Question

Brezis' exercise 8.17: the kernel of $A^*$ where $A u=u''-xu'$

score 29 · Answer 1 · 2026-03-25 21:48:42

29

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Brezis' exercise 8.17: the kernel of $A^*$ where $A u=u''-xu'$

Published on 25 Mar 2026 - 21:48

score 66 · Answer 2 · 2023-11-21 22:18:25

66

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An absolutely continuous function which is not uniformly continuous?

Published on 21 Nov 2023 - 22:18

score 73 · Answer 3 · 2026-03-28 08:50:05

73

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If $u \in L^p$ and $\Delta u \in L^p$ as well, do we necessarily have $u \in W^{2,p}$ for $1<p<\infty$?

Published on 28 Mar 2026 - 8:50

score 63 · Answer 4 · 2023-11-22 15:34:16

63

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Sobolev space on the circle (Fourier series)

Published on 22 Nov 2023 - 15:34

score 55 · Answer 5 · 2026-03-27 00:58:41

55

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Proof of $|x|^{-\alpha} \in W^{1,p}(U)$

Published on 27 Mar 2026 - 0:58

score 64 · Answer 6 · 2026-04-01 03:44:23

64

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Brezis' exercise 8.18: show that $u$ is the solution of some ODE with appropriate boundary conditions

Published on 01 Apr 2026 - 3:44

score 121 · Answer 7 · 2023-11-22 22:31:46

121

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Does $\|f_n -f\|_{L^2} \to 0$ imply $f_n \to f$ in the weak topology of $H^1 (I)$?

Published on 22 Nov 2023 - 22:31

score 41 · Answer 8 · 2026-03-27 00:59:58

41

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If $u(x) \in H^1(\mathbb R^n;\mathbb C)$, does the weak-derivative of $\frac{u(x)}{|u(x)|}$ exists?

Published on 27 Mar 2026 - 0:59

score 58 · Answer 9 · 2026-03-26 12:58:30

58

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Is this comment about weak convergence in $W^{1,2}(\Omega)$ correct?

Published on 26 Mar 2026 - 12:58

score 32 · Answer 10 · 2026-03-26 21:26:00

32

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The map $T: (H^1 (I), \|\cdot\|_{L^2}) \to \mathbb R, u \mapsto \|u\|_{H^1}$ is lower semi-continuous

Published on 26 Mar 2026 - 21:26

score 88 · Answer 11 · 2026-03-28 13:27:20

88

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Rellich-Kondrachov theorem in dimension one

Published on 28 Mar 2026 - 13:27

score 58 · Answer 12 · 2026-03-28 13:21:19

58

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Let $I$ be an open bounded interval of $\mathbb R$. The injection $W^{1,p}(I) \subset L^q(I)$ is compact for any $p, q \in [1, \infty)$

Published on 28 Mar 2026 - 13:21

score 54 · Answer 13 · 2023-11-24 15:32:08

54

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Brezis' exercise 8.20.4: prove that $\frac{u(x)}{x^2} \in L^2(I), \frac{u(x)}{x} \in H^1(I)$, and $\frac{u'(x)}{x} \in L^2(I)$

Published on 24 Nov 2023 - 15:32

score 42 · Answer 14 · 2026-03-27 17:57:29

42

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If Sobolev space is isometric to $L^2(\mathbb{R})$ then Sobolev space is complete?

Published on 27 Mar 2026 - 17:57

score 76 · Answer 15 · 2026-03-29 03:36:59

76

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The existence of a solution to $-(pU')' + qU = f$ on $(0, 1)$ with boundary condition $U(1) = 0$

Published on 29 Mar 2026 - 3:36

List Question

Brezis' exercise 8.17: the kernel of $A^*$ where $A u=u''-xu'$

An absolutely continuous function which is not uniformly continuous?

If $u \in L^p$ and $\Delta u \in L^p$ as well, do we necessarily have $u \in W^{2,p}$ for $1<p<\infty$?

Sobolev space on the circle (Fourier series)

Proof of $|x|^{-\alpha} \in W^{1,p}(U)$

Brezis' exercise 8.18: show that $u$ is the solution of some ODE with appropriate boundary conditions

Does $\|f_n -f\|_{L^2} \to 0$ imply $f_n \to f$ in the weak topology of $H^1 (I)$?

If $u(x) \in H^1(\mathbb R^n;\mathbb C)$, does the weak-derivative of $\frac{u(x)}{|u(x)|}$ exists?

Is this comment about weak convergence in $W^{1,2}(\Omega)$ correct?

The map $T: (H^1 (I), \|\cdot\|_{L^2}) \to \mathbb R, u \mapsto \|u\|_{H^1}$ is lower semi-continuous

Rellich-Kondrachov theorem in dimension one

Let $I$ be an open bounded interval of $\mathbb R$. The injection $W^{1,p}(I) \subset L^q(I)$ is compact for any $p, q \in [1, \infty)$

Brezis' exercise 8.20.4: prove that $\frac{u(x)}{x^2} \in L^2(I), \frac{u(x)}{x} \in H^1(I)$, and $\frac{u'(x)}{x} \in L^2(I)$

If Sobolev space is isometric to $L^2(\mathbb{R})$ then Sobolev space is complete?

The existence of a solution to $-(pU')' + qU = f$ on $(0, 1)$ with boundary condition $U(1) = 0$

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