# Tagged Questions

For question about weak derivatives, a notion which extends the classical notion of derivative and allows us to consider derivatives of distributions rather than functions.

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### Proof of the Helmholtz-Hodge decomposition

Let $\Omega\subseteq\mathbb R^3$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$...
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### Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
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### If $F∈H^{-1}(Ω,ℝ^d)$ and $∃p∈\mathcal D'(Ω):\left.F\right|_{\mathcal D(Ω,ℝ^d)}=∇p$, then $∃\overline p∈H^{-1}(Ω):F=∇\overline p$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $\mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d)$ $H^{-1}(\Omega):=H_0^1(\Omega)'$...
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### Is the image of gradient map from Sobolev space to Lebesgue space weakly closed?

Suppose f is a map defined between $W_0^{1,p}(\Omega)$ and $L^{p'}(\Omega)$ as follows - $u \mapsto |\nabla u|^{p-1}$. Is the range of this map weakly closed in $L^{p'}$?.
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### Product of two $W^{1,p}_0$ functions

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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### Convergence of the series $\sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha}$

Suppose $S=\{y_k\}_{k=1}^{\infty}$ is a countable dense subset of the open unit ball $B(0,1)$ in $\mathbb{R}^n$. We write u(x) = \sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha} \...
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### Weak derivative

Let $u \in C(\Omega)$ be a function with weak derivative $Du \in C(\Omega)^n$. How does one prove that $Du$ coincides with the classical derivative? Is the mean value theorem for integration helpful?...
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