Question

Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

score 586 · Answer 1 · 2026-03-31 16:57:17

586

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Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

Published on 31 Mar 2026 - 16:57

score 772 · Answer 2 · 2026-04-11 10:48:28

772

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An exercise showing that $l^1$ is not the dual of $l^\infty$

Published on 11 Apr 2026 - 10:48

score 1.7k · Answer 3 · 2026-03-31 16:57:26

1.7k

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Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Published on 31 Mar 2026 - 16:57

score 699 · Answer 4 · 2026-03-31 16:54:07

699

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Verify that the unbounded function belongs to $W^{1,n}$

Published on 31 Mar 2026 - 16:54

score 123 · Answer 5 · 2026-04-12 08:45:43

123

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is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$?

Published on 12 Apr 2026 - 8:45

score 233 · Answer 6 · 2015-01-17 15:05:17

233

Views

$f,g\in L^1(\mu)\implies fg\in L^1(\mu)$

Published on 17 Jan 2015 - 15:05

score 1.1k · Answer 7 · 2026-04-08 18:07:56

1.1k

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If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

Published on 08 Apr 2026 - 18:07

score 1k · Answer 8 · 2026-04-08 18:06:37

1k

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

Published on 08 Apr 2026 - 18:06

score 425 · Answer 9 · 2026-04-13 00:24:36

425

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Topology of $L^2$ space

Published on 13 Apr 2026 - 0:24

score 50 · Answer 10 · 2015-01-18 12:14:57

50

Views

$f_n\to f$ in $L^2$ and $fg\in L^2(\Omega)\implies f_n\,g\in L^2?$

Published on 18 Jan 2015 - 12:14

score 54 · Answer 11 · 2015-01-18 13:13:36

54

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If $f_n\to f$ in $L^1$ can we derive that the functions $f_n$ are bounded by an integrable function?

Published on 18 Jan 2015 - 13:13

score 61 · Answer 12 · 2026-04-10 07:57:32

61

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Is $f(u) := \int_\Omega u^2(x)h(x)$ weakly lower semicontinuous in $L^2(\Omega)$?

Published on 10 Apr 2026 - 7:57

score 74 · Answer 13 · 2026-04-12 19:22:16

74

Views

Boundedness of linear operators in $L^p$

Published on 12 Apr 2026 - 19:22

score 1.8k · Answer 14 · 2015-01-21 06:22:43

1.8k

Views

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Published on 21 Jan 2015 - 6:22

score 2.5k · Answer 15 · 2026-04-10 06:12:39

2.5k

Views

Adjoint of Integral Operator in $L^p$

Published on 10 Apr 2026 - 6:12

List Question

Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

An exercise showing that $l^1$ is not the dual of $l^\infty$

Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Verify that the unbounded function belongs to $W^{1,n}$

is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$?

$f,g\in L^1(\mu)\implies fg\in L^1(\mu)$

If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

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

Topology of $L^2$ space

$f_n\to f$ in $L^2$ and $fg\in L^2(\Omega)\implies f_n\,g\in L^2?$

If $f_n\to f$ in $L^1$ can we derive that the functions $f_n$ are bounded by an integrable function?

Is $f(u) := \int_\Omega u^2(x)h(x)$ weakly lower semicontinuous in $L^2(\Omega)$?

Boundedness of linear operators in $L^p$

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Adjoint of Integral Operator in $L^p$

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