votes
1
answer
1k
views
Regularity of solution to Fokker Planck equation
Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t =...
CommunityBot
- 1
3
votes
1
answer
86
views
$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line
I consider the following problem on the half real line
$$
\begin{cases}
u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm]
u|...
- 6,367
1
vote
0
answers
23
views
Uniform bound on the first moment for a perturbed advection-diffusion equation
I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line:
$$
\begin{cases}
u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x = ...
- 63.9k
1
vote
0
answers
39
views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
3
votes
1
answer
212
views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation:
$$
\begin{cases}
\nabla \times (F\times\mathbf v)=g, \\
\operatorname{div}(\mathbf v)=0,
\end{cases}\label{1}\tag{1}
$$ where $F$ and $g$ are given vector fields. The ...
- 12.9k
5
votes
0
answers
112
views
Fredholm index of degenerate elliptic PDE
We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$:
$$
L(u):= -\operatorname{div}(a \, \nabla u) = 0,
$$
where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$...
- 363
6
votes
2
answers
250
views
Minimal assumptions for existence of solutions of First order PDE
I'm looking for a reference about existence of linear homogeneous first order PDE, in particular about the minimal assumption on the data. In literature I found that one require $C^1$-regularity on ...
- 16.2k
4
votes
0
answers
92
views
Well-posedness for linear transport equations with fractional diffusion term
I have a rather applied problem where I consider an equation of the form
$$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$
...
- 206
9
votes
2
answers
493
views
Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
4
votes
1
answer
308
views
A certain solution for Sine-Gordon Equation
I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
- 21.5k
5
votes
2
answers
267
views
Positivity for the mild solution of a heat equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 6,367
11
votes
3
answers
3k
views
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a ...
- 6,367
1
vote
0
answers
120
views
Well-posedness result for a linear parabolic equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 185
0
votes
0
answers
40
views
Multivariable Variation of Parameters $\psi_1(x, k)\int^{(x,t)}e^{ik^3(t-\tau)}[M_1(\xi,k)q(\xi,\tau)d\xi-X_1(\xi,\tau,k)d\tau]+\dots$
I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here). For greater clarity, I split my question up into two ...
- 547
0
votes
0
answers
37
views
Finding a particular solution to a Lax pair on $[0, \infty)$ to solve $q_t + q_{xxx} + u(x)q = 0$ using the Fokas Method
I am trying to read through the paper titled IBV problems for linear PDEs with Variable Coefficients by P. A. Treharne and A. S. Fokas (link here).
This post is a follow up to my first that ...
- 547
0
votes
0
answers
69
views
Inside and up to boundary regularity improvement of linear differential operator
I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$
$$A=\sum a_\alpha\...
- 421
0
votes
0
answers
46
views
Uniqueness results for linear first order systems of PDEs
Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$):
$$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
- 1,205
2
votes
0
answers
238
views
What is the fundamental solution for the backward heat equation?
According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
- 63.9k
0
votes
0
answers
62
views
Uniqueness problem of constant coefficient differential operator with given boundary information on compact domain
I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have
\begin{equation}\label{...
- 421
4
votes
3
answers
473
views
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 $t$...
- 141
1
vote
0
answers
211
views
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 ...
- 383
4
votes
2
answers
481
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 ...
- 468
1
vote
0
answers
40
views
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 $...
- 515
0
votes
1
answer
217
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 $...
- 111
7
votes
2
answers
567
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 ...
- 21.5k
1
vote
0
answers
400
views
Calculating frequency of sound of ringing metal coin
I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
- 5,407
0
votes
0
answers
111
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 $...
- 119
3
votes
0
answers
65
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
$$...
- 51
1
vote
0
answers
131
views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
1
vote
0
answers
32
views
Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation
I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO).
Consider the following initial boundary value problem for the linear ...
- 6,367
1
vote
1
answer
149
views
Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
3
votes
1
answer
195
views
The maximum in the Poisson problem on the cube with constant source
Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial (...
- 1,446
2
votes
0
answers
153
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 ...
- 51
1
vote
0
answers
106
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 ...
- 11
4
votes
2
answers
283
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
238
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 ...
- 2,494
11
votes
1
answer
1k
views
Research topics in microlocal analysis
Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
- 63.9k
2
votes
1
answer
163
views
Justification for uniqueness of solutions to dispersive PDE
For the sake of concreteness, we consider the linear Schrödinger equation
$$
\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).
$$
The solution is typically obtained by taking the Fourier transform ...
- 21
2
votes
1
answer
391
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 $\dot W^{1,p}(\mathbb{R}^n)$,...
- 2,155
4
votes
2
answers
781
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 ...
- 1,904
3
votes
0
answers
62
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 ...
- 808
1
vote
0
answers
38
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 \...
- 317
1
vote
0
answers
43
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 ...
- 1,219
4
votes
1
answer
168
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{\...
- 684
2
votes
1
answer
258
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 $a$ ...
- 6,367
2
votes
1
answer
172
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 \...
- 108k