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4 votes
0 answers
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Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
11 votes
0 answers
317 views
+50

Sobolev's PDE Scottish Book Problem (Problem 188)

In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution. In 2015, when the second edition of the Scottish Book with updates and commentary on ...
Mark Lewko's user avatar
1 vote
0 answers
84 views

Does sets of positive capacity rule out constant functions?

Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...
Guy Fsone's user avatar
  • 1,101
3 votes
0 answers
74 views

Reference for PDEs from system of SDEs

I'm working with a system of SDEs \begin{align*} dX_t &= b(X_t, t) + \sigma dB_t\\ dY_t &= c(X_t, Y_t, t) + \sigma dB_t. \end{align*} Here, the Brownian motion is the same. I know that ...
optimal_transport_fan's user avatar
6 votes
1 answer
168 views

Laplacian is surjective from $\mathcal{C}^{\infty}(B)$ to $\mathcal{C}^{\infty}(B)$

Let $B$ denote the open unit ball in $\mathbb{R}^n$. Let $\mathcal{C}^{\infty}(B)$ represent the space of smooth functions on $B$. Is the Laplacian operator $\Delta$ surjective as a map from $\mathcal{...
Jessie L's user avatar
1 vote
1 answer
73 views

N-soliton, The Lax operator and the transmission coefficient

I'm interested in the soliton stability result given in HERBERT KOCH and DANIEL TATARU's paper "MULTISOLITONS FOR THE CUBIC NLS IN 1-D AND THEIR STABILITY", published in IHES. However, I ...
sorrymaker's user avatar
1 vote
1 answer
85 views

$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains

Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form $$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$ are there any elliptic ...
BBB's user avatar
  • 93
9 votes
2 answers
493 views

Reference Request for global Hölder continuity of solutions to elliptic PDEs

This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
DarkViole7's user avatar
1 vote
0 answers
52 views

High order parabolic PDEs on manifolds: Reference request

I recently became interested in parabolic PDEs of order 4 on surfaces. However, I have a very little background in parabolic PDEs. I discovered Lunardi's book (Analytic semigroups and optimal ...
Dorian's user avatar
  • 363
8 votes
0 answers
115 views

optimal regularity for the Neumann heat equation on Lipschitz domains

$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
leo monsaingeon's user avatar
2 votes
1 answer
66 views

How many non-trivial solutions can a semilinear elliptic equation have on a smooth star-shaped bounded domain with 0-Dirichlet boundary conditions?

I am not an expert in elliptic partial differential equations, but while studying the attractor structure of evolutionary PDEs, I frequently encounter problems related to elliptic equations. ...
monotone operator's user avatar
1 vote
0 answers
73 views

Convexity and subdifferential monotonicity

Do you know any reference where I can find some results in this sense: Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
Bogdan's user avatar
  • 1,759
9 votes
2 answers
418 views

Reference request: Parabolic Equations

I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
Falcon's user avatar
  • 452
1 vote
2 answers
237 views

Calderón–Zygmund/$L^p$ estimates for the linear heat equation

Let $C_r$ denote the open cylinder $$ C_r = \{(x,t) \in \mathbb R^{n+1} : |x| < r, -r^2 < t < 0\} $$ and consider a classical $C^{2,1}_{x,t}(C_1)$-solution to the linear heat equation $$ \...
Desura's user avatar
  • 233
1 vote
0 answers
120 views

Well-posedness result for a linear parabolic equation on the torus

Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
kumquat's user avatar
  • 185
7 votes
1 answer
580 views

Sobolev spaces are smooth? Their dual is strictly convex?

Do you know any reference which says something about the: Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton. ...
Bogdan's user avatar
  • 1,759
5 votes
2 answers
364 views

Euler-Lagrange equations for minimizer of energy with indicator function

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
BBB's user avatar
  • 93
1 vote
0 answers
55 views

References for the generalized Dirichlet problem for plurisubharmonic functions on the bidisc

In the paper "On a Generalized Dirichlet Problem for Plurisubharmonic Functions and Pseudo-Convex Domains. Characterization of Silov Boundaries" Theorem 5.3, the following result is obtained ...
Gamabunto's user avatar
3 votes
0 answers
131 views

Uniqueness for a nonlinear kinetic PDE-system with heat transfer coupling in one dimension

I am currently trying to understand the following article "Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit" (2021) by Favre, ...
kumquat's user avatar
  • 185
8 votes
1 answer
584 views

Reference request: Software for producing sounds of drums of specified shapes

Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
Michael Hardy's user avatar
2 votes
0 answers
86 views

Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space

Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
SMS's user avatar
  • 1,407
0 votes
0 answers
46 views

Uniqueness results for linear first order systems of PDEs

Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$): $$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
Samuele's user avatar
  • 1,205
0 votes
1 answer
100 views

Spatially localised solution to the Schrödinger equation with potential is a combination of eigenfunctions

In this Terry Tao's blog post, he claims that if one has a solution to the Schrödinger equation $$i\,\partial_t u +\Delta u=Vu $$ with a "reasonably smooth and localised $V$", $u$ has ...
Earl Jones's user avatar
2 votes
0 answers
238 views

What is the fundamental solution for the backward heat equation?

According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
Andrew's user avatar
  • 2,715
4 votes
1 answer
172 views

Viscosity solutions of eikonal equation on Riemannian manifolds

It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$ admits the unique viscosity ...
ChesterX's user avatar
  • 235
11 votes
3 answers
727 views

Application of Lie group analysis of PDE (beyond calculation of exact solutions)

I am learning the Lie symmetry group method for PDEs. In my reading, all of the applications of this method are to calculate the exact solutions of PDEs. Are there any good references which provide ...
KWSK's user avatar
  • 111
2 votes
0 answers
56 views

Stability on manifold with boundary

Let $(X,\partial X)$ a smooth Kahler manifold with boundary, i.e. the interior of $X$ is Kahler, Donaldson proved that: Given a smooth vector bundle $E$ over $X$ such that $E$ is holomorphic over the ...
TaiatLyu's user avatar
  • 395
1 vote
1 answer
110 views

Looking for definition of function spaces appearing in article of DiPerna & Lions

I am looking for the definition of various function spaces appearing in the following article, preferably with references to other sources where such spaces are discussed in greater detail: Article: ...
Simon's user avatar
  • 21
0 votes
0 answers
99 views

Two-sided estimates of fundamental solutions of second-order parabolic equations

I am reading the paper Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications by F.O. Porper and S.D. Eidel'man. Below, the cited paper is [2] :S.D. ...
Akira's user avatar
  • 835
0 votes
1 answer
203 views

Perturbation methods for stochastic/partial differential equations

I'm asking for a good reference on perturbation methods for stochastic and/or partial differential equations. Something like this: Perturbation of a stochastic differential equation I'm familiar with ...
Math_Day's user avatar
  • 131
2 votes
1 answer
160 views

Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term

I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying $$ \begin{...
l'étudiant's user avatar
2 votes
0 answers
138 views

Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ ...
Laithy's user avatar
  • 969
3 votes
3 answers
252 views

Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?

If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...
Perelman's user avatar
  • 163
4 votes
0 answers
97 views

Techniques to estimate PDE which are elliptic in some directions and degenerate in others

I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
Aidan Backus's user avatar
0 votes
1 answer
77 views

$L^\infty$ estimate for elliptic PDE with mixed boundary conditions

Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint. Consider the problem $$\Delta u = f \quad\text{in $\...
BBB's user avatar
  • 93
1 vote
0 answers
207 views

Specific type of PDE

While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices): $$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \...
Gennaro Marco Devincenzis's user avatar
2 votes
2 answers
241 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", ...
Sarthak's user avatar
  • 87
4 votes
3 answers
473 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 $t$...
Math2024's user avatar
  • 141
5 votes
0 answers
878 views

A fourth-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
  • 141
2 votes
0 answers
111 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
  • 835
7 votes
1 answer
531 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
  • 969
3 votes
0 answers
203 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
  • 809
3 votes
0 answers
153 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 ...
Y. Paka's user avatar
  • 131
3 votes
0 answers
161 views

Lebesgue measure of the boundary of the positivity set of a function is zero?

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
10 votes
0 answers
422 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 \...
Akira's user avatar
  • 835
0 votes
0 answers
25 views

Reference request heat equation with moving interface

Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$: $$ \begin{cases} u_t = \sigma_1^2u_{xx} ~~~~\...
NancyBoy's user avatar
  • 393
0 votes
0 answers
56 views

Godunov splitting convergence research

The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
Redsbefall's user avatar
0 votes
0 answers
52 views

Properties of "potential vector field" in Helmholtz decomposition

It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as $$ F= \nabla V+ \nabla \times R$$ with $V$ a potential and $R$ another vector field. These components ...
tommy1996q's user avatar
1 vote
0 answers
82 views

For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
Akira's user avatar
  • 835
1 vote
0 answers
114 views

Global existence for large data in $H^{-1/2}(\mathbb R)$ of viscous Burgers' equation with external forcing

First, a quick summary of what to know about viscous (or dissipative) Burgers' equation $$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$ Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev ...
Lorenzo Pompili's user avatar

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