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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.
32
votes
Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theore...
Yes, there is a standard way. Take any nonzero $u\in W^{1,p}(U)$, with support in a ball, say w.l.o.g $\operatorname{supp}(u)\subset B(0,r)\subset U$, and consider
$$ u_\epsilon(x):= u\big(\frac{x}{\ …
- 60.6k
29
votes
Accepted
A Hölder continuous function which does not belong to any Sobolev space
Your guess is indeed right. Following a similar idea gives you the Takagi or blancmange function.
It is even quasi-Lipschitz (it has a modulus of continuity $\omega(t)=ct(|\log(t)|+1)$ for a suitable …
19
votes
Accepted
Basis for the space of Harmonic homogeneous polynomial in N variables.
Let $K$ denote the Kelvin transform, and let $|\alpha|:=\sum_{j=1}^n\alpha_j$ denote the weight of the multi-index $\alpha\in\mathbb{N}^n$. Then, an explicit base for the space of homogeneous harmonic …
- 60.6k
13
votes
Arzelà-Ascoli theorem and Hölder spaces
For completeness, let's mention a simpler and more general statement: For $\Omega\subset\mathbb{R}^n$ a bounded open set, $k\in\mathbb{N}$ and $0<\beta<\alpha\le1$ there is a compact embedding
$$ C^{ …
- 60.6k
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
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
8
votes
The characteristic (indicator) function of a set is not in the Sobolev space H¹
One reason is this: if $f$ is in $H^1({\mathbb R^n})$, you have $\int_{\mathbb R^n}|f(x+h)-f(x)|^2 dx\le C|h|^2$ for all $h \in{\mathbb R^n}$ . Now in the case of $f:=\chi_E$ the integral is just the …
- 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
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
$$S_0:= -\big(a^2-|x|^ …
- 60.6k
5
votes
Accepted
Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions
More generally: if $\omega$ is a modulus of continuity with $\omega'(0)=\infty$ there is an $\omega$-continuous, smooth function $f$ on $\mathbb{R}_+$, with prescribed derivative $p_k\in \mathbb{R}$ a …
- 60.6k
4
votes
Solving a differential system
For simplicity, let's assume initially that the support of $\mu$ is the whole real line, so that $F$ is a homeomorphism $ \mathbb{R} \to (0,1) $. Let's denote $b:=F(0)=\mu(-\infty,0]=1-\mu[0,+\infty) …
- 60.6k
4
votes
Lebesgue Riemann Theorem.
Actually it is quicker to sketch the proof than checking a reference.
Assume $u:\mathbb{R}^n\to \mathbb{R}$ is measurable and periodic w.r.to $x_i$ with period $b_i - a_i$, for $1\le i\le n$. Then …
- 60.6k
4
votes
How to show continuity and monotonicity of solutions to this parametrized equation?
Put $t=1-\sqrt{s}\in[0,1/2)$ so the equation writes
$$ \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
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
4
votes
Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $
Extension is a well studied topic in the theory of Sobolev spaces, included in any treatise on the topic. Have a look e.g. to Adams' Sobolev spaces, Chapter 4 (Interpolation and Extension theorems). F …
- 60.6k