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.
4,466 questions
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Does this variant coincide with the usual Hölder space?
$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$
Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$.
The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of ...
- 825
7
votes
1
answer
411
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Limit of zero sets of harmonic functions
Let $u_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly ...
- 73
1
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0
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138
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Wave equation on 2D semi-infinite plane
I'm trying to understand what a correct procedure is for solving the following wave equation on a semi-infinite plane $-\infty < x < \infty$, $0 \le z < \infty$:
$$
\begin{cases}
\nabla^2 \...
- 11
1
vote
0
answers
109
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$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
- 93
8
votes
4
answers
2k
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How to interpret couplings in optimal transport?
Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable ...
- 309
1
vote
0
answers
66
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Solutions for a system of PDE's
Let $ \Omega_s(x) $ solve the system
$$s \dfrac{\partial^2}{\partial s^2}\Omega_s(x)=\pm x\dfrac{\partial}{\partial x}\Omega_s(x) $$
$$2\sqrt{s}\frac{\partial}{\partial s} \sqrt{\pm\Omega_s(x)}=\sqrt{...
- 383
1
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0
answers
113
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Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,115
3
votes
1
answer
305
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Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?
Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...
- 1,759
2
votes
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Understanding the Bochner space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ in terms of the Fréchet derivative
In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$.
...
- 3,477
3
votes
1
answer
183
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Looking for an electronic copy of Kato's old paper
I would like to know if anyone has an electronic copy of the following paper:
MR0279626 (43 #5347)
Kato, Tosio
Linear evolution equations of "hyperbolic'' type.
J. Fac. Sci. Univ. Tokyo Sect. I ...
- 509
12
votes
0
answers
398
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A model of pillows
(The same system with slightly different questions has been asked in MSE.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
- 134
4
votes
1
answer
372
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Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
- 1,759
2
votes
0
answers
143
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Mathematical study of dispersive PDEs [closed]
My understanding is that there have been a lot of activities in harmonic analysis and PDEs that use sophisticated tools to study dispersive PDEs like the Schrodinger's equation. E.g. the Strichartz ...
- 265
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0
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113
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Solving $\frac{\partial}{\partial t}f = h f + h \int h f$
Is there a closed form solution to the following differential equation?
$$\frac{\partial}{\partial t}f(i, t) = a h(i) f(i, t) + b h(i) \int \mathrm{d}i\ h(i) f(i, t)$$
Where $h(i)=C (i+1)^{-p}$ with $...
- 2,252
39
votes
5
answers
5k
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Explicit eigenvalues of the Laplacian
Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...
- 1,101
2
votes
1
answer
237
views
Approximation of Hölder functions by Fourier series
Let $Q$ be a bounded domain in $\mathbf R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$.
Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2\...
- 21
5
votes
1
answer
307
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A detail in one step in a theorem from a paper of Brezis and Merle
In paper [1] Brezis and Merle prove theorem 3 by using the following fact. Let $w_n=u_n-v_n$, $\Delta w_n=0$ on $\Omega$ (a bounded domain in $\mathbb{R^2}$) and $w_n^+$ is bounded in $L^{\infty}_\...
- 53
2
votes
1
answer
181
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Parabolic Schwarz lemma
Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
- 59
2
votes
2
answers
755
views
Derivative of the absolute value
Let $f \in W^{1,p}(U)$, then how to prove that $|f| \in W^{1,p}(U)$, where $W$ means the sobolev space over some open subset $U \in \mathbb{R}^n$.
In Lieb's Analysis he prove that Let $f$ be in $W^{1,...
- 23
9
votes
2
answers
777
views
Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
- 5,380
4
votes
1
answer
487
views
Nonsmooth version of Hopf boundary point lemma
Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\...
- 5,380
2
votes
0
answers
68
views
Existence of Weak solution of a PDE with singular term
On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
- 677
2
votes
1
answer
164
views
Optimal assumption on H^2 regularity
In many text book (Evans, Gilbarg-Trudinger for example) there is a classical result of interior regularity for weak solutions to a elliptic divergence problem $\rm{div}(A(x)u)=f$ in $\Omega\subset\...
- 81
2
votes
0
answers
78
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Verify the explicit solution formula for a degenerate Fokker-Planck equation $\partial_t u = \nabla\cdot(D\,\nabla u + Cxu)$
Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$
$$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$
where $D \in \mathbb{R}^{d \times d}$ ...
- 730
2
votes
0
answers
124
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Feynman-Kac for PIDEs: to jump or not to jump?
Consider the following Cauchy problem for a $\mathscr{C}^2$ function $F$ characterized by a PIDE:
\begin{align}
\begin{cases}
& F_t(t,x)+\alpha(t,x)F_x(t,x)+\frac{1}{2}\beta^2(t,x)F_{xx}(t,x)
\\
&...
- 171
2
votes
0
answers
197
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Trouble understanding Lax method for KDV equation for inverse scattering method
I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
- 21
6
votes
1
answer
410
views
Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrodynamics?
Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics?
Tried my luck with Google's search engine, didn't show much info.
I guess you can try to use these features ...
- 1,594
3
votes
2
answers
248
views
What is standard continuity argument for well-posedness?
Motivation: I'm trying to understand the proof of Theorem 3.1 in Antonelli, Saut, and Sparber - Well-Posedness and averaging of NLS with time-periodic dispersion management. Though in the following I'...
- 325
5
votes
0
answers
151
views
weak maximum principle for weighted laplacian
Consider a radial weight $w=|x|^2 \geq 0$ for all $x\in \mathbb{R}^n$ and consider the operator
$$Lu= \frac{1}{w}\operatorname{div}(w\nabla u).$$
Then if $-Lu\leq 0$ on a smooth bounded domain $\Omega$...
- 537
4
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0
answers
137
views
Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
- 2,095
2
votes
1
answer
483
views
What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?
Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE,
$$
d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0
$$
where $\left(W_t\right)_{t \geqslant 0}$ ...
- 537
3
votes
1
answer
318
views
Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?
I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) ...
- 547
1
vote
0
answers
88
views
Pointwise convergence of Schrodinger's equation with potential term
A famous problem of Carleson asks if $f\in H^s(\mathbb{R}^n)$, under what condition of $s$ do we have almost everywhere pointwise convergence of the solution to the Schrodinger's equation
$$iu_t-\...
- 265
2
votes
0
answers
141
views
A question about Gauss-Green formula - a weaker assumption
The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place
$$\...
- 1,759
1
vote
1
answer
147
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comparison principle for the minimal surface equation
Consider the (inhomogeneous) minimal surface equation for functions $u,f:D\to \mathbb{R}$ for some smooth domain $D\subset \mathbb{R}^n$
$$Lu:=\operatorname{div} \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=...
- 537
5
votes
1
answer
222
views
Domains with discrete Laplace spectrum
Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
- 443
1
vote
0
answers
82
views
Finiteness of theta vanishing in the KP direction for locally planar curves
I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...
- 318
3
votes
0
answers
101
views
Reference on Pohozaev identities?
I know Pohozaev-type identities are important tools for many PDEs including nonlinear, nonlocal problems, for showing nonexistence and uniqueness. However, is there any elementary reference on ...
2
votes
1
answer
197
views
Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
- 43
1
vote
0
answers
224
views
For the solvability of the poisson equation $\Delta u = f$ on manifold with boundary
For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...
- 809
2
votes
0
answers
82
views
An adaptive stepsize approach to solve numerically ODE with stiffness using complementarity conditions
Let us considered the following system of ODEs
\begin{align*}
\dfrac{dX}{dt} = f(X), \tag{1.1}
\end{align*}
where the unknown $X\in \mathcal{D} \subset \mathbb{R}^l$ and it is stiff. However, for ...
- 121
1
vote
0
answers
293
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...
- 61
2
votes
0
answers
72
views
Doubt on regularity at "Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition"
In the paper Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition by Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, in Chapter 2 there is a construction of a ...
0
votes
1
answer
414
views
Sufficient conditions for an asymptotic compactness
This question relates a theory of Mosco convergence.
Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$.
A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
- 721
0
votes
0
answers
176
views
Difficulty in understanding one step in Ciraolo-Figalli-Maggi paper
I am in trouble in understanding one step of theorem 1.1 in the paper by Ciraolo-Figalli-Maggi: arXiv link, link at Figalli's page.
Namely in equation \eqref{1}, it is written: "Since $u=\sigma+\...
- 337
3
votes
0
answers
102
views
Can Sobolev space be characterized by spectral decomposition?
Consider a homogeneous Carnot group $\mathbb{G}$ with step $r$. Let $X_1,\cdots,X_m$ be the first layer of its Lie algebra. Denote by $\mathcal{L}=-\sum_{i=1}^m X_i^2$ the sub-Laplacian on $\mathbb{G}$...
- 561
15
votes
2
answers
2k
views
Reference request: the theory of currents
I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
- 151
1
vote
0
answers
94
views
Nonlocal elliptic problem - what is its associated energy?
It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...
- 1,759
3
votes
1
answer
260
views
$W^{1,p}$ ($1\le p<2$) uniqueness of elliptic equations
Let $B$ denote the $n$-dimensional unit ball. Assume $u\in\bigcap_{1\le p<2} W_0^{1,p}(B)$ satisfies $$
\int a_{ij}\partial_j u \partial_iv =0,$$ for any $v\in C_c^{\infty}(B)$, where we assume $a_{...
- 435
0
votes
0
answers
81
views
Is it possible to continuously embed $C^\infty(\mathbb{T}^n)$ as a vector space into $\mathcal{D}(\mathbb{R}^n)$ by some "inverse" of periodization?
Let $\mathbb{T}^n$ be the $n-$dimensional torus and $C^\infty(\mathbb{T}^n)$ be the Frechet space of smooth periodic functions on $\mathbb{R}^n$.
According to p.298 of Folland "Real Analysis"...
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