# Tagged Questions

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### Given a solution flow to find periodic solutions

Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any ...
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### Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: $$u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3.$$I ...
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### is this intengral bounded?

Im just stuck beacause I dont know if this integral in bounded, I was trying to make a change of variable but I cant get to anything: (edited what need is that f is bounded for a fixed x) ...
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### weak derivative questions

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
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### Solutions to Dirichlet problem on the half space with $L^{\infty}$ boundary data.
Consider the Dirichlet problem with boundary data $f(x)\in L^{\infty}(\mathbb{R}^{d-1})$ on the halfspace $\mathbb{R}^{d}_{+}$, where $y>0$ and $x\in\mathbb{R}^{d-1}.$ One can prove that ...