Some elements of Sobolev spaces
This page collects some elements of the theory of Lebesgue and Sobolev spaces. For further information, see e.g. L. C. Evans: Partial Differential Equations, R. A. Adams: Sobolev spaces or G. B. Folland: Real analysis.
Preliminaries
A multi-index is a vector \(\alpha\in\mathbb{N}_0^d\) with components \(\alpha_1,\dots,\alpha_d\in\mathbb{N}_0 = \{0,1,2,\dots\}\). We write \(|\alpha| = \alpha_1+\alpha_2+\dots+\alpha_d\). We use multi-indices as a shorthand notation for mixed partial derivatives: If \(u:\Omega\to\mathbb{R}\) is a given, smooth function on some domain \(\Omega\subset\mathbb{R}^d\) then \[ D^\alpha u := \frac{\partial^{|\alpha|}u}{\partial x_1^{\alpha_1}\cdots\partial x_d^{\alpha_d}}. \] For instance, if \(d=2\) and \(\alpha=(2,1)\) then \(D^\alpha u = \dfrac{\partial^3 u}{\partial x_1^2 \partial x_2}\).
We denote by \(C^\infty(\Omega)\) the space of all infinitely differentiable functions \(\phi:\Omega\to\mathbb{R}\), and by \(C_c^\infty(\Omega)\) those \(\phi\) in \(C^\infty(\Omega)\) with compact support – that is, there exists a compact set \(K\subset\Omega\) such that \(u\equiv 0\) outside of \(K\). Note that if \(\Omega\) is open then this implies that \(\phi\) and all its derivatives vanish close to the boundary of \(\Omega\).
Recall that if \(\Omega\subset\mathbb{R}^d\) is open then by the integration-by-parts formula (or Gauss' theorem, or the divergence theorem) we have \[ \int_\Omega u\nabla v\ dx = \int_{\partial\Omega} uvn\ dx - \int_\Omega v\nabla u\ dx \qquad \forall\ u,v\in C^\infty(\Omega), \] where \(n\) is the unit normal on \(\partial\Omega\). If either \(u\) or \(v\) has compact support then the boundary integral drops out, and we can generalize the above to the following:
Integration by parts
If \(u\) and \(v\) are \(k\) times differentiable and at least one of them has compact support, then \begin{equation}\label{eq:intbyparts} \int_\Omega u D^\alpha v\ dx = (-1)^{|\alpha|}\int_\Omega vD^\alpha u\ dx \end{equation} for every multi-index \(\alpha\) of size \(|\alpha|\leq k\).
The Lebesgue space \(L^2(\Omega)\)
Definition
For a set \(\Omega \subset \mathbb{R}^d\) we define \[L^2(\Omega) := \left\{ u:\Omega\to\mathbb{R}\ :\ \int_\Omega |u(x)|^2\ dx < \infty\right\}.\] We denote the \(L^2\) norm of a function \(u\in L^2(\Omega)\) by \(\|u\|_{L^2(\Omega)} := \left(\int_\Omega |u(x)|^2\ dx\right)^{1/2}\). Functions \(u\in L^2(\Omega)\) are called square integrable functions.
It can be shown that \(L^2(\Omega)\) is a Hilbert space when equipped with the inner product \((u,v) := \int_\Omega u(x)v(x)\ dx\). The fact that \((u,v)\) is a finite number follows from Hölder's inequality: \[ \int_\Omega u(x)v(x)\ dx \leq \|u\|_{L^2(\Omega)}\|v\|_{L^2(\Omega)} \qquad \forall\ u,v\in L^2(\Omega) \] (also called the Cauchy-Schwartz inequality for \(L^2(\Omega)\)). Note in particular that \((u,u) = \|u\|_{L^2(\Omega)}^2\).
Examples
\(L^2\) contains a very large number of functions, including highly discontinuous and/or singular functions. If \(u(x)=|x|^\gamma\) for \(x\) in the unit interval \(\Omega=(-1,1)\), then \(u\in L^2(\Omega)\) provided \(\gamma>-1/2\). More generally, in the \(d\)-dimensional unit ball \(\Omega=\{x\in\mathbb{R}^d\ :\ |x|\leq 1\}\), the function \(u(x)=|x|^\gamma\) satisfies (using polar coordinates) \[ \int_\Omega |u(x)|^2\ dx = \int_\Omega |x|^{2\gamma}\ dx = \omega_n\int_0^1 r^{2\gamma} r^{d-1}\ dr < \infty \qquad \text{provided } \gamma>-d/2. \] (Here, \(\omega_n\) is the area/volume/hypervolume of the unit sphere in \(\mathbb{R}^d\).) Thus, singularities of order less than \(d/2\) are square integrable. For example, the function \(u(x) = \frac{1}{|x|}\) is square integrable in three dimensions or greater, but not in one or two dimensions.
Sobolev spaces
Definitions
We let now \(\Omega\subset\mathbb{R}^d\) be an open, bounded, connected set.
Definition
The Sobolev space \(H^k(\Omega)\) is the space of all functions \(u\in L^2(\Omega)\) such that for every multi-index \(\alpha\in\mathbb{N}_0^d\) of length \(|\alpha|\leq k\) exists a function \(v_\alpha\in L^2(\Omega)\) such that \[ \int_\Omega u(x) D^\alpha \phi(x)\ dx = (-1)^{|\alpha|}\int_\Omega v_\alpha(x)\phi(x)\ dx \qquad \forall\ \phi\in C_c^\infty(\Omega). \] The functions \(v_\alpha\) are the weak derivatives of \(u\). At the risk of confusion, we denote \(D^\alpha u = v_\alpha\). For the sake of convenience we will denote \(H^0(\Omega) = L^2(\Omega)\).
Remark
Note the similarity between the definition of the weak derivatives and the integration-by-parts formula. In fact, if \(u\) also lies in \(C^\infty(\bar{\Omega})\) then all of its weak derivatives exist and coincide with its classical derivatives. It is straighforward to show that the weak derivatives of a function are unique (if they exist).
The space \(H^k(\Omega)\) is equipped with the norm \[ \|u\|_{H^k(\Omega)} := \Biggl[\sum_{|\alpha|\leq k} \|D^\alpha u\|_{L^2(\Omega)}^2\Biggr]^{1/2}, \] where the sum goes over all multi-indices of length at most \(k\). It can be shown that \(H^k(\Omega)\) with the norm \(\|\cdot\|_{H^k(\Omega)}\) is a Banach space. Equipped with the inner product \[ (u,v)_{H^k} := \sum_{|\alpha|\leq k} (D^\alpha u, D^\alpha v) \] (where \((u,v)\) is the \(L^2\) inner product), the Sobolev space \(H^k(\Omega)\) is in fact a Hilbert space.
The \(H^k\) semi-norm is the number \[ |u|_{H^k(\Omega)} := \Biggl[\sum_{|\alpha|=k} \|D^\alpha u\|_{L^2(\Omega)}^2\Biggr]^{1/2}. \] Thus, \(\|u\|_{H^k(\Omega)} = \left(\sum_{r=0}^k |u|_{H^r(\Omega)}^2\right)^{1/2}\).
Definition
The space \(H_0^k(\Omega)\) is defined as the closure of \(C_c^\infty(\Omega)\) in \(H^k(\Omega)\), or in other words, the space of all \(u\in H^k(\Omega)\) for which there exists a sequence \(\phi_n\in C_c^\infty(\Omega)\) such that \(\|u-\phi_n\|_{H^k(\Omega)} \to 0\) as \(n\to\infty\).
It can be shown that an equivalent definition is \[ H_0^k(\Omega) = \{u\in H^k(\Omega)\ :\ D^\alpha u = 0 \text{ on } \partial\Omega \text{ for every } |\alpha|\leq k-1\}. \] Thus, for instance, \[ H_0^1(\Omega) = \{u\in H^1(\Omega)\ :\ u = 0 \text{ on } \partial\Omega\}. \]
Inequalities and embeddings
Theorem (the Poincaré inequality)
Let \(\Omega\) be open and bounded. Then there exists a constant \(C>0\), only dependent on \(\Omega\), such that \[ \|u\|_{L^2(\Omega)} \leq C|u|_{H^1(\Omega)} \qquad \forall\ u\in H_0^1(\Omega). \] More generally, for every \(k\in\mathbb{N}\) there exists a constant \(C>0\) such that \[ |u|_{H^{r-1}(\Omega)} \leq C|u|_{H^r(\Omega)} \qquad \forall\ r=1,\dots,k \text{ and } u\in H_0^k(\Omega). \] The result also holds if \(H_0^k(\Omega)\) is replaced by \(H_{\Gamma}^k(\Omega) := \{u\in H^k(\Omega)\ :\ D^\alpha u = 0 \text{ on } \Gamma \text{ for all } |\alpha|\leq k-1\}\), where \(\Gamma\subset\partial\Omega\) is only a part of the boundary of \(\Omega\).
Remark
As a consequence of the Poincaré inequality, there is a constant \(C>0\) such that \[ \|u\|_{H^1(\Omega)} \leq C|u|_{H^1(\Omega)} \qquad \forall\ u\in H_0^1(\Omega), \] and similarly for the \(H^k\) norm. This inequality is used extensively for proving coercivity of bilinear forms on \(H_0^1(\Omega)\).
Definition
Let \(\Omega\subset\mathbb{R}^d\) be an open set. We say that \(\Omega\) has a \(C^1\) boundary if \(\partial\Omega\) it can be parametrized locally by a differentiable function. More precisely, for every \(x_0\in\partial\Omega\) there exists an open neighborhood \(U\subset\mathbb{R}^d\) of \(x_0\), an open set \(V\subset\mathbb{R}^{d-1}\) and a differentiable, bijective function \(\phi:V\to U\cap\partial\Omega\).
Similarly, \(\Omega\) has Lipschitz boundary if \(\partial\Omega\) can be parametrized locally by a Lipschitz function.
Intuitively, a domain has \(C^1\) (or Lipschitz) boundary if for every \(x_0\in\partial\Omega\), after rotating the domain, the boundary \(\partial\Omega\) in the vicinity of \(x_0\) is the graph of a \(C^1\) (or Lipschitz continuous) function. Note that domains with Lipschitz boundaries can have "kinks", so any polygon (or polyhedron in 3D) has Lipschitz boundary.
Theorem (Sobolev embedding theorem)
Let \(\Omega\subset\mathbb{R}^d\) be an open, bounded subset with a \(C^1\) boundary, and let \(m\) be a nonnegative integer. Then \(H^k(\Omega) \subset C^m(\Omega)\) provided \(k > m + d/2\).
For instance, \(H^1\) functions on the real line are continuous (\(k=1\), \(d=1\) and \(m=0\) in the above theorem), and \(H^3\) functions in the plane are differentiable (\(k=3\), \(d=2\) and \(m=1\)).
Remark
To gain some intuition into the Sobolev embedding theorem, consider the function \(u(x)=|x|^\gamma\) for \(x\) in a neighborhood of the origin. This function is continuous if \(\gamma\geq0\) and discontinuous for \(\gamma<0\). Moreover, its \(r\)-th partial derivatives all behave like \(|x|^{\gamma-r}\). Hence, \[ \gamma \geq m \quad \Rightarrow \quad u\in C^m. \] On the other hand, assume that \(u\in H^k\) for some integer \(k\). Then all \(k\)-th partial derivatives of \(u\) lie in \(L^2\), which (by the example in the section on Lebesgue spaces) happens if and only if \(\gamma-k > -d/2\). Thus, if we ensure that \(k > m+d/2\) then \[ \gamma > k-d/2 > m+d/2-d/2 = m \quad \Rightarrow \quad u\in C^m. \]
Approximation by smooth functions
Note that, by definition, \(C_c^\infty(\Omega)\) is dense in \(H^k_0(\Omega)\) for any \(k\in\mathbb{N}\). A similar result holds for \(H^k(\Omega)\):
Theorem
Let \(\Omega\subset\mathbb{R}^d\) be an open, bounded subset with Lipschitz boundary (see the definition in the previous section). Then \(C^\infty(\bar{\Omega})\) is dense in \(H^k(\Omega)\) for any \(k\in\mathbb{N}\). In other words, for any \(u\in H^k(\Omega)\) there are functions \(\phi_1,\phi_2,\ldots \in C^\infty(\bar{\Omega})\) such that \(\|u-\phi_n\|_{H^k(\Omega)} \to 0\) as \(n\to\infty\).