Question

Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

score 109 · Answer 1 · 2026-04-15 16:05:35

109

Views

Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

Published on 15 Apr 2026 - 16:05

score 80 · Answer 2 · 2026-04-10 21:24:35

80

Views

Proof that estimate holds

Published on 10 Apr 2026 - 21:24

score 111 · Answer 3 · 2026-04-12 20:27:26

111

Views

Construct extension of function

Published on 12 Apr 2026 - 20:27

score 553 · Answer 4 · 2026-04-12 02:04:57

553

Views

Schwartz functions dense?

Published on 12 Apr 2026 - 2:04

score 49 · Answer 5 · 2026-04-14 21:00:00

49

Views

assigning boundary values to a weakly differentiable function

Published on 14 Apr 2026 - 21:00

score 80 · Answer 6 · 2026-03-28 01:26:19

80

Views

If $\{\nabla u_j\}$ is Cauchy in $L^p(\mathbb{R}^n)$ and $\int_{B(0,1)} u_j dx = 0$, does $\{u_j\}$ converge in $L^p_{\text{loc}}(\mathbb{R}^n)$?

Published on 28 Mar 2026 - 1:26

score 1k · Answer 7 · 2026-04-17 00:31:47

1k

Views

Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Published on 17 Apr 2026 - 0:31

score 154 · Answer 8 · 2026-04-12 15:18:19

154

Views

Can we extend a function $u \in H_0^1(\Omega)$ to $\overline{u} \in H_0^1(\widetilde{\Omega})$ with $\Omega \subset \widetilde{\Omega}$?

Published on 12 Apr 2026 - 15:18

score 198 · Answer 9 · 2026-04-16 01:44:46

198

Views

Determine if a function belongs to the sobolev space $W^{1,p}(\mathbb{R})$ and not to $L^q(\mathbb{R})$

Published on 16 Apr 2026 - 1:44

score 371 · Answer 10 · 2026-04-15 10:56:50

371

Views

Why is the Black-Scholes PDE called degenerate

Published on 15 Apr 2026 - 10:56

score 88 · Answer 11 · 2026-04-13 14:07:49

88

Views

Convergence of the inverse in Sobolev spaces

Published on 13 Apr 2026 - 14:07

score 161 · Answer 12 · 2026-04-14 05:39:29

161

Views

Reference request: $W^{2,2}$ estimates of elliptic PDE with measurable coefficients

Published on 14 Apr 2026 - 5:39

score 1k · Answer 13 · 2026-04-12 20:22:15

1k

Views

Generalized Poincaré Inequality on H1 proof

Published on 12 Apr 2026 - 20:22

score 18 · Answer 14 · 2026-04-17 04:27:07

18

Views

$||D_hu||_{L^2(Q_+)} \leq ||\nabla u||_{L^2(Q_+)}$ for $u \in H_0^1(Q_+)$

Published on 17 Apr 2026 - 4:27

score 237 · Answer 15 · 2026-04-13 02:52:27

237

Views

Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

Published on 13 Apr 2026 - 2:52

List Question

Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

Proof that estimate holds

Construct extension of function

Schwartz functions dense?

assigning boundary values to a weakly differentiable function

If $\{\nabla u_j\}$ is Cauchy in $L^p(\mathbb{R}^n)$ and $\int_{B(0,1)} u_j dx = 0$, does $\{u_j\}$ converge in $L^p_{\text{loc}}(\mathbb{R}^n)$?

Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Can we extend a function $u \in H_0^1(\Omega)$ to $\overline{u} \in H_0^1(\widetilde{\Omega})$ with $\Omega \subset \widetilde{\Omega}$?

Determine if a function belongs to the sobolev space $W^{1,p}(\mathbb{R})$ and not to $L^q(\mathbb{R})$

Why is the Black-Scholes PDE called degenerate

Convergence of the inverse in Sobolev spaces

Reference request: $W^{2,2}$ estimates of elliptic PDE with measurable coefficients

Generalized Poincaré Inequality on H1 proof

$||D_hu||_{L^2(Q_+)} \leq ||\nabla u||_{L^2(Q_+)}$ for $u \in H_0^1(Q_+)$

Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

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