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# Questions tagged [sobolev-spaces]

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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### Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
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### Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
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### dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
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### The dual of the Sobolev space $W^{k,p}$

The dual of the Sobolev space if defined to be $$(W^{k,p}(\Omega))' = W_0^{-k,p'}(\Omega)$$ where $\frac 1 p + \frac 1 {p'} = 1$. Why makes this definition sense, especially why do we have $L^{p'}$-...
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### Elliptic Regularity Theorem

I want to collect some results on elliptic regularity. The problem I consider is \begin{align} Lu&=f,&in \quad U,\\ u&=g,&on \quad \partial U.\tag{1} ...
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### prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
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### Product rule of weak derivatives

I am working on proving the following proposition: If $u,v\in {W^1(\Omega)}$ and $uv,uDv+vDu\in L^1_{\operatorname{loc}}(\Omega)$, then we have the product formula $$D(uv)=uDv+vDu.$$ The definition I ...
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### Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
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### Differences between $-\Delta: H_0^1(\Omega)\to H^{-1}(\Omega)$ and $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$

I'll try to explain what I want to know: Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. When we look to $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$, the meaning of $-\Delta$ is ...
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### Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to contain ...
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### Understanding a theorem concerning Sobolev spaces

I have two doubts in the proof of the theorem below. If you want the detaIls can be found here. Theorem A Let $q$ a given real number $q>p$. Let $u \in W^{1,p}$ be a solution to (E2). Assume ...
Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt$$ where $P$ is a strictly hyperbolic linear operator. For ...
What is the intuition behind the Sobolev-type inequality (The Gagliardo-Nirenberg-Sobolev Inequality) Assume $1\leq p <n$ and $U$ a be a bounded open subset of $\mathbb{R}^{n}$, and suppose \$\...