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Nate River
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Let $\Omega$$\mathbb B^n$ be a boundedthe open connected subset ofunit ball in $\mathbb R^n$, and $g: \Omega \to \mathbb R^n$$g: \mathbb B^n \to \mathbb R^n$ a measurable function with $|g| \in L^p (\Omega)$$|g| \in L^p (\mathbb B^n)$. Does there exist some function $f$ in the Sobolev space $W^{1, p} (\Omega)$$W^{1, p} (\mathbb B^n)$ such that $\nabla f = g$ a.e.?

Let $\Omega$ be a bounded open connected subset of $\mathbb R^n$, and $g: \Omega \to \mathbb R^n$ a measurable function with $|g| \in L^p (\Omega)$. Does there exist some function $f$ in the Sobolev space $W^{1, p} (\Omega)$ such that $\nabla f = g$ a.e.?

Let $\mathbb B^n$ be the open unit ball in $\mathbb R^n$, and $g: \mathbb B^n \to \mathbb R^n$ a measurable function with $|g| \in L^p (\mathbb B^n)$. Does there exist some function $f$ in the Sobolev space $W^{1, p} (\mathbb B^n)$ such that $\nabla f = g$ a.e.?

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Nate River
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Let $g: \mathbb R^n \to \mathbb R^n$$\Omega$ be a bounded open connected subset of $\mathbb R^n$, and $g: \Omega \to \mathbb R^n$ a measurable function with $|g| \in L^p (\mathbb R^n)$$|g| \in L^p (\Omega)$. Does there exist some function $f$ in the Sobolev space $W^{1, p}$$W^{1, p} (\Omega)$ such that $\nabla f = g$ a.e.?

Let $g: \mathbb R^n \to \mathbb R^n$ be a measurable function with $|g| \in L^p (\mathbb R^n)$. Does there exist some function $f$ in the Sobolev space $W^{1, p}$ such that $\nabla f = g$ a.e.?

Let $\Omega$ be a bounded open connected subset of $\mathbb R^n$, and $g: \Omega \to \mathbb R^n$ a measurable function with $|g| \in L^p (\Omega)$. Does there exist some function $f$ in the Sobolev space $W^{1, p} (\Omega)$ such that $\nabla f = g$ a.e.?

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Nate River
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  • 99

Can every $L^p$ function be written as the weak derivative of a Sobolev function?

Let $g: \mathbb R^n \to \mathbb R^n$ be a measurable function with $|g| \in L^p (\mathbb R^n)$. Does there exist some function $f$ in the Sobolev space $W^{1, p}$ such that $\nabla f = g$ a.e.?