# 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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### Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}^{N}$ is open ...
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### If $f$ satisfies $\int_{\Bbb{R}}\frac{f(x)\varphi(x)}{1+x^2}\,dx=0$ for some test functions $\varphi$, is $f$ zero?

Let $D$ be the set of functions $\Bbb{R} \to \Bbb{C}$ of the form $$D = \Big\{ \sum_{k=1}^{n} c_k \mathrm{e}^{it_k x} : c_1, \cdots, c_n \in \Bbb{R}, t_1, \cdots, t_n \in \Bbb{R}, n \geq 1 \Big\}.$$...
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### Bound on Rayleigh quotient on $H^{1}_{\text{per}}[0,1]^{n}$

Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows: A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it ...
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### Tensor product space dense in $H_0^1$?

Given $C_c^{\infty}((a_i,b_i))$ for $i \in \{1,2\}.$ Is it then true that $$C_c^{\infty}(a_1,b_1) \otimes C_c^{\infty} (a_2,b_2)$$ is dense in $H_0^1((a_1,b_1) \times (a_2,b_2))$? Clearly, by ...
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### Find the Minimum of $F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R)$.

Let $F: BV(\mathbb R) \to \mathbb R$ be a functional defined as: $F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R).$ Show that there is no minimum on $W^{1,1}$, but the ...
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### How to find out if a function belongs to $H^2$ or $H^1$

I'm beginning with Sobolev spaces and I found out, that $$H^k = W^{k,2}.$$ I've also seen the following exercise recently: $$\frac{1}{2}u'' = 1$$ And here I'm supposed to find out if $u$ ...
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