# Tagged Questions

A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

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### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $R^d$. (Say, a ball, say a cube...) For which classes $\cal C$ of functions, every function $f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
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### $W^{1, p}(M, N)$ path-connected if and only if $C^0(M, N)$ is path-connected

I'm asked to show that for compact, smooth Riemmanian manifolds $M$, $N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from "...
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### Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
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### Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$...