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.

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An expansion for 2d Euler equation

Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$: $$ -\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
Davidi Cone's user avatar
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80 views

An asymmetric quadrilinear estimate

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
Medo's user avatar
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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann,...
Elio Li's user avatar
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What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?

I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
Julian Chaidez's user avatar
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189 views
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Do the eigenvalues of a thin strip around a semicircle converge to the respective eigenvalues of a semicircle?

Background: Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
Student's user avatar
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Growth/Decay of conformal Killing fields in cone metrics

Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric $$g= dr^2 + r^2 \gamma$$ on $[1,\infty) \times S^2$. Does there exist a nontrivial conformal Killing field vanishing ...
Laithy's user avatar
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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 $...
BigbearZzz's user avatar
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2 votes
1 answer
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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 \...
CComp's user avatar
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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? $$|...
Nate River's user avatar
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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 ...
Ali's user avatar
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1 answer
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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-...
DerGalaxy's user avatar
2 votes
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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 ...
Luis Yanka Annalisc's user avatar
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2 answers
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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", ...
Sarthak's user avatar
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3 answers
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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 $t$...
Math2024's user avatar
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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) \\ ...
elmas's user avatar
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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{...
chrisv's user avatar
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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 ...
Jacaré's user avatar
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1 answer
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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 $...
Stack_Underflow's user avatar
3 votes
0 answers
66 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-\|...
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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 ...
G. Blaickner's user avatar
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4 votes
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384 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? ...
Math2024's user avatar
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 ...
BBB's user avatar
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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 ...
António Borges Santos's user avatar
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1 answer
117 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 ...
António Borges Santos's user avatar
6 votes
0 answers
194 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 $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
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1 vote
0 answers
45 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 ...
yak's user avatar
  • 11
2 votes
0 answers
101 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 ...
Akira's user avatar
  • 749
7 votes
1 answer
489 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 ...
Laithy's user avatar
  • 865
1 vote
1 answer
92 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 ...
vent de la paix's user avatar
3 votes
1 answer
141 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 ...
Sebastian Bechtel's user avatar
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0 answers
83 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 ...
0x11111's user avatar
  • 473
2 votes
0 answers
162 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)}+\...
Elio Li's user avatar
  • 719
2 votes
0 answers
93 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 ...
Student's user avatar
  • 601
2 votes
0 answers
138 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 ...
kobeahibe's user avatar
7 votes
2 answers
332 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 ...
B.Hueber's user avatar
  • 987
1 vote
2 answers
197 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 ...
Jingeon An-Lacroix's user avatar
1 vote
1 answer
120 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 ...
JMK's user avatar
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0 answers
59 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 ...
Elio Li's user avatar
  • 719
1 vote
1 answer
81 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(...
maxematician's user avatar
1 vote
0 answers
55 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 ...
Evelina Shamarova's user avatar
1 vote
0 answers
52 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 $T_+>0$ ...
Student's user avatar
  • 601
0 votes
1 answer
105 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 ...
Ma Joad's user avatar
  • 1,591
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 $\Delta u=0$ ...
User 11111's user avatar
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0 answers
49 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 ...
GJC20's user avatar
  • 1,210
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 ...
Ma Joad's user avatar
  • 1,591
7 votes
2 answers
452 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 ...
G. Blaickner's user avatar
  • 1,137
1 vote
0 answers
94 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, \...
onamoonlessnight's user avatar
2 votes
1 answer
137 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 ...
Physics user's user avatar
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0 answers
79 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?
ABIM's user avatar
  • 4,989
2 votes
0 answers
81 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 ...
B.Hueber's user avatar
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