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.
4,275
questions
2
votes
1
answer
108
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
2
votes
1
answer
83
views
Gradient flows and particle representations
I was looking into gradient flows and their particle representations, mostly in the context of probability.
A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
- 123
4
votes
0
answers
74
views
Solution to the Eikonal equation with almost everywhere continuous derivative
Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?
$$|...
- 4,832
4
votes
2
answers
339
views
Nontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by
$$ v(r,\theta) = u(\frac{1}{r},\theta)$$
is also harmonic for $r>0$. Note that the Kelvin ...
- 4,089
1
vote
1
answer
80
views
Sufficient initial conditions for "Non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 61
2
votes
0
answers
89
views
Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
- 1,133
2
votes
2
answers
182
views
A Inequality in the paper by Kenig, Ponce and Vega
I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle",
...
- 75
3
votes
3
answers
316
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 $t$...
- 71
1
vote
0
answers
94
views
How using the standard Galerkin method
I am attempting to solve the following evolution problem using the standard Galerkin method
$$\begin{cases}
\dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\
...
- 55
0
votes
0
answers
24
views
Metric entropy of mixed norm spaces with exponent-free bounds
Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
- 21
3
votes
0
answers
59
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
0
votes
1
answer
159
views
About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
- 111
3
votes
0
answers
69
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
- 161
3
votes
0
answers
100
views
On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,147
4
votes
0
answers
443
views
A 4th-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$x^3 f_{xxxt}+ f =0$
Does anyone know if this type of PDE already appeared in the literature? ...
- 71
2
votes
0
answers
50
views
Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
- 31
4
votes
1
answer
259
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
0
votes
1
answer
119
views
Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
6
votes
0
answers
196
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
1
vote
0
answers
48
views
Using connection form for unknown frame field
I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
- 11
2
votes
0
answers
104
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 851
7
votes
1
answer
496
views
Conformal Killing fields satisfy a third order PDE
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
- 885
1
vote
1
answer
95
views
Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator
I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time.
Consider the following ...
- 187
3
votes
1
answer
157
views
PDE: compactness vs blowup
There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following:
Solve (easier) approximate problems, show some form of compactness for the approximate ...
0
votes
0
answers
87
views
Fourier integral operators and parametrix
Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary.
Question: Is there an expression for the ...
- 493
2
votes
0
answers
170
views
Question about the formula of Green function of Laplacian on sphere
I'm reading a paper which said that
the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
- 729
2
votes
0
answers
100
views
Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 653
2
votes
0
answers
141
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 111
7
votes
2
answers
363
views
Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
- 987
1
vote
2
answers
205
views
Interpretation of the singular integral for the definition of fractional Laplacian in classical case $s=1$
Fractional Laplacian is often defined via next principal value integral (assume dimension one for the simplicity)
$$(-\Delta)^su(x):=c_sP.V.\int_\mathbb{R} \frac{u(x)-u(y)}{|x-y|^{1+2s}}\, dy.$$
This ...
1
vote
1
answer
126
views
Existence of solution to nonlinear first order PDE with C^1 bounds
I'm looking for general existence of a PDE of the form
$$ f: U \times [0, \delta) \to \mathbb{R}$$
$$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$
where $f(p,0)$ is prescribed and $F$ is non-linear ...
- 299
0
votes
0
answers
61
views
Some questions about the concept of stable solution of elliptic PDE
For $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is stable if
$$
Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall ...
- 729
1
vote
1
answer
86
views
How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let
$$
\begin{matrix}
F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\
(x,z,p) \mapsto F(...
- 197
1
vote
0
answers
56
views
One more question on biharmonic functions
Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $C=\{w>0\}$;
$w$ is subharmonic ...
1
vote
0
answers
55
views
$L^p$ estimates for critical heat equation on $\mathbb{R}^n$
Background:
Let $u:\mathbb{R}^n\times [0,T_+)\to \mathbb{R}$ be a solution to
$$\partial_t u = \Delta u + |u|^{p-1}u$$
where $p=2^{*}-1=\frac{2n}{n-2}-1=\frac{n+2}{n-2}$, $n\geq 3$ and $T_+>0$ ...
- 653
0
votes
1
answer
106
views
Controlling convolutions with maximal functions
For $f\in L^1(\mathbb R^n),$ let $Mf$ be the (Edited: changed the type of maximal function) Stein spherical maximal function. Let $\varphi\in C_c^\infty.$ Then, can we have a pointwise estimate of the ...
- 1,641
1
vote
0
answers
41
views
Behaviour of higher order Laplacian in punctured domain
Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
- 11
0
votes
0
answers
50
views
Question on the modelling of (viscous) fluid in a bag with holes
Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)
What is the corresponding PDE to model the ...
- 1,230
0
votes
0
answers
38
views
Conservation law for generic linear hyperbolic PDEs?
Consider the wave equation:
$$
u_{tt} = \Delta u, \text{ on }U\times [0,T], u=0\text{ on }\partial U\times[0,T].
$$
To prove the only solution for the zero initial condition is zero, we only need to ...
- 1,641
7
votes
2
answers
466
views
Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...
- 1,147
1
vote
0
answers
96
views
criticality for nonlinear wave equations on manifolds
On $\mathbb{R}^{1+n}$, the initial value problem for the homogeneous wave equation
$$ \Box \phi = \partial_t^2 \phi - \Delta_{\mathbb{R}^n} \phi = 0, \\ (\phi, \partial_t \phi)|_{t=0} = (\phi_0, \...
- 894
2
votes
1
answer
142
views
Prove if the fractional Laplacian of a function is bounded
Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...
- 123
0
votes
0
answers
80
views
Elliptic PDEs in BSDEs and in optimal control
This soft/reference question is related to this MO post of a similar nature.
What are some examples of elliptic PDEs appearing in control and BSDEs?
- 4,969
2
votes
0
answers
82
views
Question on Cauchy problem on manifolds
Let $(M,g)$ be a globally-hyperbolic manifold, i.e. $M=\mathbb{R}\times\Sigma$ and $g=-\beta^2 \, dt^2+h_{t}$. Furthermore, let $E$ be a vector bundle over $M$ and $\square$ a normally-hyperbolic ...
- 987
0
votes
0
answers
64
views
Examples of symmetry-breaking solitons which retain a subgroup symmetry
There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions.
However, all symmetry breaking soliton examples I have seen go from the ...
- 49
4
votes
1
answer
220
views
Closed-form solution to hyperbolic PDE
Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesgue integrable. Consider the hyperbolic PDE
$$
\begin{cases}
\partial_{x,y}u & = A\...
- 4,969
1
vote
0
answers
55
views
Pohozaev type obstruction for higher order elliptic operators
I was reading about Pohazaev type obstructions: precisely, I mean the following kind of results. Let $h\in C^1(\mathbb{R}^n)$ and consider the following Dirichlet problem
$$
\begin{cases}
\Delta u + ...
- 75
0
votes
0
answers
75
views
What is the PDE corresponding to this weak formulation?
Consider a flow $(\mu_t)_{t\ge 0}$ such that
every $\mu_t$ is a probability on $\mathbb R_+$;
$\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
- 1,230
10
votes
0
answers
395
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 851
1
vote
1
answer
109
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 851