# Tagged Questions

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

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### positive and negative parts of a function?

I have this question that I found in a demonstration of a theorem: Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $u^+,u^-$ the positive and negative parts of $u$ respectively. Why do ...
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### A $W^{1,p}$ function that is unbounded on any open subset of $B_1(0)$.

I'm currently studying the properties of Sobolev Spaces in calculus of variations and functional analysis and was wondering if there is a function, that is $W^{1,p}$ but is unbounded on any open ...
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### Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)

For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ ...
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### Continuous representative for functions in $W^{1,2}(\mathbb{R})$

I want to prove that $K(x,y) = \frac{1}{2}e^{-|x-y|}$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$ and as a hint I have given that for $f\in W^{1,2}(\mathbb{R})$ I should use its continuous ...
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### Density in sobolev spaces?

Is the Sobolev space $H_0^1(I)$ dense in $L^2(I)$, where $I\subset\mathbb R$? If so, how do I prove it?
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### Are solutions of nondegenerate PDE positive if the data are the right sign?

Let $\Omega$ be a bounded domain and take Lipschitz function $f:\mathbb{R} \to \mathbb{R}$ be smooth and $0 < c_1 \leq f' \leq c_2$. With the equation $$u' - \Delta f(u) = g$$ $$u(0) = u_0$$ with ...
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### The existence of solution of wave equation with initial/boundary data in $H^{-1}$ and $L^2$

In a paper by Zuazua, "Propagation, Observation, Control, and Numerical Approximation of Waves", which you can find here http://www.sissa.it/fa/am/DCS2003/reading_mat/zuazuajpg.pdf he considers the ...
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### convolutions and the mollification of functions in $L^1_{\hbox{loc}}(\Omega)$

The following is from the appendix in Leonin's First Course in Sobolev Spaces: Here is my question: Could anybody explain what is the meaning of "$u_\varepsilon$ is well-defined"? What is ...
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### A limit about measure set

Assume $\Psi\in W^{1,\frac{3}{2}}_{loc}(\mathbb{R^2})$, satisfies: $$\lim\limits_{r\to 0}\frac{1}{r}\int_{B_r(x)}|\nabla\Psi(y)|^{3/2}dy=0$$ Then for each $(x_0,y_0)$, and for every $\epsilon>0$ ...
Let's consider the one-dimensional ODE: $$u_{,xx}(x)+1=0,\quad \forall x\in ]0,\ell[$$ together with the two Dirichlet boundary conditions $u(0)=0$ and $u(\ell)=0$. The corresponding weak ...