List questions from sobolev-spaces

Published on 11 Apr 2026 - 9:33

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Are there $L^2$ functions whose Laplacian is in $L^2$ yet its gradient is not in $\mathbf{L}^2$?

Published on 02 Jul 2015 - 14:36

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Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Published on 02 Jul 2015 - 22:09

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Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Published on 06 Jul 2015 - 3:01

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A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Published on 28 Mar 2026 - 3:03

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Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

Published on 07 Jul 2015 - 4:59

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Composition operators on fractional-order Sobolev spaces

Published on 10 Apr 2026 - 16:45

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Continuous Sobolev Embedding

Published on 09 Apr 2026 - 9:09

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Why L belongs to the dual space $H^{-1}$

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Closedness of the range of differential operator first order

Published on 07 Jul 2015 - 19:22

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non existence of weak derivative evans pde chapter 5 example 2

Published on 09 Jul 2015 - 7:25

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a vector calculus problem when coping with problem 9 chapter 5 evans

Published on 11 Apr 2026 - 2:43

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$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

Published on 10 Jul 2015 - 8:32

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the equivalence of a absolute value function $|D^2 u|$ in problem 10(b) evans pde chapter 5

Published on 11 Jul 2015 - 6:51

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How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

Published on 28 Mar 2026 - 19:35

List Question

show that $uv\in W^{1,r}(\Omega)$

Are there $L^2$ functions whose Laplacian is in $L^2$ yet its gradient is not in $\mathbf{L}^2$?

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

Composition operators on fractional-order Sobolev spaces

Continuous Sobolev Embedding

Why L belongs to the dual space $H^{-1}$

Closedness of the range of differential operator first order

non existence of weak derivative evans pde chapter 5 example 2

a vector calculus problem when coping with problem 9 chapter 5 evans

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

the equivalence of a absolute value function $|D^2 u|$ in problem 10(b) evans pde chapter 5

How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

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