Question

How does derivative transfer to test function?

score 86 · Answer 1 · 2026-03-25 23:18:39

86

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How does derivative transfer to test function?

Published on 25 Mar 2026 - 23:18

score 38 · Answer 2 · 2026-03-28 17:40:27

38

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If $\Omega\subseteq\mathbb R^d$ is sufficiently nice, does it hold $L^\infty(\Omega)\subseteq H^1(\Omega)$?

Published on 28 Mar 2026 - 17:40

score 124 · Answer 3 · 2026-03-27 11:44:05

124

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Computing Hodge Laplacian

Published on 27 Mar 2026 - 11:44

score 461 · Answer 4 · 2021-06-09 10:31:56

461

Views

If a matrix $A$ is orthogonal, show that $\Delta(f \circ A) = (\Delta f)\circ A$

Published on 09 Jun 2021 - 10:31

score 54 · Answer 5 · 2026-03-26 20:41:15

54

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Showing $\int_{\Bbb S^{d-1}} \Delta_{\Bbb S^{d-1}} g(\Omega)\ j(\Omega) d^{d-1}\Omega = 0$

Published on 26 Mar 2026 - 20:41

score 281 · Answer 6 · 2021-06-10 04:07:49

281

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Neumann's problem implies that the mean value of the normal derivative at the boundary is zero.

Published on 10 Jun 2021 - 4:07

score 26 · Answer 7 · 2021-06-11 19:56:03

26

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Show that the solution to the Neumann problem is $u(x,y)=c+\frac{1}{2\pi}\int_{-\infty}^{\infty}{f(\xi)\ln[y^2+(x-\xi)^2]} d\xi$

Published on 11 Jun 2021 - 19:56

score 200 · Answer 8 · 2021-06-11 21:50:04

200

Views

Question about uniqueness of solutions of Laplace equation

Published on 11 Jun 2021 - 21:50

score 92 · Answer 9 · 2026-03-28 02:24:24

92

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Is the "Dirichlet Laplacian" an extension of $\Delta$ on $C^2(Ω)$ or only on $\{u\in C(\overline Ω)\cap C^2(Ω):\left.u\right|_{\partial\Omega}=0\}$?

Published on 28 Mar 2026 - 2:24

score 101 · Answer 10 · 2026-03-28 03:34:40

101

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Can we show that the normal derivative of the first eigenfunction of $-\Delta$ is negative?

Published on 28 Mar 2026 - 3:34

score 50 · Answer 11 · 2021-06-26 08:20:44

50

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Solution to $\Delta u = 0$ on $B, u=f$ on $\partial B$, which is of class $C^0(\bar{B})\cap C^2(B)\cap W_2^1(B)$

Published on 26 Jun 2021 - 8:20

score 794 · Answer 12 · 2021-06-29 08:43:34

794

Views

Fourier transform of Laplacian in polar coordinates

Published on 29 Jun 2021 - 8:43

score 162 · Answer 13 · 2026-03-26 23:10:39

162

Views

Show that the solution of $u_t=\Delta u$ satisfies $u(t_0,x)\ge c\operatorname{dist}(x,\partial\Omega)$ for some $t_0,c>0$

Published on 26 Mar 2026 - 23:10

score 1.8k · Answer 14 · 2021-07-01 13:59:30

1.8k

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Explaining the integral of Laplacian of $r \mapsto \frac1r$

Published on 01 Jul 2021 - 13:59

score 61 · Answer 15 · 2026-03-25 12:26:14

61

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Isomorphism of Differential operator involving the Laplacian

Published on 25 Mar 2026 - 12:26

List Question

How does derivative transfer to test function?

If $\Omega\subseteq\mathbb R^d$ is sufficiently nice, does it hold $L^\infty(\Omega)\subseteq H^1(\Omega)$?

Computing Hodge Laplacian

If a matrix $A$ is orthogonal, show that $\Delta(f \circ A) = (\Delta f)\circ A$

Showing $\int_{\Bbb S^{d-1}} \Delta_{\Bbb S^{d-1}} g(\Omega)\ j(\Omega) d^{d-1}\Omega = 0$

Neumann's problem implies that the mean value of the normal derivative at the boundary is zero.

Show that the solution to the Neumann problem is $u(x,y)=c+\frac{1}{2\pi}\int_{-\infty}^{\infty}{f(\xi)\ln[y^2+(x-\xi)^2]} d\xi$

Question about uniqueness of solutions of Laplace equation

Is the "Dirichlet Laplacian" an extension of $\Delta$ on $C^2(Ω)$ or only on $\{u\in C(\overline Ω)\cap C^2(Ω):\left.u\right|_{\partial\Omega}=0\}$?

Can we show that the normal derivative of the first eigenfunction of $-\Delta$ is negative?

Solution to $\Delta u = 0$ on $B, u=f$ on $\partial B$, which is of class $C^0(\bar{B})\cap C^2(B)\cap W_2^1(B)$

Fourier transform of Laplacian in polar coordinates

Show that the solution of $u_t=\Delta u$ satisfies $u(t_0,x)\ge c\operatorname{dist}(x,\partial\Omega)$ for some $t_0,c>0$

Explaining the integral of Laplacian of $r \mapsto \frac1r$

Isomorphism of Differential operator involving the Laplacian

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