Question

Can $f\in L^2(\Omega)$ imply $\nabla f\in H^{-1}(\Omega):=(H_0^1(\Omega))^*$?

score 86 · Answer 1 · 2026-04-14 09:34:47

86

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Can $f\in L^2(\Omega)$ imply $\nabla f\in H^{-1}(\Omega):=(H_0^1(\Omega))^*$?

Published on 14 Apr 2026 - 9:34

score 100 · Answer 2 · 2026-04-14 00:57:46

100

Views

How is this inequality called? (And how to improve this process)

Published on 14 Apr 2026 - 0:57

score 824 · Answer 3 · 2026-04-09 01:14:35

824

Views

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

Published on 09 Apr 2026 - 1:14

score 138 · Answer 4 · 2026-04-15 12:14:22

138

Views

Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Published on 15 Apr 2026 - 12:14

score 131 · Answer 5 · 2026-04-09 02:11:45

131

Views

Existence of operator

Published on 09 Apr 2026 - 2:11

score 49 · Answer 6 · 2026-04-13 20:10:27

49

Views

Can $f\in L^2(\Omega)$ imply $\nabla f\in [(H^1(\Omega))^*]^n$?

Published on 13 Apr 2026 - 20:10

score 94 · Answer 7 · 2026-04-15 08:12:28

94

Views

Aproximating a Sobolev function by $p$-subharmonic functions.

Published on 15 Apr 2026 - 8:12

score 3k · Answer 8 · 2026-04-16 19:16:05

3k

Views

Generalized Poincaré Inequality on H1 proof.

Published on 16 Apr 2026 - 19:16

score 415 · Answer 9 · 2026-03-27 14:03:35

415

Views

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

Published on 27 Mar 2026 - 14:03

score 141 · Answer 10 · 2026-04-16 03:04:31

141

Views

Scalar property of $ C(\Omega)=\sum_{|\alpha|\leq m}\color{blue}{\big|\Omega\big|^{\frac{2|\alpha|-n}{n}}} \int_{\Omega}|D^\alpha f|^2\ dx $

Published on 16 Apr 2026 - 3:04

score 363 · Answer 11 · 2026-04-14 12:31:11

363

Views

Hadamard counterexample Dirichlet-problem

Published on 14 Apr 2026 - 12:31

score 195 · Answer 12 · 2026-03-27 04:58:06

195

Views

can I get weak convergence in sobolev spaces from convergence of distributions

Published on 27 Mar 2026 - 4:58

score 410 · Answer 13 · 2026-04-13 19:30:06

410

Views

Removable singularities in Sobolev spaces

Published on 13 Apr 2026 - 19:30

score 2.3k · Answer 14 · 2026-04-15 02:52:25

2.3k

Views

$H_m(\mathbb{R}^n)$ , the completion of $C_C^{\infty}(\mathbb{R}^n)$

Published on 15 Apr 2026 - 2:52

score 46 · Answer 15 · 2026-04-16 13:07:58

46

Views

If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u $ in $W^{1,p}(\bar{\Omega})$

Published on 16 Apr 2026 - 13:07

List Question

Can $f\in L^2(\Omega)$ imply $\nabla f\in H^{-1}(\Omega):=(H_0^1(\Omega))^*$?

How is this inequality called? (And how to improve this process)

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Existence of operator

Can $f\in L^2(\Omega)$ imply $\nabla f\in [(H^1(\Omega))^*]^n$?

Aproximating a Sobolev function by $p$-subharmonic functions.

Generalized Poincaré Inequality on H1 proof.

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

Scalar property of $ C(\Omega)=\sum_{|\alpha|\leq m}\color{blue}{\big|\Omega\big|^{\frac{2|\alpha|-n}{n}}} \int_{\Omega}|D^\alpha f|^2\ dx $

Hadamard counterexample Dirichlet-problem

can I get weak convergence in sobolev spaces from convergence of distributions

Removable singularities in Sobolev spaces

$H_m(\mathbb{R}^n)$ , the completion of $C_C^{\infty}(\mathbb{R}^n)$

If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u $ in $W^{1,p}(\bar{\Omega})$

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