Sobolev space

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

The spaces $W^{0,p}(\mathrm{\Omega })$ are simply defined to be the spaces $L^p(\mathrm{\Omega })$ of Lebesgue $p$-summable functions.We then define the space $W^{m,p}(\mathrm{\Omega })$ to be the space of functions $u\in L^p(\mathrm{\Omega })$ which have weak derivatives $g=(g_1,\mathrm{\dots },g_n)$ such that $g_i\in W^{m-1,p}(\mathrm{\Omega })$.

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

$${\parallel u\parallel }_{W^{m,p}}=\sum _{k=0}^m\sum _{i_1=1}^n\mathrm{\cdots }\sum _{i_k=1}^n{\left[{\int }_{\mathrm{\Omega }}{\left|\frac{{\partial }^{k}u(x)}{\partial x_{i_1}\mathrm{\cdots }\partial x_{i_k}}\right|}^p𝑑x\right]}^{\frac{1}{p}}$$

i.e. 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 $H^m(\mathrm{\Omega }):=W^{m,2}(\mathrm{\Omega })$ which turn out to be Hilbert spaces with the scalar product given by

$${(u,v)}_{H^m(\mathrm{\Omega })}=\sum _{k=0}^m\sum _{i_1=1}^n\mathrm{\cdots }\sum _{i_k=1}^n{\int }_{\mathrm{\Omega }}\frac{{\partial }^{k}u(x)}{\partial x_{i_1}\mathrm{\cdots }\partial x_{i_k}}\frac{{\partial }^{k}v(x)}{\partial x_{i_1}\mathrm{\cdots }\partial x_{i_k}}𝑑x.$$