# Sobolev space

We define the Sobolev spaces of functions $W^{m,p}(\Omega)$ where $\Omega$ is an open subset of $\mathbf{R}^{n}$, $m\geq 0$ is an integer and $p\in[1,+\infty]$.

The spaces $W^{0,p}(\Omega)$ are simply defined to be the spaces $L^{p}(\Omega)$ of Lebesgue $p$-summable functions. We then define the space $W^{m,p}(\Omega)$ to be the space of functions $u\in L^{p}(\Omega)$ which have weak derivatives $g=(g_{1},\ldots,g_{n})$ such that $g_{i}\in W^{m-1,p}(\Omega)$.

The space $W^{m,p}$ turns out to be a Banach space when endowed with the norm

 $\|u\|_{W^{m,p}}=\sum_{k=0}^{m}\sum_{i_{1}=1}^{n}\cdots\sum_{i_{k}=1}^{n}\left[% \int_{\Omega}\left|\frac{\partial^{k}u(x)}{\partial x_{i_{1}}\cdots\partial x_% {i_{k}}}\right|^{p}\,dx\right]^{\frac{1}{p}}$

i.e.Β the sum of the $L^{p}$ norms of $u$ and of all weak derivatives of $u$ up to the $m$-th order.

Of particular interest are the spaces $H^{m}(\Omega):=W^{m,2}(\Omega)$ which turn out to be Hilbert spaces with the scalar product given by

 $(u,v)_{H^{m}(\Omega)}=\sum_{k=0}^{m}\sum_{i_{1}=1}^{n}\cdots\sum_{i_{k}=1}^{n}% \int_{\Omega}\frac{\partial^{k}u(x)}{\partial x_{i_{1}}\cdots\partial x_{i_{k}% }}\frac{\partial^{k}v(x)}{\partial x_{i_{1}}\cdots\partial x_{i_{k}}}\,dx.$
Title Sobolev space SobolevSpace 2013-03-22 14:54:55 2013-03-22 14:54:55 paolini (1187) paolini (1187) 10 paolini (1187) Definition msc 46E35 Sobolev function WeakDerivative