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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.
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
8
votes
Accepted
Asymptotic behavior of a certain oscillatory integral
We can evaluate $I(x)$ explicitly, and then asymptotically.
Indeed, using the substitution $s=ru/x$, we get
\begin{equation*}
I(x)=\frac1{\sqrt x}\lim_{R\to\infty}J_R(x), \tag{1}\label{1}
\end{equ …
- 128k
8
votes
Accepted
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
$\newcommand\al\alpha\newcommand\de\delta\newcommand\R{\Bbb R}\newcommand\B{\mathrm B}$It follows from the ODE
$$f''=f^{-\al} \tag{1}\label{10} $$
and the condition $f>0$ that $f$ is (strictly) convex …
- 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
7
votes
Accepted
Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)...
$\newcommand{\Om}{\Omega}\newcommand{\de}{\delta}\newcommand{\ep}{\varepsilon}\newcommand{\R}{\mathbb R}\newcommand{\pOm}{\partial\Om}$Let $n:=N$. Consider the following "cone" condition:
the boundar …
- 128k
6
votes
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\ma...
This inequality cannot hold because of the homogeneity matter. Indeed, take any $u$ with $\int u^3\in(0,\infty)$ and $\int u^2+\int(u')^2<\infty$. Then replace $u$ by $cu$ for a real number $c$, and l …
- 128k
6
votes
Accepted
Log-concavity of function
Direct calculations show that
$$(f_2*f_0)(y)=\frac{1}{4} \sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}} \left(y^2+1\right),
$$
$$(f_1*f_0)(y)=\frac{1}{2} \sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}} y,
$$
$$(f …
- 128k
6
votes
Accepted
Lipschitz property of the symmetric rearrangement
$\newcommand{\R}{\mathbb R}$By the continuity of measure, the nonincreasing function $\mu$ is right-continuous. The function $u^*$ is radial, that is,
\begin{equation*}
u^*(x)=U(|x|)
\end{equation …
- 128k
6
votes
Accepted
A constant ratio of integrals? Part I
No. E.g., if $n=2$ and $u(x,y)=x + x^2 - y^2$ for $(x,y)\in\mathbb R^2$, then $\dfrac{r^2A(r)}{B(r)}=2\dfrac{1+2r^2}{1+r^2}$.
- 128k
6
votes
Accepted
Asymptotic behavior of a double oscillatory integral
$\newcommand{\R}{\mathbb R}\newcommand\sgn{\operatorname{sgn}}\newcommand{\vpi}{\varphi}$Obviously, for $a:=\sqrt{2\pi}\,\psi(0)$ we have
\begin{equation*}
\psi(0)=f(0),
\end{equation*}
where
\be …
- 128k
6
votes
Accepted
Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}...
The answer is no. E.g., let $t_1\sim t_2\downarrow 0$ and $|x|\sim\sqrt{t_2}$.
- 128k
6
votes
Accepted
Poincaré-type Inequality
For $k$ to make sense, we should assume that $\|u\|_p\ne0$.
Let $l(x)$ and $r(x)$ denote the left- and right-hand sides of your displayed inequality. Then, by Tonelli's theorem, for any real $p>0$
$$ …
- 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
5
votes
Accepted
Linear transport equation with unbounded coefficients
No. If e.g. $n=1$, $p=0$, and $Bq(x)=1$ for all $x$, then $f(t,x)=f_0(t+x)$, which does not decay along the lines $\{(t,x)\colon t+x=c\}$ for real $c$.
The OP has changed the question, now looking fo …
- 128k
5
votes
Accepted
A constant ratio of integrals? Part II
In fact, this is true for any homogeneous polynomial $u$ (not identically $0$), be $u$ harmonic or not.
Indeed, let $m\ge1$ be the degree of such a polynomial $u$. Then
$$u(tx)=t^m u(x)$$
and
$$v(tx)= …
- 128k