List questions from sobolev-spaces

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Decreasing of energy functional

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Inner product in Sobolev space

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Lp convergence in sobolev space implies sobolev convergence?

Published on 22 Feb 2017 - 9:48

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How can we prove $\langle(Ψ\cdot∇)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}+\frac12\langle(∇\cdotΨ)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}=0$ for $Φ,Ψ\in H_0^1(Λ,ℝ^d)$?

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Evaluating a lift operator

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How to prove $H^1_0(\Omega)$ is strictly contained in $H^1(\Omega)$ when $\Omega$ is open and bounded?

Published on 24 Feb 2017 - 15:30

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$\left | \int_{\mathbb R^d} \nabla f \nabla g \right| \leq \|\Delta f \|_{L^2} \|g\|_{L^{2}}$?

Published on 27 Feb 2017 - 4:58

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Does the Gagliardo–Nirenberg interpolation inequality also hold for surfaces?

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The functional derivative , Fréchet and Gâteaux differentiability of the energy operator

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the multiplication in Sobolev spaces

Published on 28 Feb 2017 - 9:29

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$\langle \nabla g_n, \nabla h \rangle \to \langle \nabla \phi, \nabla h \rangle \implies \langle g_n, \psi \rangle \to \langle \phi, \psi\rangle $?

Published on 01 Mar 2017 - 12:16

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The homogeneous Sobolev space-inclusion

Published on 29 Mar 2026 - 2:36

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R.H.S. of Poisson equation localized $\Rightarrow$ Solution localized

Published on 26 Mar 2026 - 9:41

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Fractional differential operator

Published on 03 Mar 2017 - 18:57

List Question

Sobolev Spaces and periodic functions

Decreasing of energy functional

Inner product in Sobolev space

Lp convergence in sobolev space implies sobolev convergence?

How can we prove $\langle(Ψ\cdot∇)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}+\frac12\langle(∇\cdotΨ)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}=0$ for $Φ,Ψ\in H_0^1(Λ,ℝ^d)$?

Evaluating a lift operator

How to prove $H^1_0(\Omega)$ is strictly contained in $H^1(\Omega)$ when $\Omega$ is open and bounded?

$\left | \int_{\mathbb R^d} \nabla f \nabla g \right| \leq \|\Delta f \|_{L^2} \|g\|_{L^{2}}$?

Does the Gagliardo–Nirenberg interpolation inequality also hold for surfaces?

The functional derivative , Fréchet and Gâteaux differentiability of the energy operator

the multiplication in Sobolev spaces

$\langle \nabla g_n, \nabla h \rangle \to \langle \nabla \phi, \nabla h \rangle \implies \langle g_n, \psi \rangle \to \langle \phi, \psi\rangle $?

The homogeneous Sobolev space-inclusion

R.H.S. of Poisson equation localized $\Rightarrow$ Solution localized

Fractional differential operator

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