Paweł Strzelecki | Lectures 3 and 4

On October 24 and 31, we covered the following:

Boundary behaviour of harmonic functions and the notion of a barrier:
- if the external ball condition is satisfied at $x_0\in\partial\Omega$ , then there exists a (harmonic) barrier at $x_0$ ;
- if there exists a barrier at $x_0$ , then Poincaré’s méthode de balayage gives a solution of $\Delta u =0$ which is continuous up to the boundary at $x_0$ ;
Sobolev spaces $W^{m,p}$ (of functions in $L^p$ with weak derivatives up to order $m$ that are also in $L^p$ ): two definitions;
A few examples;
Theorem: $W^{m,p}$ is a Banach space;
Density of smooth functions in $W^{m,p}$ ; the Meyers-Serrin theorem ` $H=W$ ’;
Poincaré’s inequality for $u \in W_0^{1,p}$ on a bounded domain $\Omega$ ;
Some remarks on the one dimensional case and on the case $p=n$ ;
Sobolev’s imbedding theorem for $1\le p < n$ on $\mathbb{R}^n$ , with Gagliardo’s proof:

\displaystyle \left(\int_{\mathbb{R}^n} |u|^{p^\ast}\, dx\right)^{1/p^\ast} \le C(n,p) \biggl(\int_{\mathbb{R}^n} |\nabla u|^{p}\, dx\biggr)^{1/p}\, , \qquad p^\ast=\frac{np}{n-p};

The estimate of oscillations of a $C^1$ function $u$ on a convex domain $\Omega$ by Riesz potentials:

\displaystyle |u(x)-u_S| \le \frac{\text{diam}(\Omega)^n}{n|S|} \int_{\Omega} |x-y|^{1-n}|\nabla u(y)|\, dy\, , \qquad x\in \Omega, \ S\subset \Omega;

As a corollary of the above: Morrey’s imbedding of $W^{1,p}$ , $p>n$ , into $C^\alpha$ , $\alpha=1-\frac np$ ;
As another corollary: in dimension $n$ , functions of class $W^{m,p}$ , where $mp>n$ , are continuous (upon a modification on a set of measure zero).

You can find this material in Gilbarg and Trudinger (Chapter 2 on harmonic functions and Chapter 7 on Sobolev spaces), Evans (Chapter 5) and my lecture notes in Polish (Chapter 6, Sobolev spaces).