List questions from sobolev-spaces

Published on 16 Apr 2026 - 3:45

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If $f\in L^2$, has the equation $u_{xx}=f$ an unique solution $u\in H^2\cap H_0^1$?

Published on 16 Apr 2026 - 3:27

129

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There are $u$ in $W^{1,p}(D)$ and a weakly converging subsequence $\left\{ u_{m_{k}}\right\} $ to $u$.

Published on 16 Apr 2026 - 4:39

140

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Is $\nabla u \in L^{\infty}$ if $u$ is bounded $C^{0}$?

Published on 14 Apr 2026 - 15:28

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Fractional Sobolev embedding into $L^\infty$

Published on 10 Apr 2026 - 18:04

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Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

Published on 17 Apr 2026 - 1:05

277

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A version of Rellich-Kondrachov's theorem

Published on 13 Apr 2026 - 22:08

47

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Characteristic Method for linear equation

Published on 12 Apr 2026 - 6:10

40

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$T: H^{-\infty}(R^n) \to H^\infty(R^n)$ continuous iff $T: H^{-r}(R^n) \to H^s(R^n)$ bounded for all $r,s>0$?

Published on 12 Apr 2026 - 23:28

139

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What is the Frechet derivative of $(u^+)^q$?

Published on 12 Apr 2026 - 6:51

236

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Showing PDE solution map (takes initial condition and gives solution) is continuous?

Published on 13 Apr 2026 - 11:03

683

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PDE uniqueness by energy method contradicts non-uniqueness???

Published on 11 Apr 2026 - 18:39

48

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$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Published on 28 Mar 2026 - 23:58

1k

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Is $H^2\cap H^1_0$ dense in $H_0^1$?

Published on 15 Apr 2026 - 20:04

102

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Is $\{f\in H^1;\;\int f=0\}$ dense in $\{f\in L^2;\;\int f=0\}$?

Published on 15 Apr 2026 - 22:42

List Question

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

If $f\in L^2$, has the equation $u_{xx}=f$ an unique solution $u\in H^2\cap H_0^1$?

There are $u$ in $W^{1,p}(D)$ and a weakly converging subsequence $\left\{ u_{m_{k}}\right\} $ to $u$.

Is $\nabla u \in L^{\infty}$ if $u$ is bounded $C^{0}$?

Fractional Sobolev embedding into $L^\infty$

Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

A version of Rellich-Kondrachov's theorem

Characteristic Method for linear equation

$T: H^{-\infty}(R^n) \to H^\infty(R^n)$ continuous iff $T: H^{-r}(R^n) \to H^s(R^n)$ bounded for all $r,s>0$?

What is the Frechet derivative of $(u^+)^q$?

Showing PDE solution map (takes initial condition and gives solution) is continuous?

PDE uniqueness by energy method contradicts non-uniqueness???

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Is $H^2\cap H^1_0$ dense in $H_0^1$?

Is $\{f\in H^1;\;\int f=0\}$ dense in $\{f\in L^2;\;\int f=0\}$?

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