Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,468 questions
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Riesz transform after linear transformation
I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation
$$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$
I ended up with ...
- 188k
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Asymptotic behaviors of different types of equations with capillarity or viscous terms
Let us consider
$$
\begin{align}
u_t + \left(\frac{u^2}{2}\right)_{\!\!x} = 0 \\
v_t + \left(\frac{v^2}{2}\right)_{\!\!x} - v_{xx} = 0 \\
\tilde v_t -\tilde v_{xx} = 0 \\
w_t + \left(\frac{w^2}{2}\...
- 6,367
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"Euler system" in Christodoulou's The Action Principle and PDEs
In The Action Principle and PDEs Christodoulou spends some time describing what he calls the Euler system associated to a system of variational PDEs (sections 2.5-7, 6.2). Briefly, given a bundle $E\...
- 1,446
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The module generated by kernel of an elliptic differential operator
Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\...
- 356
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votes
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answers
141
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Parabolic maximum principle for non-compact manifold with boundary
Let $M:=\mathbb{R}^n\setminus \mathbb{D}^n$, where $\mathbb{D}^n$ is the open unit ball in $\mathbb{R}^n$ and $u\colon M\times [0,\infty)\rightarrow \mathbb{R}$ is solution of the following PDE
\begin{...
- 632
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Reference request: a PDE related to Kahler–Ricci flow
I was reading the survey by Imbert and Silvestre where I noticed the PDE
$$
\frac {\partial u} {\partial t} = \ln(\det (D^2u))
$$
for the study of the Kahler–Ricci flow (Eq (2.2) at page 10 in ...
- 6,367
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Semilinear elliptic equation
Assume $u$ is a smooth solution for
$$
\Delta u + f(u)=0\qquad \hbox{in}\quad \Omega
$$
and $\Omega$ is a smooth convex domain in $\mathbb{R}^n$.
Is there a conjecture which are the weakest conditions ...
- 329
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votes
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answers
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Estimates for higher order derivatives of the Airy Kernel
Consider the kdv equation (from here)
$$\left\{\begin{array}{l}
\partial_{t} v+\partial_{x}^{3} v=0 \\
v(x, 0)=v_{0}(x)
\end{array}\right.$$
Its solution can be written as $v(t,x)=S_t*v_0(x),$ where $...
- 537
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Gradient estimate and $L^1$ theory for the Laplace operator
Let $\psi \in C^{\infty}_{c}(\Omega)$ where $\Omega$ is a bounded smooth domain, and $\phi$ the solution to
\begin{equation*}
-\Delta \phi =\psi, ~\phi|_{\partial \Omega}=0.
\end{equation*}
My ...
- 63.9k
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votes
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answers
210
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Bounding integral expression with total variation of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$
for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
- 128k
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Sign changing elliptic problem
Take $B_1$ the unit ball in Euclidean $N$ dimensional space and suppose $3 \le N \le 10$ and take $ 1<p< \frac{N+2}{N-2}$. By some abstract theory there is a infinite sequence of smooth radial ...
- 1,385
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Is there a general method for determing the domain of dependence of (higher-order) PDEs?
Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around,...
- 39.1k
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votes
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Kernel for an equation involving the Ornstein-Uhlenbeck operator
Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$
\begin{align}
\Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\
u&=0 \text{ on }\partial \Omega
\end{align}
Are ...
- 6,035
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Literature on reaction diffusion equations
My research area is age structure modelling, basically when applied to reaction diffusion equations. We mainly discuss the existence of travelling wave solutions; I want to work on the stability of ...
- 6,367
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150
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Analyticity of solutions to Schrödinger's equation
Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
- 439
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369
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Hamiltonian, energy, and conservation laws of nonlinear PDEs
In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
- 188k
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Asking for reference about a relation related to Fourier transform [closed]
Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$
Could ...
- 3,808
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Viscous approximation of Eikonal equation
Consider the Eikonal equation
\begin{align*}
\begin{cases}\left|D u\right|^{2}=1 & \text { on } \Omega \\ u \equiv 0 & \text { on } \partial \Omega\end{cases}
\end{align*}
and the viscous ...
- 595
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Solution to the differential equation $ \langle \nabla g(X), \overline{\nabla g(X)} \rangle =-\frac{\partial g(X)}{\partial x_{1}} $?
If $ \vec{u} $ (respectively $ \vec{v} $) is an element of $ \mathbb{C}^{n} $ with components $ u_{i} $ (respectively $ v_{i} $) and $ \langle \vec{u},\vec{v} \rangle $ is the Hermitian inner product $...
- 39.1k
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votes
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answers
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Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
CommunityBot
- 1
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125
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Bounding integral expression with Sobolev norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
CommunityBot
- 1
1
vote
0
answers
33
views
Limiting absorption principle for higher powers of resolvents
Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator ...
- 141
3
votes
1
answer
217
views
The energy of a semilinear ODE
I'm currently reading Caffarelli, Gidas, Spruck's paper "Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth". For some background, we ...
- 39.1k
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An inequality involving weight $|x|^\alpha$
Background:
Let $u:\mathbb{R}^n\to \mathbb{R}$. Then the paper considers the following problem
\begin{align*}
-\operatorname{div}(w(x) \nabla u) &= w(x) \text{ in } \Omega \\
...
- 537
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vote
0
answers
104
views
Regularity of the Robin function
I consider an analytic bounded domain $\Omega\subset \mathbb R^3$ and an the operator $L_a=-\Delta +a$ where $a$ is a function from $\Omega$ to $\mathbb R$. I assume the operator to be coercive, in ...
- 914
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votes
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answer
240
views
Bernstein's corollary for the case of half space
The early seminal result of Bernstein in 1914 for $n=2$ is the well-known Bernstein theorem:
The only entire solutions to the minimal surface equation in $\mathbb R^3$ are the affine functions
$$u(x,...
- 153
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votes
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answers
71
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Fundamental solutions for weighted laplace equation
Consider the equation $L_w u = \frac{1}{w}\operatorname{div}(w\nabla u) =f(x)$, on $\mathbb{R}^n$ with radial weights $w(x)=w(|x|).$ Then I am interested in the fundamental solutions for the operator $...
- 537
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votes
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A constant ratio of integrals? Part II
This question is a follow up on my latest MO post which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context ...
- 128k
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votes
1
answer
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A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
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Question about higher order mean field equation $\left(-\Delta_{g}\right)^{m} u+\lambda=\lambda \frac{e^{2 m u}}{\int_{M} e^{2 m u} d \mu_{g}}$
I'm reading Dr.Luca Martinazzi's paper
Existence of solutions to a higher dimensional
mean-field equation on manifolds
which proves that for $m \geq 1$, there is an existence result for the equation
$$...
- 809
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votes
1
answer
631
views
Uniqueness of Kantorovich potentials?
$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth.
Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
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votes
1
answer
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views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 89
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
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votes
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answers
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$H^s$ norm of non-integer power of functions
Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
My ...
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Parseval-Plancherel identity involving two functions and Riesz transform
Let $J^s = (\mathbb I - \Delta)^{s/2}$, and, $\hat{\mathcal{R}_x u}(\xi)=\frac{-i \xi_1}{|\xi|} \hat{u}(\xi)$, and $u \in H^\infty(\mathbb{T}^2)$, where $u:(\mathbb R, \mathbb{T}^2) \to \mathbb C.$
...
- 2,459
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66
views
Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463
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votes
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241
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Dense subspaces of the Hardy space $H^1$
Let $H^1$ be the Hardy space on $\mathbb{R}^n$, defined e.g. as the set of $u\in L^1$ such that $Ru\in L^1$, where $R$ is the Riesz transform on $\mathbb{R}^n$. It seems to me that simple functions ...
- 9,007
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votes
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answers
135
views
Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
- 235
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votes
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answer
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Asymptotics of integral representation of distribution
I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
- 870
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answers
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Existing work on $\Delta u=c-h e^{u}$ on compact manifold with dimension n, I have read J.Kazdan's work, the condition $c > 0, n \ge 3$ is not solved
I'm reading Prof. Kazdan's lectures
At page 69, Prof. Kazdan describes the research on the $\Delta u=c-h e^{u}$ PDE on a compact $n$-dimensional manifold before 1983. (Here $c$ is a constant while $h$ ...
- 809
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votes
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answer
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A question about the commutator $[J^s,u]\partial_x u$
I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that this paper contains a possible typo. Here is the archive version which ...
- 159
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votes
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answers
641
views
Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
- 9,007
-1
votes
1
answer
87
views
Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$
I am having the following integral:
$$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$
where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$,
$...
- 39.1k
3
votes
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answers
188
views
Laplace equation in a domain with a hole
I want to know if there is some reference on the well-posedness of the following problem :
For $\Omega$ a bounded domain (what results depending on the regularity of the domain?) of $\mathbb{R}^2$ (...
7
votes
1
answer
405
views
Convex solutions of the Poisson equation
Let $D$ be a planar, bounded, convex open domain. Given a positive function $f:D\to(0,+\infty)$, let us consider the Poisson equation
$$\Delta u=f\quad\hbox{in }D.$$
Not specifying any boundary ...
- 52.3k
2
votes
0
answers
104
views
Relations between different "propagation of chaos" type results?
My questions come from the paper Logarithmic Sobolev inequalities for some
nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...
- 730
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votes
1
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380
views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
CommunityBot
- 1
1
vote
1
answer
472
views
Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space
I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...
- 6,045
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votes
0
answers
158
views
Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
- 1,334
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votes
0
answers
106
views
Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods
Consider the Cahn-Hilliard equation
$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$
defined on your favorite domain. I'm looking for a literature reference that formally ...
- 53