Question

Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$

score 18 · Answer 1 · 2026-03-19 08:34:39

18

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Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$

Published on 19 Mar 2026 - 8:34

score 56 · Answer 2 · 2024-03-19 13:47:45

56

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Fundamental lemma of calculus of variations for higher order-derivatives

Published on 19 Mar 2024 - 13:47

score 218 · Answer 3 · 2024-03-20 13:00:03

218

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Derivative of expected value

Published on 20 Mar 2024 - 13:00

score 68 · Answer 4 · 2024-03-23 08:22:10

68

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Derivative of $H(c)=\mathbb E[(c-X)\chi_{X\leq c}]$

Published on 23 Mar 2024 - 8:22

score 22 · Answer 5 · 2024-03-27 13:38:04

22

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$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

Published on 27 Mar 2024 - 13:38

score 1.2k · Answer 6 · 2026-03-27 02:39:51

1.2k

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chain rule for weak derivatives

Published on 27 Mar 2026 - 2:39

score 3.1k · Answer 7 · 2015-01-18 17:21:44

3.1k

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first order weak derivative of function $ f(x) = |x| $

Published on 18 Jan 2015 - 17:21

score 873 · Answer 8 · 2015-01-30 21:43:34

873

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Difference quotient bounded in $L^1$ does not imply the weak partial derivative exists and is in $L^1$

Published on 30 Jan 2015 - 21:43

score 83 · Answer 9 · 2015-01-31 16:07:18

83

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Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

Published on 31 Jan 2015 - 16:07

score 2.1k · Answer 10 · 2015-01-31 21:35:23

2.1k

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How to prove the chain rule with respect to weak derivatives?

Published on 31 Jan 2015 - 21:35

score 2k · Answer 11 · 2015-02-08 08:11:05

2k

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How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?

Published on 08 Feb 2015 - 8:11

score 201 · Answer 12 · 2015-02-08 21:56:13

201

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Question about this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

Published on 08 Feb 2015 - 21:56

score 1.8k · Answer 13 · 2015-02-10 07:48:11

1.8k

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Does the existence of weak derivatives require the lower order derivatives also to exist?

Published on 10 Feb 2015 - 7:48

score 181 · Answer 14 · 2015-02-21 20:03:15

181

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A step of change of variables I don't understand when show derivative of a function in $L^n$?

Published on 21 Feb 2015 - 20:03

score 183 · Answer 15 · 2015-03-20 18:18:37

183

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Weak derivative (Sobolev spaces)

Published on 20 Mar 2015 - 18:18

List Question

Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$

Fundamental lemma of calculus of variations for higher order-derivatives

Derivative of expected value

Derivative of $H(c)=\mathbb E[(c-X)\chi_{X\leq c}]$

$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

chain rule for weak derivatives

first order weak derivative of function $ f(x) = |x| $

Difference quotient bounded in $L^1$ does not imply the weak partial derivative exists and is in $L^1$

Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

How to prove the chain rule with respect to weak derivatives?

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?

Question about this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

Does the existence of weak derivatives require the lower order derivatives also to exist?

A step of change of variables I don't understand when show derivative of a function in $L^n$?

Weak derivative (Sobolev spaces)

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