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Results tagged with ap.analysis-of-pdes
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user 36721
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
Accepted
Lemma from Donnelly-Fefferman's paper
$\newcommand{\al}{\alpha}
\newcommand{\G}{\mathcal{G}}
\newcommand{\PP}{\mathcal{P}}$
You are missing condition (ii) in formula (5.9) in that paper. That formula is
\begin{equation}
\begin{aligned}
& …
- 128k
2
votes
An inequality involving weight $|x|^\alpha$
For your question (or statement) about a particular piece of a paper to be more effective, you should refer, not just to the paper, but to that particular piece of it.
It appears that the piece in que …
- 128k
1
vote
Accepted
An estimate of the integral of the higher order derivative of a bump function
$\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\na}{\nabla}$Note that
\begin{equation}
(\na^n\rho_\ep)(x)=\ep^{-d-n}(\na^n\rho)(\ep^{-1}x).
\end{equation}
So,
\begin{equation …
- 128k
3
votes
Accepted
Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$
We have
$$\nabla^2 g(t,x)=g(t,x)\Big(\frac{xx^\top}{(2t)^2}-\frac{I_d}{2t}\Big),$$
where $I_d\in\mathbb R^{d\times d}$ is the identity matrix and $\mathbb R^d$ is identified with $\mathbb R^{d\times1} …
- 128k
4
votes
Solving $X$ for prescribed $\operatorname{div}(X)$ of compact support
This is impossible in general. Indeed, if $X$ has a compact support, then the flux of $X$ through a large enough sphere is $0$, whereas the integral of $f$ over the corresponding ball (say $B$) will n …
- 128k
3
votes
Doubts in first lemma in the paper of Adams regarding sharp Moser inequality
$\newcommand\la\lambda$To answer your first question, write
$$\int_{-\infty}^\infty|E_\la|e^{-\la}\,d\la
=\int_{-\infty}^\infty e^{-\la}\,d\la\int_0^\infty dt\,1(F(t)\le\la)
=\int_0^\infty dt\,\int_{- …
- 128k
5
votes
How to show continuity and monotonicity of solutions to this parametrized equation?
With $r:=p/2\in[1/2,1)$ and $y:=1-\sqrt s\in[0,1/2)$, rewrite your equation (1) as
$$G(r,y):=2^r y^{2 r-1} (r+y-r y)-1=0. \tag{2}$$
For any $r\in(1/2,1)$, $G(r,0)=-1\ne0$, so that $y=0$ is not a solut …
- 128k
7
votes
Accepted
Convergence rate for $L^2$ convergence
$\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\ …
- 128k
0
votes
Bounded weak derivative
$\newcommand{\R}{\mathbb{R}}
\newcommand{\al}{\alpha}$
This is a partial answer: Assuming additionally that $D^\al f$ exists in $C(\R^n)$ for $|\al|=1$, let us show that $D^\al f$ is bounded.
Indeed, …
- 128k
3
votes
Estimating singular double integral
$\newcommand\de\delta\newcommand\De\Delta$Let $a:=\alpha\in(0,2)$ and $w:=uv$. Then all we can get from your conditions on $u,v$ is that $w\in L^2$. The integral in question is
$$I:=\int_{B_\de(1/2)}d …
- 128k
3
votes
Riesz rearrangement inequality
$\newcommand\R{\mathbb R}$The Riesz rearrangement inequality
$$\iint_{\mathbb{R}^n\times \mathbb{R}^n} f(x) g(x-y) h(y) \, dx\,dy \\
\le \iint_{\mathbb{R}^n\times \mathbb{R}^n} f^*(x) g^*(x-y) h^*(y) …
- 128k
2
votes
Showing that a Gaussian achieves equality in a logarithmic Sobolev inequality
$\newcommand{\na}{\nabla}\newcommand{\R}{\mathbb R}$Without loss of generality, $C=1$, so that
\begin{equation}
f(x)\equiv\exp\Big\{-\frac{\pi |x|^2}{2a^2}\Big\},
\end{equation}
where it is assum …
- 128k
3
votes
Accepted
Extremizers of the Sobolev inequality
Take any extremizer $u$ as in the paper you linked.
From the conditions $u(r)=r^{1-m/2}v(r)$, $u(r)=o(r^{1-m/2})$, and $u'(r)=o(r^{-m/2})$ (as $r\to\infty$), we deduce $v(r)=r^{m/2-1}u(r)$, $v(r)=o(1) …
- 128k
2
votes
Accepted
Convergence of a level set when $f^n\to f$ in $C^1$ sense
$\newcommand\de\delta$Apparently, (i) by $\epsilon\to0$ you meant $n\to\infty$ and (ii) by "$f^n\to f$ in $C^1$ sense" you meant that
$$\sup_x|f^n(x)-f(x)|+\sup_x|\nabla f^n(x)-\nabla f(x)|\to0. \tag{ …
- 128k
8
votes
Accepted
Forcing the uniqueness of a solution of an ODE
$\newcommand\ep\varepsilon$First, the conditions that $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ and $f_n(x)\ge\sqrt{x}$ for $x\in[0,1]$ imply $f_n(0)>0$. Since
\begin{equation*}
\begin{cases}
y_n(0) …
- 128k