Tagged Questions

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Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
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Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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Weak Derivative Heaviside function

I have to prove that the Heaviside function $$H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases}$$ doesn't admit weak derivative in ...
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some sobolev norm estimation

I would like to show this inequality. I need help to show this inequality Let $F(\Phi)=\left|\Phi\right|^{\alpha}\Phi$ with even integer $\alpha>0$. Let $k$ be a positive integer satisfying ...
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Basic Query On Sobolev Space

Okay, I am very new to the premise of Sobolev Spaces and there is one exercise here that's really grinding my gears. The premise is this- to what Sobolev spaces for each real number $\alpha$ does ...
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If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
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weak differentiability of log log function

I want to understand why the following function has a weak derivative in two or three dimensions: $w(x) = \ln |\ln|x|| , x \in B_{1/2}(0)$. Can I say that if I have a strong derivative (except for ...
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For which real values of $\alpha$ PDE $\Delta u(x,y)+2u(x,y)=x-\alpha$ has at least one weak solution?

Problem. Consider boundary value problem: \begin{cases} \Delta u(x,y)+2u(x,y)=x-\alpha, & \text{in $\Omega$,} \\ u(x,y)=0, & \text{on $\partial\Omega$,} \\ \end{cases} where $\alpha$ is ...