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Results tagged with ap.analysis-of-pdes
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user 6953
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
1
answer
443
views
The comparison between the square of the functional value and the sum of squares of the $L^2...
I was reading a paper where I came across the following argument :
For any $x$ in $M$ and for a geodesic ball $B(x; \varepsilon)$ in a compact Riemannian
manifold $M$ with injectivity radius bigger …
- 1,471
4
votes
2
answers
1k
views
Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $
The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ?
For detail :
Fix $p \geq 1. …
- 1,471
1
vote
2
answers
917
views
Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a posi …
- 1,471
3
votes
Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ f...
It is true, you have to use Kellog-Warschowski's theorem (wikipidea, will give link soon),for $C^{0,\alpha}$ -maps on $S^1$, and note that $|t-\zeta|^{\alpha}$ is a $C^{0,\alpha}(S^1)$ map, with the H …
- 1,471
4
votes
1
answer
727
views
A quick and elementary question from Hubbard's Teichmuller Theory : Volume I
Hi,
On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :
Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U \b …
- 1,471
4
votes
1
answer
600
views
A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$
Hello,
This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :
If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p …
- 1,471
1
vote
1
answer
289
views
Boundary regularity of quasiconformal homeomorphisms of the unit disk ?
Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving quasi …
- 1,471
2
votes
1
answer
461
views
Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ f...
I asked the question before, but didn't get any reply, so I took the liberty to ask again.
Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is th …
- 1,471
0
votes
2
answers
288
views
Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet...
Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following
Theorem: …
- 1,471
1
vote
1
answer
199
views
On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis
Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
what …
- 1,471
0
votes
1
answer
180
views
What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\a...
Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $ …
- 1,471
2
votes
3
answers
750
views
How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets
$Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak deri …
- 1,471
2
votes
0
answers
163
views
Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D}...
This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = \frac{1}{2\p …
- 1,471
3
votes
0
answers
430
views
Boundary regularity of the solution to the Beltrami equation
Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:
Let us consider the orientation-preserving homeomorphic solutions $f: D …
- 1,471