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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.
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
Lebesgue Riemann Theorem.
Actually it is quicker to sketch the proof than checking a reference.
Assume $u:\mathbb{R}^n\to \mathbb{R}$ is measurable and periodic w.r.to $x_i$ with period $b_i - a_i$, for $1\le i\le n$. Then …
- 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
2
votes
Harmonic Functions
To reduce the case of a continuous $f:\mathbb{R}^2\to\mathbb{R}$ to, say the $C^2$ or $C^3$ case, one can simply mollify $f$ via convolution with a smooth kernel with compact support. Then $f_\epsilon …
- 60.6k
3
votes
Accepted
$L^2$ boundeness of a sequence
A counterexample in one dimension: take $\Omega:=(0,1)$ and $f_n(x):=\frac{\sqrt 2}{x+\frac{1}{n}}$. Then $f''_n(x)-f_n^3(x)=0$ while $\| f''_n \| _{2,\Omega}=\|f^3_n\|_{2,\Omega}=O(n^{5/2}) \, .$
- 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
19
votes
Accepted
Basis for the space of Harmonic homogeneous polynomial in N variables.
Let $K$ denote the Kelvin transform, and let $|\alpha|:=\sum_{j=1}^n\alpha_j$ denote the weight of the multi-index $\alpha\in\mathbb{N}^n$. Then, an explicit base for the space of homogeneous harmonic …
- 60.6k
9
votes
Accepted
How to prove this Poincare Inequality
Just assume that $g$ is a bounded function with $\int_B g < 0$, and positive somewhere in th ball $B$ . Then the set
$$\Big \{u\in H^1(B)\, : \|u\|_2= 1\, ,\, \int_B g u^2\ge 0 \Big \}$$
is not em …
- 60.6k
29
votes
Accepted
A Hölder continuous function which does not belong to any Sobolev space
Your guess is indeed right. Following a similar idea gives you the Takagi or blancmange function.
It is even quasi-Lipschitz (it has a modulus of continuity $\omega(t)=ct(|\log(t)|+1)$ for a suitable …
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
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
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
$$u+\Big( \frac2p -1\ …
- 60.6k
3
votes
Accepted
A Sobolev-type inequality with weights
To complete your computation, let's treat the case of a function supported in interval $(0,1)$. Indeed, for $ f\in C^\infty_c(0,1)$ there is an inequality
$$ \int_0^1 r^{-3}f(r)^6 dr\le C\left(\int_ …
- 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
3
votes
Is there a good metric under which a sequence of compact sets can converge to an infinite di...
As a general idea, without any other information on a sequence, it seems rather unlikely to find a natural metric in which it converges. Yet a metric in which it has a convergent subsequence is a mor …
- 60.6k