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Results tagged with ap.analysis-of-pdes
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user 6101
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
13
votes
Arzelà-Ascoli theorem and Hölder spaces
For completeness, let's mention a simpler and more general statement: For $\Omega\subset\mathbb{R}^n$ a bounded open set, $k\in\mathbb{N}$ and $0<\beta<\alpha\le1$ there is a compact embedding
$$ C^{ …
- 60.6k
4
votes
How to show continuity and monotonicity of solutions to this parametrized equation?
Put $t=1-\sqrt{s}\in[0,1/2)$ so the equation writes
$$ \Big(1-\frac p2\Big)\, t^p+ \frac p2\, t^{p-1}=2^{-\frac p2}$$
Now if we put $u:=t^{p-1}$ the equation takes the form
$$u+\Big( \frac2p -1\ …
- 60.6k
7
votes
Showing integrability of a locally integrable function on a bounded domain under some additi...
Let $(g_k)_{k\ge0}$ be a sequence of smooth functions such that $g_k(x)=1$ if $\text{dist}(x,\partial\Omega)\ge 2^{-k}$,
$g_k (x)=0$ if $\text{dist}(x,\partial\Omega)\le 2^{-k-1}$ and $0\le g_k\ …
- 60.6k
3
votes
Analytic approximations of smooth vector fields
Here is how to derive both results from the Stone-Weierstras theorem. As you say, it's not direct, but not a long way either. Recall these simple applications of the S-W theorem, to be used in PB1 res …
- 60.6k
1
vote
Accepted
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
The flow $X:[0,T]\times \mathbb{R}\to\mathbb{R}$ defined by $X(t,x)= x+t\chi_{\mathbb{R}_+}(x)$, for all $(t,x)\in [0,T]\times \mathbb{R}$, is a regular Lagrangian flow solution to $(\star)$ in the se …
- 60.6k
3
votes
Dense set in Sobolev space ${H^1}\left( {0,1} \right)$
Consider, for $m\ge2$, the function $\varphi_m(x):=x(1-x)^m$ . So
$$\varphi_m(0)=0\qquad \varphi_m(1)=0$$ $$ \varphi_m'(0)=1\qquad \varphi_m'(1)=0\ .$$ It is also easy to see that $\| \varphi_m'\ …
- 60.6k
2
votes
Accepted
How to prove the following Whittaker formula
If you start from the expression for $W_{k,m}(z)$ given here,
$$W_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}U\left(m-k+\frac{1}{2}, 1+2m, z\right)$$
and express the Tricomi function $ …
- 60.6k
2
votes
Accepted
Difference quotient for solutions of ODE and Liouville equation
Question 1: Denoting $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ the components of $\tilde \Phi(x,y,t)$ , we have $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\fra …
- 60.6k
3
votes
Version of Banach-Steinhaus theorem
I like Jochen Wengenroth's approach, and I think there is a point that it is worth to clarify. If we want to make a norm out of $A$, we need it to be a balanced set, so we'd like to pass to the bounde …
- 60.6k
5
votes
Accepted
Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions
More generally: if $\omega$ is a modulus of continuity with $\omega'(0)=\infty$ there is an $\omega$-continuous, smooth function $f$ on $\mathbb{R}_+$, with prescribed derivative $p_k\in \mathbb{R}$ a …
- 60.6k
6
votes
A curious determinant of quadratic forms
Let's put $x:=(0,X)\in k^{n+1}$ and $y:=-ae_1+x\in k^{n+1}$. Then $S$ writes as a symmetric rank-$2$ perturbation of a multiple of the identity, $S=S_0+ \lambda I_{n+1},$ with
$$S_0:= -\big(a^2-|x|^ …
- 60.6k
1
vote
Property Sobolev space
Actually one can find a larger space of $g$, taking into account the Sobolev inequalities: if $f\in W^{k,p}$ and $g\in W^{k,q}$, then for any order of derivation $0\le i\le k$, one has $D^if\in L^{p_i …
- 60.6k
4
votes
Solving a differential system
For simplicity, let's assume initially that the support of $\mu$ is the whole real line, so that $F$ is a homeomorphism $ \mathbb{R} \to (0,1) $. Let's denote $b:=F(0)=\mu(-\infty,0]=1-\mu[0,+\infty) …
- 60.6k
8
votes
Trying to solve a linear PDE... I thought it was simple
The Ansatz
$$f(x,y)=x^pu(x^\alpha y^{-2}) $$
yields to a linear second order ODE for $u(t)$
$$2t^2 u''+ (4\alpha+2p)tu'+(z_1+z_2t)u=0\ , $$
which can be immediately solved by series in terms of h …
- 60.6k
4
votes
Accepted
Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$
It's Strauss embedding theorem for radially symmetric functions, proven here:
W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math.
Phys. 55 (1977), 149-162.
- 60.6k