Sobolev spaces measure integrability of a function and its weak derivatives. They supply complete energy spaces in which weak PDE formulations are well posed and compactness is available.
Definitions and completeness
Definition. For \(k\in\mathbb N\) and \(1\le p\le\infty\), \[ W^{k,p}(\Omega) =\{u\in L^p(\Omega):D^\alpha u\in L^p,\ |\alpha|\le k\}, \] with norm \[ \|u\|_{W^{k,p}} =\left(\sum_{|\alpha|\le k}\|D^\alpha u\|_p^p\right)^{1/p} \] for \(p<\infty\). \(H^k=W^{k,2}\) is Hilbert.
Theorem. \(W^{k,p}(\Omega)\) is Banach; \(H^k(\Omega)\) is Hilbert.
Proof. A Cauchy sequence has \(D^\alpha u_n\to v_\alpha\) in \(L^p\). Let \(u=v_0\). For every test \(\varphi\), \[ \int v_\alpha\varphi =\lim_n\int D^\alpha u_n\,\varphi =(-1)^{|\alpha|}\lim_n\int u_nD^\alpha\varphi =(-1)^{|\alpha|}\int uD^\alpha\varphi. \] Hence \(v_\alpha=D^\alpha u\) weakly, and convergence holds in the Sobolev norm.
Poincaré inequality
Theorem. If \(\Omega\) is bounded and connected with suitable boundary regularity, then for \(u\in W_0^{1,p}(\Omega)\), \[ \|u\|_{L^p}\le C\|\nabla u\|_{L^p}. \] For general \(u\), \[ \|u-u_\Omega\|_{L^p}\le C\|\nabla u\|_{L^p}, \qquad u_\Omega=|\Omega|^{-1}\int_\Omega u. \]
A compactness proof argues by contradiction: otherwise normalized functions would have gradients tending to zero. Rellich compactness gives a strong \(L^p\) limit that is constant; zero trace or zero mean makes that constant zero, contradicting normalization.
Sobolev embeddings
For \(1\le p<d\), \[ W^{1,p}(\mathbb R^d) \hookrightarrow L^{p^*}(\mathbb R^d), \qquad p^*=\frac{dp}{d-p}. \] On bounded Lipschitz domains, analogous embeddings hold, and the embedding into \(L^q\) is compact for \(q<p^*\). If \(p>d\), Morrey's inequality gives a Hölder representative: \[ |u(x)-u(y)| \le C\|\nabla u\|_{L^p}|x-y|^{1-d/p}. \]
Trace theorem
Theorem. On a bounded Lipschitz domain, restriction of smooth functions to the boundary extends uniquely to a bounded surjective map \[ \operatorname{Tr}:H^1(\Omega)\to H^{1/2}(\partial\Omega). \] Moreover, \[ H_0^1(\Omega)=\ker(\operatorname{Tr}). \]
A trace is not pointwise evaluation of an arbitrary \(L^2\) equivalence class. It is a continuous boundary value determined by interior regularity. The fractional space \(H^{1/2}\) is the exact natural boundary regularity for \(H^1\).
Compactness
Theorem (Rellich--Kondrachov). For a bounded Lipschitz domain, \[ H^1(\Omega)\hookrightarrow L^2(\Omega) \] compactly.
Bounded energy sequences therefore have strongly \(L^2\)-convergent subsequences. Weak convergence controls derivatives; compact embedding upgrades the function itself, enabling passage through many nonlinear lower-order terms.