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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Looking for an electronic copy of Holmgren's old paper
I would like to know if anyone has an electronic copy of the following paper:
"Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen. Översigt Vetensk. Akad. Handlingar 58, ...
2
votes
0
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From some priori estimates can we estimate higher Sobolev norm?
Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that
$$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$
where $u|_{\partial\Omega}=\phi$.
Can we ...
- 39.1k
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vote
0
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Regularity results for non uniform elliptic equation
I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
3
votes
0
answers
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A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
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0
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Regularity with explicit bound
Let $\Omega$ be an open, bounded with $C^2$ boundary (or smooth as we want). A result about elliptic regularity is given as follows. If $\Omega_0\subset\subset \Omega$ and $u$ is a weak solution of $...
- 375
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Conditions of parameters to have bounded solution of Dynkin's equation in exit problem
Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
- 35.5k
3
votes
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Temporal decay of solutions to fractional porous medium type equations
I am interested in the following family of gradient flows on $\mathbb{R}^d$, $d\geq 1$:
$$\begin{cases}\partial_t u = \nabla\cdot(u\nabla|\nabla|^{-\alpha} u) \\ u\geq 0,\end{cases} \qquad d>\alpha&...
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1
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Explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)$
Do we know an explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)?$
- 34.7k
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Reaction-diffusion systems treated as dynamical systems
I wonder if there is a good reference on reaction-diffusion systems on $\mathbb{R}^N$, that treats them as dynamical systems.
I have the books of Alain Haraux – Systèmes dynamiques dissipatifs et ...
- 1,759
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Existence of the derivative of functionals of Brownian motion
Let $v(x, t) = \mathbb E [f(x + W_t)]$ with a Brownian motion $W$. Then, Malliavin calculus leads to the sensitivity in $x$:
$$\partial_x v(x, t) = \frac{1}{t} \mathbb E [ f(x + W_t) W_t ].$$
I am ...
- 128k
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2
answers
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Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 17.2k
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votes
1
answer
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Log-concavity of the modified Bessel function of a second kind
I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
- 17.2k
3
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Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
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How to show that $\Delta W \leq −2(n − 4)V$?
I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below,
...
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1
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Plummer and Coulomb kernel for the Poisson equation
Consider the $d$-dimensional Coulomb "kernel" defined by:
\begin{equation}
x \in \mathbb{R}^{d} \mapsto g(x):=\left\{\begin{array}{ll}
\log \frac{1}{|x|} & \text { if } d=2 \\
\frac{1}{|...
- 91.8k
2
votes
1
answer
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Liouville theorem for fractional Laplacian
Is there any Liouville type theorem for the half space problem
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s v &= f(v) &&\text{in } \mathbb R^N_+\\
v & =0 &&\...
- 407
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0
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Difference between two fractional Schrödinger equations
Let us consider the fractional Schrödinger equation with periodic boundary conditions
$$
\begin{cases}
iu_t\mathbf{+}(-\Delta)^{\alpha}u= \pm |u|^2u,\; x \in \mathbb{T}, t \in \mathbb{R}_+\\
u(x,0)=...
- 63.9k
6
votes
1
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Space of solutions to a fourth order wave equation
I'm interested in finding solutions a fourth order version of the standard wave equation in $d$ dimensional Minkowski spacetime $\mathcal{M}^d$. Defining $\Box := \partial_0^2 - \sum_{i = 1}^{d-1} \...
- 21.6k
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votes
1
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Two types of limits of viscosity solutions
I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to ...
CommunityBot
- 1
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0
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Existence and properties of the solution of a type of PDE
In doing optimal control of Parabolic PDE's we often have to solve a problem like this:
$$\begin{cases} \dfrac{\partial y}{\partial t}-d\Delta y(t,x)=f(y(t,x),p(t,x)) & (t,x)\in (0,T)\times\Omega ...
- 1,759
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A counterexample to regular boundary points for minimizers of variational integrals under subquadratic growth
Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\...
- 347
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1
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Interpolation for Sobolev spaces
How one can identify the following (complex) interpolation space
$$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$
where $\Omega$ is a regular domaine. After research, it seems that ...
CommunityBot
- 1
3
votes
1
answer
302
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Convergence of a level set when $f^n\to f$ in $C^1$ sense
Let $f^n$ be a family of $C^1$ functions and $f(x)=-|x|^2+1$ such that
$$f^n\to f$$
in $C^1$ sense as $\varepsilon\to 0$. I want to ask that does the level set $\{f^n=0\}$ converges to $\{f=0\}$ in ...
CommunityBot
- 1
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votes
1
answer
337
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Alternative to well-known trace estimate in Riemannian geometry?
Let $g,\hat{g}$ be two Riemannian metrics with volume forms $dv_g$, $dv_{\hat{g}}$, respectively. A standard estimate in the subject is the following: $$\text{tr}_g(\hat{g}) \leq \text{tr}_{\hat{g}} (...
CommunityBot
- 1
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Results about Schrödinger equations
Does anyone know any paper or book that deals with Schrodinger equations, specifically on asymptotic properties like blowup or limitation of solution when time goes to infinity using Schrödinger ...
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1
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uniform convergence of $H^r$ projectors on compact sets?
Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...
- 6,035
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2
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Linear hyperbolic PDE on compact two dimensional domain
Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
- 21.6k
3
votes
1
answer
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Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
- 12.6k
7
votes
0
answers
209
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Li-Yau inequality on $\mathbb R^2$ for functions that are somewhat close to $1$
Let $u:\mathbb R^2\times \mathbb R_{>0}\to \mathbb R_{>0}$ be a positive solution to the heat equation on $\mathbb R^2$ ($u_{xx}+u_{yy}=u_t$, no constants). The Li-Yau inequality in this case ...
- 7,193
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2
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Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 128k
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2
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Definition of entropy and entropy flux of conservation laws: component-wise reasoning
Consider the conservation law
$$\DeclareMathOperator{\dvg}{\operatorname{div}}
\partial_t u(x,t) + \dvg G(u(x,t)) =0, \\
u \in U\subseteq \mathbb R^m, x\in X\subseteq \mathbb R^n, G \subseteq \mathbb ...
- 6,367
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Counting number of distinct eigenvalues
Let $\Omega$ be a Lipschitz domain in $\mathbb{R}^n$, and let $N(\lambda)$ be the number of Dirichlet Laplacian eigenvalues less than or equal to $\lambda$. The famous Weyl's law says that as $\...
- 1,350
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Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,115
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votes
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Is every minimal graph smooth?
The following result was taken from the book of Gilbarg-Trudinger:
In particular, if the graph is minimal, then $u$ is smooth.
Now comes my question: does the same conclusion hold for graphs over ...
- 2,095
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140
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Computing the fractional laplacian of a logarithm function
Are there explicit formulas to compute the fractional laplacian $(-\Delta)^{s/2}\log |x|$ in $\mathbb{R}^n$ for $s\in (0,2)$?
- 547
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1
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Asymptotic behaviour of solution of Kazdan-Warner equations
Let $X$ be a closed manifold. $g:X\rightarrow \mathbb{R}$ be a smooth function ,$\alpha$ a section of a line bundle with discrete zeros and $c>0$ a constant, then Kazdan-Warner's work says that the ...
- 954
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2
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The difference between the nonlocal and local conditions problems
In some problems involving ordinary differential equations, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness ...
CommunityBot
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What does it mean by "converges boundedly"?
On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says
Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
- 29.8k
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Does the weak formulation of a parabolic PDE applies to a (good) non-test function?
Let $\rho:\mathbb R^d\times[0,\infty)\to(0,\infty)$ such that $\int \rho_t(x)\,dx=1$ for all $t\geq0\,$, $\rho$ is Holder-continuous (in both variables) and $\rho_t\in W^{1,1}(\mathbb R^d)$ for a.e. $...
- 311
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Solve $(A(x).\nabla)u+cu=0$
ِDoes the equation
$y\partial_x u(x,y)-x\partial_y u(x,y)+cu=0$
have complex-valued compact-supported or vanishing-at-infinity $C^1$ solution defined on the whole plane without any singularity? Here $...
- 39.1k
1
vote
1
answer
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Assumptions on the flux of a conservation law required to obtain an entropy inequality
On page 87 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem which I summarise as follows
Theorem. (Theorem 4.5.2 in the book.) Let $U$ be a weak ...
- 39.1k
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vote
0
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251
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Regularity of a Fokker-Planck PDE with unbounded coefficient
Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\
- 2,494
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How to prove this this integral equality which contain nonlocal operator, $(-\partial_{xx})^{1/2}$?
Suppose that $\theta(t,x)$ is even about $x$ and is smooth. $0\le \gamma<1/2$, $0<\delta<1-2\gamma$. $\Lambda=(-\partial_{xx})^{1/2}$
My Question: How to prove that
$$
\int_0^{\infty} \frac{(\...
- 171
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votes
0
answers
67
views
Asking a reference for a fact about nonlocal operators
Let assume that $u$ is smooth enough and $ -\Delta (u \phi) \in L^1(\Omega)$ for any $\phi \in C_c^{\infty}(\Omega)$. Then it easily follows that $ -\Delta u \in L^1_{\mathrm{loc}}(\Omega)$ by ...
- 371
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0
answers
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Are solutions to this elliptic PDE uniformly bounded in $\mathbb{R}^n?$
Given a fixed value $\lambda>0$ let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution (or eigenfunction with eigenvalue $\lambda$) in dimension $n\geq 3$ to the following PDE,
$$-\Delta u = \lambda ...
- 6,367
3
votes
1
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269
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Is this equation of hyperbolic type?
I want to now whether this equation is of hyperbolic type:
$$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$
with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$
I would say that the answer is yes. By ...
- 52.4k
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0
answers
61
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Tuning parameters of PDEs given a set of data
I am interested in doing statistical inference in the context of PDEs. Loosely speaking, the kind of problem I have in mine is the following.
Problem setting
Let $(t_i, x_i, y_i) \in \mathbb{R} \...
- 21.6k
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vote
0
answers
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How to check that $f(t,x)\ge 0$ for any $x\ge 0$?
Suppose $f(t,x)$ satisfices $\partial_t f=\Lambda^{-\alpha}f\partial_xf$, for $0<\alpha<1$ and where $\Lambda=(-\partial_{xx})^{1/2}$, $f_0(x)=f(0,x)$ is odd, and $f_0(x)\ge 0$, $\forall x\ge 0$....
- 3,574
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1
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Finiteness of Schatten $p$-norm of truncated free resolvent
Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$.
For $p \geq 1$, let ...
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