Sobolev spaces measure functions and their weak derivatives in \(L^p\). They are the bridge between distributional differentiation and the Hilbert or Banach methods used in PDE.
Weak differentiability
Sobolev spaces begin by replacing classical derivatives with weak derivatives. This keeps differentiation available after passing to \(L^p\) limits.
Definition. For open \(U\subset\mathbb R^d\), \(W^{k,p}(U)\) consists of \(f\in L^p(U)\) whose weak derivatives \(\partial^\alpha f\in L^p(U)\) for all \(|\alpha|\le k\). The norm is \(\|f\|_{W^{k,p}}=\sum_{|\alpha|\le k}\|\partial^\alpha f\|_{L^p}\). The space \(H^k(U)\) means \(W^{k,2}(U)\).
Completeness
Completeness is essential because Sobolev arguments take limits of Cauchy sequences of functions and derivatives simultaneously.
Theorem. For \(1\le p\le\infty\), \(W^{k,p}(U)\) is Banach. For \(p=2\), \(H^k(U)\) is Hilbert with inner product \(\sum_{|\alpha|\le k}\langle\partial^\alpha f,\partial^\alpha g\rangle_{L^2}\).
Proof. Let \((f_n)\) be Cauchy in \(W^{k,p}\). Then for every \(|\alpha|\le k\), \((\partial^\alpha f_n)\) is Cauchy in \(L^p\), hence converges to some \(g_\alpha\in L^p\). Let \(f=g_0\). For every test function \(\varphi\), the weak derivative identity for \(f_n\) gives \(\int \partial^\alpha f_n\,\varphi=(-1)^{|\alpha|}\int f_n\,\partial^\alpha\varphi\). Passing to the \(L^p\)-limits using H?lder duality with compactly supported smooth functions gives \(\int g_\alpha\varphi=(-1)^{|\alpha|}\int f\,\partial^\alpha\varphi\). Thus \(g_\alpha=\partial^\alpha f\), and \(f_n\to f\) in \(W^{k,p}\). The Hilbert claim for \(p=2\) follows from completeness and the displayed inner product.
Mollification
Mollification approximates weakly differentiable functions by smooth functions while preserving the relevant norm estimates.
Proposition. If \(f\in W^{k,p}_{\mathrm{loc}}(U)\) and \(\rho_\varepsilon\) is a standard mollifier, then \(\rho_\varepsilon*f\in C^\infty\) on compact subsets \(K\Subset U\), and \(\partial^\alpha(\rho_\varepsilon*f)=\rho_\varepsilon*(\partial^\alpha f)\) for \(|\alpha|\le k\). Moreover \(\rho_\varepsilon*f\to f\) in \(W^{k,p}(K)\).
Proof. The derivative identity is the distributional convolution identity. Approximation in \(L^p(K)\) for each derivative follows from the standard \(L^p\)-continuity of translations and the fact that mollification is an average of small translations. Summing over \(|\alpha|\le k\) gives convergence in the Sobolev norm on compact subsets.
Poincar? inequality on an interval
Poincar? inequalities say that boundary or mean conditions let derivatives control functions. They are basic coercivity estimates.
Theorem. If \(u\in W^{1,p}_0(0,L)\), \(1\le p\le\infty\), then \(\|u\|_{L^p(0,L)}\le L\|u'\|_{L^p(0,L)}\).
Proof. For smooth compactly supported \(u\), \(u(x)=\int_0^x u'(t)\,\mathrm{d}t\), so \(|u(x)|\le\int_0^L|u'(t)|\,\mathrm{d}t\). A sharper \(L^p\) estimate follows from Hardy's inequality, and the rough bound \(\|u\|_p\le L\|u'\|_p\) follows by H?lder and integration. For general \(u\in W^{1,p}_0\), take smooth compactly supported approximations in \(W^{1,p}\) and pass to the limit.
One-dimensional Sobolev embedding
In one dimension, Sobolev control gives actual continuity. This theorem is the simplest form of a Sobolev embedding.
Theorem. If \(u\in W^{1,p}(a,b)\) with \(p>1\), then \(u\) has a H?lder continuous representative with exponent \(1-1/p\), and \(|u(x)-u(y)|\le\|u'\|_{L^p}|x-y|^{1-1/p}\).
Proof. For smooth \(u\), \(u(x)-u(y)=\int_y^x u'(t)\,\mathrm{d}t\). H?lder inequality gives the estimate. Approximate a Sobolev function by smooth functions locally or use the absolutely continuous representative theorem; the estimate passes to the limit and produces a continuous representative.
Weak elliptic formulation
Weak formulations move derivatives onto test functions and place elliptic equations inside Hilbert or Banach spaces.
Example. The Dirichlet problem \(-\Delta u=f\) on \(U\) is written: find \(u\in H_0^1(U)\) such that \(\int_U\nabla u\cdot\nabla v\,\mathrm{d}x=\int_Ufv\,\mathrm{d}x\) for all \(v\in H_0^1(U)\). Poincar? gives coercivity of the left side, and Lax-Milgram gives existence and uniqueness when \(f\in H^{-1}(U)\).