Question

Find $\alpha$ so $x\mapsto ||x||_2^{-\alpha} \in L_{\text{loc}}^1(\mathbb{R}^d),W^{1,p}([-1,1]^d)$

score 79 · Answer 1 · 2019-10-23 13:23:29

79

Views

Find $\alpha$ so $x\mapsto ||x||_2^{-\alpha} \in L_{\text{loc}}^1(\mathbb{R}^d),W^{1,p}([-1,1]^d)$

Published on 23 Oct 2019 - 13:23

score 101 · Answer 2 · 2019-10-23 22:41:13

101

Views

Sobolev Regularity means esssup can be replaced with regular sup?

Published on 23 Oct 2019 - 22:41

score 37 · Answer 3 · 2026-03-29 08:35:39

37

Views

All values of $\beta \in R$ : $u(x):= \max\{||x||^\beta -5,0\} \in W^{1,2}(R^2)$

Published on 29 Mar 2026 - 8:35

score 75 · Answer 4 · 2019-10-24 17:46:32

75

Views

Is $f$ weakly lower semicontinuous?

Published on 24 Oct 2019 - 17:46

score 140 · Answer 5 · 2019-10-24 18:29:58

140

Views

Is the duality pairing $\langle \phi, \psi \rangle_{H^{1/2}\times H^{-1/2}}$ simply the $L^2$ inner product?

Published on 24 Oct 2019 - 18:29

score 34 · Answer 6 · 2026-03-29 22:14:36

34

Views

Approximation of a Sobolev hyper-surface by a smooth one

Published on 29 Mar 2026 - 22:14

score 183 · Answer 7 · 2026-03-26 04:53:01

183

Views

Intuition for Sobolev spaces $H^{1}(U)$ and $H_{0}^{1}(U)$ where $U\subset\mathbb{R}^{d}$ is open (and Laplace operator)

Published on 26 Mar 2026 - 4:53

score 42 · Answer 8 · 2019-10-25 13:58:41

42

Views

Integral of $ \partial_{x_i} f $ over $ \mathbb{R}^N $ is $0$ if $f, \partial_{x_i} f \in L^1(\mathbb{R}^N)$

Published on 25 Oct 2019 - 13:58

score 44 · Answer 9 · 2019-10-27 09:31:51

44

Views

Linear continuos map

Published on 27 Oct 2019 - 9:31

score 142 · Answer 10 · 2019-10-27 13:18:00

142

Views

Showing that a function from Sobolev space $W^{1,2}$ decays to zero

Published on 27 Oct 2019 - 13:18

score 135 · Answer 11 · 2026-03-29 07:22:25

135

Views

Strong convergence from weak one. Correct reasoning?

Published on 29 Mar 2026 - 7:22

score 91 · Answer 12 · 2026-03-26 02:57:53

91

Views

Show $||u-\hat{u}||_{H^1(\Omega)} \leq c||a-\hat{a}||_{L^{\infty}}$ for $−\text{div}(a\nabla{u})=f, \text{div}(\hat{a}\nabla{u})=f$

Published on 26 Mar 2026 - 2:57

score 149 · Answer 13 · 2026-03-30 23:54:42

149

Views

How big is $ H^{\frac{1}{2}}_{per} (0,2\pi) $?

Published on 30 Mar 2026 - 23:54

score 236 · Answer 14 · 2026-03-29 07:19:39

236

Views

A little help to show strong convergence in Sobolev space $H^1$

Published on 29 Mar 2026 - 7:19

score 78 · Answer 15 · 2026-03-29 08:45:07

78

Views

How to make sense of Sobolev spaces on bounded domain from the point of view of distributions.

Published on 29 Mar 2026 - 8:45

List Question

Find $\alpha$ so $x\mapsto ||x||_2^{-\alpha} \in L_{\text{loc}}^1(\mathbb{R}^d),W^{1,p}([-1,1]^d)$

Sobolev Regularity means esssup can be replaced with regular sup?

All values of $\beta \in R$ : $u(x):= \max\{||x||^\beta -5,0\} \in W^{1,2}(R^2)$

Is $f$ weakly lower semicontinuous?

Is the duality pairing $\langle \phi, \psi \rangle_{H^{1/2}\times H^{-1/2}}$ simply the $L^2$ inner product?

Approximation of a Sobolev hyper-surface by a smooth one

Intuition for Sobolev spaces $H^{1}(U)$ and $H_{0}^{1}(U)$ where $U\subset\mathbb{R}^{d}$ is open (and Laplace operator)

Integral of $ \partial_{x_i} f $ over $ \mathbb{R}^N $ is $0$ if $f, \partial_{x_i} f \in L^1(\mathbb{R}^N)$

Linear continuos map

Showing that a function from Sobolev space $W^{1,2}$ decays to zero

Strong convergence from weak one. Correct reasoning?

Show $||u-\hat{u}||_{H^1(\Omega)} \leq c||a-\hat{a}||_{L^{\infty}}$ for $−\text{div}(a\nabla{u})=f, \text{div}(\hat{a}\nabla{u})=f$

How big is $ H^{\frac{1}{2}}_{per} (0,2\pi) $?

A little help to show strong convergence in Sobolev space $H^1$

How to make sense of Sobolev spaces on bounded domain from the point of view of distributions.

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