# Tagged Questions

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### Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$\mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx}$$ ...
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Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$\inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ?$$ If ...
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### Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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### Some ideas about $H=W$

Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ . Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$ where ...
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### How can I integrate this?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where ...
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### Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
While studying a nonlinear PDE arising from quantum mechanics, I met a statement that I cannot prove easily. Let us write $E=W^{1,2}(\mathbb{R}^3)$ for the usual Sobolev space, and define the ...