3
votes
1answer
55 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
4
votes
2answers
130 views

How to define difference quotients for Sobolev functions

There is a definition for difference quotients of Sobolev functions I do not understand. Let $U\subset\Omega\subset\mathbb{R}^n$ be open sets such that the closure of $U$ is compact in $\Omega$ and ...
2
votes
0answers
126 views

How can we glue Sobolev functions?

Let $u:A\cup B\to R$ be a function (where $A$ and $B$ are disjoint connected sets and $A\cup B$ is connected) such that $u$ restrict to $A$ and to $B$ are in $W^{1, p}$. Which result guarantees me ...
3
votes
1answer
377 views

Intuition behind Sobolev norm

This morning I was thinking at the following (simple) fact. Let us consider $[0, 1] \to \mathbb{R}$ functions and define a linear functional $$F(u)=u(1)-u(0).$$ $F$ is not continuous on $L^2(0, 1)$ ...