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.
4,271
questions
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Looking for a proof of a geometric regularity criteria: generalization of the exterior cone condition / Zaremba's criterion
The topic is Perron's method for the Dirichlet problem.
I am looking for a proof of the following statement:
Let $\Omega$ be an open bounded set in $\mathbb{R}^n$ with $n \geq 3$ and $0 \in \partial \...
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votes
0
answers
75
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elliptic pde question
Suppose $\Omega$ is a smooth bounded domain in $ R^N$ and suppose we have
$$-\Delta u(x) = (u(x)_+)^3 \qquad \mbox{ in } D$$
and $$ -\Delta u(x) = (u(x)_-)^3 \qquad \mbox{ in } \Omega \backslash \...
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vote
0
answers
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Time dependent reaction-diffusion semigroup
I'm interested in the following linear reaction-diffusion equation
\begin{align*}
&\partial_tu(t,x) = \sigma(t)\Delta u(t,x),\\
& u(0)=u_0\in X
\end{align*}
where $X$ is a Banach space and $\...
- 60.2k
1
vote
0
answers
74
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Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 2,758
7
votes
1
answer
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A diffeomorphism of the torus with constant singular values
Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Does there exist an area-preserving ...
- 6,621
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Global existence of $L^p$-solutions to a quasilinear diffusion equation
We consider the diffusion problem
$$\begin{cases}
\partial_t u = \nabla \cdot (a(u)\nabla u), \quad t>0, x \in \mathbb{R}^n \\
u(0) = u_0
\end{cases}$$
for functions $u \colon [0,T] \times \mathbb{...
- 101
1
vote
0
answers
64
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wave equation with $H^{-1}$ source
Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation
$$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$
with $u|_{(0,T)\times ...
- 4,089
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vote
0
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Fractional Laplacian extension problem and uniqueness question
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
- 11.4k
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Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
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1
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0
answers
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A question about extension problem related to fractional laplacian
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...
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Eigenfunctions of the fractional Laplacian are smooth?
Let $\Omega\subset\mathbb{R}^n$ open, bounded with smooth boundary, let $s\in(0,1)$. I know that the fractional Laplacian has a sequence of eigenfunctions $\{e_k\}_{k\in\mathbb{N}}\subset H^s(\mathbb{...
- 51.6k
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1
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The uniqueness of Barycenters in the Wasserstein space
I am reading the paper Barycenters in the Wasserstein space by Martial Agueh and Guillaume Carlier. In Proposition 3.5, they prove the existence and uniqueness of
$$\nu \mapsto \sum_{i=1}^p \frac{\...
2
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0
answers
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Solvability of a PDE involving the Dirichlet-to-Neumann operator
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric).
Let $N: L^2(\...
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vote
1
answer
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Solving an equation with fractional laplacian [closed]
Let $s\in (0,1)$, how i can solve the equation:
$$ (-\Delta)^su=1,\quad\text{in}\quad(-1,1)?$$
I have no idea, any help would be appreciated.
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votes
1
answer
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Is there a metric on Euclidean space that turns the Helmholtz equation into the Laplace equation?
Is there a Riemannian metric $\tilde g$ on $\mathbb R^d$ such that
$$\tag{1}
\Delta_{\tilde g}=e^f(\Delta +1),$$
for some $f\in C^\infty(\mathbb R^d)$? Here $\Delta=\partial_{x_1}^2+\ldots+\partial_{...
- 672
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0
answers
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Continuity of weak solutions to wave equation with time-dependent coefficients
Consider the following second-order wave equation
$$
u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega
$$
with boundary conditions
$$
u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
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0
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Reference request: PDE of the form $(\Delta - |u|^2)f = F(u)$
I am interested in equations of the form
$$(\Delta -|u|^2)f = F(u)$$
where $F$ depends on $u$ and preferably on its derivative, too. $u$ is supposed to be given and $f$ the unknown. More precisely I ...
- 469
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Determination of the nature of stationary values in variational calculus
In variational calculus, when we solve the Euler-Lagrange equation $\frac{d}{dx}L_p(u',u,x)-L_z(u',u,x)$, where $L=L(p,z,x)$, to find stationary inputs of the functional
$$
I[u]=\int_0^1 L(u',u,x)dx,
$...
- 1,641
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votes
3
answers
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Posing Cauchy data for the heat equation: $t=0$ a characteristic surface?
When solving the heat equation on say $\mathbb{R}$ (or $[0,2\pi]$, whichever is easier to talk about) we are posing Cauchy data on the surface $t=0$. My understanding is that $t=$constant are ...
- 51.6k
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votes
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299
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Comparison of the spectrum decomposition
In the spectral theorem, we learned that the spectrum of a linear operator $A$ is a disjoint union of: point spectrum (eigenvalues), continuous spectrum (the kernel of $zI-A$ is zero, the closure of $\...
- 11.6k
4
votes
1
answer
510
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Decay estimate on wave equation
In this paper here it is claimed in (1.3) that it is classical and immediate from the explicit solution of the wave equation in 3D
$$(\partial_t^2 -\Delta )u(t,x)=0$$
with $u(0,x)=0$ and $u_t(0,x)=g(x)...
- 37.6k
1
vote
0
answers
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views
Reference request: existence/uniqueness of solutions to convection diffusion equations
I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form
$$
\frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...
- 111
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2
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Examples of applications of hyperbolic conservation laws
I am giving a talk in front of my applied PDE research group on hyperbolic conservation laws, the most basic form of which is the PDE $$ u_t + f(u)_x = 0 $$ where $u$ is the conserved quantity and $f$ ...
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3
votes
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answer
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wave equation with vanishing trace at infinity
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
- 37.6k
1
vote
0
answers
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views
Do the solutions of parabolic PDE problems with different initial conditions converge to each other?
Let's say we have a parabolic PDE system:
$$
(PDE) \hspace{0.5cm} u_t+f(u)_x=\mu \cdot u_{xx},
$$
where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n, \: n\geq 2$. And let's ...
- 647
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answers
108
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Examples/applications of parabolic PDEs that are not posed on domains or manifolds
Are there any examples of parabolic PDEs
$$u' - Au = f$$
posed in a Gelfand triple setting $V \subset H \subset V^*$ with $V$ and $H$ chosen NOT as spaces of functions (or distributions) over a domain ...
0
votes
0
answers
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Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
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Semiclassical analysis and reflection law
I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
- 2,758
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votes
1
answer
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Gradient of a function defined on a Riemannian-manifold
If I have a smooth positive scalar function $h$ defined on a 2-dimensional manifold $M$, then $h:M\rightarrow (0, \infty)$, where the metric of $M$ is $g=\frac{dx^2+dy^2}{y^2}$.
$h$ must satisfy the ...
- 106k
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votes
1
answer
265
views
Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...
- 123
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votes
0
answers
235
views
Singularity of L^1-solutions to elliptic PDEs on the puntured ball
Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^n$. Then it is true that if $u\in L^1(\mathbb{B})$ such that $\Delta(u)=0$ on $\mathbb{B}\backslash\{0\}$, then $\Delta(u)$, as a distribution on $\...
- 562
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votes
3
answers
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Fractional Laplacian of radially symmetric functions
For a "good" function $u$, I consider its (Gagliardo) fractional Laplacian ($0<s<1$)
$$
(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy,
$$
at least as a ...
- 2,758
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votes
0
answers
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views
Elliptic equations in semi-infinite strips
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
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Green operator of elliptic differential operator and radius of convergence
Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
- 1,373
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votes
1
answer
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An inequality involving fractional Laplacian
I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$
...
- 16.3k
7
votes
2
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Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs
Where can I find a (readable and self-contained) proof of the following result?
Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
- 5,632
2
votes
1
answer
170
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Positive subharmonic functions with constant integral blowing up at boundary
Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying
$\Delta f_n\ge 0$ (subharmonic)
$f_n\ge 0$
$\int_\Omega f_n=I>0$ ...
- 88.8k
2
votes
0
answers
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Mixed boundary value problems for Heat equation
This might be a very simple question, but basically I am looking for a good reference for studying the heat equation on Riemannian manifolds with boundary, specifically when data is put on lateral ...
- 4,089
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vote
0
answers
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Well-posedness for a wave equation with degenerate coefficients
Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem:
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
t\partial_t(t\partial_t u)-\...
- 4,089
3
votes
1
answer
126
views
Boundary regularity of rectifiable multiplicity 1 hypercurrents
Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
- 4,968
4
votes
0
answers
109
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Auxiliary spaces/conditions for orbital stability of traveling waves
In the context of orbital stability, probably one of the most used theorem to show the orbital stability of traveling waves is the one from Grillakis-Shatah-Strauss "Stability theory of solitary ...
2
votes
1
answer
203
views
Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)
Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...
- 5,632
2
votes
0
answers
181
views
Lions/diPerna type commutator estimates for differential operator in Fokker-Planck type equation
I have a question about a particular commutator estimate as it occurs in the study of Fokker-Planck equations with low regularity data, see e.g. [1,2].
Denote by $\rho_\varepsilon$ some usual ...
- 2,195
2
votes
1
answer
197
views
Unique solution of a 1-D ODE with a bounded positive right-hand-side
Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^...
- 5,632
3
votes
1
answer
236
views
Discovery of norm in PDE
We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $C^k$, i.e. $C^{k,\alpha}$. Roughly speaking, under appropriate norm, we could capture the ...
- 22.6k
6
votes
2
answers
2k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 153
3
votes
0
answers
215
views
Does the minimal surface system in the plane have the weak unique continuation property?
Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...
- 1,141
4
votes
1
answer
321
views
Some question about the spectral function of Laplace operator on $\mathbb{R}^n$
I am studying the paper of Seeley, R., A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of (R^3), Adv. Math. 29, 244-269 (1978). ZBL0382.35043. There are some ...
- 52.4k
-1
votes
1
answer
113
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Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
- 117k
3
votes
0
answers
226
views
Singular integral operator
I am working on a problem involving the Biot-Savart law in fluid dynamics. I found a theorem of singular integral which is intimately related to my research.
Assume that $K(x)$ is a classical Calderon-...
- 151