Partial Differential Equations/Sobolev spaces

There are some partial differential equations which have no solution. However, some of them have something like ‘almost a solution’, which we call a weak solution. Among these there are partial differential equations whose weak solutions model processes in nature, just like solutions of partial differential equations which have a solution.

These weak solutions will be elements of the so-called Sobolev spaces. By proving properties which elements of Sobolev spaces in general have, we will thus obtain properties of weak solutions to partial differential equations, which therefore are properties of some processes in nature.

In this chapter we do show some properties of elements of Sobolev spaces. Furthermore, we will show that Sobolev spaces are Banach spaces (this will help us in the next section, where we investigate existence and uniqueness of weak solutions).

The fundamental lemma of the calculus of variations
But first we shall repeat the definition of the standard mollifier defined in chapter 3.

Example 3.4: The standard mollifier $$\eta$$, given by
 * $$\eta: \mathbb R^d \to \mathbb R, \eta(x) = \frac{1}{c}\begin{cases} e^{-\frac{1}{1-\|x\|^2}}& \text{ if } \|x\|_2 < 1\\

0& \text{ if } \|x\|_2 \geq 1 \end{cases}$$ , where $$c := \int_{B_1(0)} e^{-\frac{1}{1-\|x\|^2}} dx$$, is a bump function (see exercise 3.2).

The following lemma, which is important for some theorems about Sobolev spaces, is known as the fundamental lemma of the calculus of variations:

Lemma 12.2:

Let $$S \subseteq \mathbb R^d$$ and let $$f, g: S \to \mathbb R$$ be functions such that $$f, g \in L^1_{\text{loc}}(S)$$ and $$\mathcal T_f = \mathcal T_g$$. Then $$f = g$$ almost everywhere.

Proof:

We define
 * $$h: \mathbb R^d \to \mathbb R, h(x) := \begin{cases}

f(x) - g(x) & x \in S \\ 0 & x \notin S \end{cases}$$

Weak derivatives
Remarks 12.2: If $$f \in L^p(S)$$ is a function and $$\alpha \in \mathbb N_0^d$$ is a $$d$$-dimensional multiindex, any two $$\alpha$$th-weak derivatives of $$f$$ are equal except on a null set. Furthermore, if $$\partial_\alpha f$$ exists, it also is an $$\alpha$$th-weak derivative of $$f$$.

Proof:

1. We prove that any two $$\alpha$$th-weak derivatives are equal except on a nullset.

Let $$g, h \in L^p(S)$$ be two $$\alpha$$th-weak derivatives of $$f$$. Then we have
 * $$\mathcal T_g = \partial_\alpha \mathcal T_f = \mathcal T_h$$

Notation 12.3 If it exists, we denote the $$\alpha$$th-weak derivative of $$f$$ by $$\partial_\alpha f$$, which is of course the same symbol as for the ordinary derivative.

Proof:

Definition and first properties of Sobolev spaces
In the above definition, $$\partial_\alpha f$$ denotes the $$\alpha$$th-weak derivative of $$f$$.

Proof:

1.

We show that
 * $$\| f \|_{\mathcal W^{n, p}(O)} = \sum_{|\alpha| \le n} \left\| \partial_\alpha f \right\|_{L^p(O)}$$

is a norm.

We have to check the three defining properties for a norm:
 * $$\|f\|_{\mathcal W^{n, p}(O)} = 0 \Leftrightarrow f = 0$$ (definiteness)
 * $$\|c f\|_{\mathcal W^{n, p}(O)} = |c|\|f\|_{\mathcal W^{n, p}(O)}$$ for every $$c \in \mathbb R$$ (absolute homogeneity)
 * $$\|f + g\|_{\mathcal W^{n, p}(O)} \le \|f\|_{\mathcal W^{n, p}(O)} + \|g\|_{\mathcal W^{n, p}(O)}$$ (triangle inequality)

We start with definiteness: If $$f = 0$$, then $$\|f\|_{\mathcal W^{n, p}(O)} = 0$$, since all the directional derivatives of the constant zero function are again the zero function. Furthermore, if $$\|f\|_{\mathcal W^{n, p}(O)} = 0$$, then it follows that $$\|f\|_{L^p(O)} = 0$$ implying that $$f=0$$ as $$\|f\|_{L^p(O)}$$ is a norm.

We proceed to absolute homogeneity. Let $$c \in \mathbb R$$.
 * $$\begin{align}

\|cf\|_{\mathcal W^{n, p}(O)} & := \sum_{|\alpha| \le n} \left\| \partial_\alpha c f \right\|_{L^p(O)} & \\ & = \sum_{|\alpha| \le n} \left\| c \partial_\alpha f \right\|_{L^p(O)} & \text{ theorem 12.4} \\ & = \sum_{|\alpha| \le n} |c| \left\| \partial_\alpha f \right\|_{L^p(O)} & \text{ by absolute homogeneity of } \| \cdot \|_{L^p(O)} \\ & = |c| \sum_{|\alpha| \le n} \left\| \partial_\alpha f \right\|_{L^p(O)} & \\ & =: |c| \|f\|_{\mathcal W^{n, p}(O)} \end{align}$$

And the triangle inequality has to be shown:
 * $$\begin{align}

\|f + g\|_{\mathcal W^{n, p}(O)} & := \sum_{|\alpha| \le n} \left\| \partial_\alpha (f + g) \right\|_{L^p(O)} & \\ & = \sum_{|\alpha| \le n} \left\| \partial_\alpha f + \partial_\alpha g \right\|_{L^p(O)} & \text{ theorem 12.4} \\ & \le \sum_{|\alpha| \le n} \left( \left\| \partial_\alpha f \right\|_{L^p(O)} + \left\| \partial_\alpha g \right\|_{L^p(O)} \right) & \text{ by triangle inequality of } \| \cdot \|_{L^p(O)} \\ & = \|f\|_{\mathcal W^{n, p}(O)} + \|g\|_{\mathcal W^{n, p}(O)} \end{align}$$

2.

We prove that $$\mathcal W^{n, p}(O)$$ is a Banach space.

Let $$(f_l)_{l \in \mathbb N}$$ be a Cauchy sequence in $$\mathcal W^{n, p}(O)$$. Since for all $$d$$-dimensional multiindices $$\alpha \in \mathbb N_0^d$$ with $$|\alpha| \le n$$ and $$m, l \in \mathbb N$$
 * $$\|\partial_\alpha f_l - \partial_\alpha f_m)\|_{L^p(O)} = \|\partial_\alpha(f_l - f_m)\|_{L^p(O)} \le \sum_{|\alpha| \le n} \left\| \partial_\alpha (f_l - f_m) \right\|_{L^p(O)}$$

since we only added non-negative terms, we obtain that for all $$d$$-dimensional multiindices $$\alpha \in \mathbb N_0^d$$ with $$|\alpha| \le n$$, $$(\partial_\alpha f_l)_{l \in \mathbb N}$$ is a Cauchy sequence in $$L^p(O)$$. Since $$L^p(O)$$ is a Banach space, this sequence converges to a limit in $$L^p(O)$$, which we shall denote by $$f_\alpha$$.

We show now that $$f := f_{(0, \ldots, 0)} \in \mathcal W^{n, p}(O)$$ and $$f_l \to f, l \to \infty$$ with respect to the norm $$\| \cdot \|_{\mathcal W^{n, p}(O)}$$, thereby showing that $$\mathcal W^{n, p}(O)$$ is a Banach space.

To do so, we show that for all $$d$$-dimensional multiindices $$\alpha \in \mathbb N_0^d$$ with $$|\alpha| \le n$$ the $$\alpha$$th-weak derivative of $$f$$ is given by $$f_\alpha$$. Convergence then automatically follows, as
 * $$\begin{align}

f_l \to f, l \to \infty & \Leftrightarrow \| f_l - f \|_{\mathcal W^{n, p}(O)} \to 0, l \to \infty & \\ & \Leftrightarrow \sum_{|\alpha| \le n} \left\| \partial_\alpha (f_l - f) \right\|_{L^p(O)} \to 0, l \to \infty & \\ & \Leftrightarrow \sum_{|\alpha| \le n} \left\| \partial_\alpha f_l - \partial_\alpha f \right\|_{L^p(O)} \to 0, l \to \infty & \text{by theorem 12.4} \\ \end{align}$$ where in the last line all the summands converge to zero provided that $$\partial_\alpha f = f_\alpha$$ for all $$d$$-dimensional multiindices $$\alpha \in \mathbb N_0^d$$ with $$|\alpha| \le n$$.

Let $$\varphi \in \mathcal D(O)$$. Since $$\partial_\alpha f_l \to f_\alpha$$ and by the second triangle inequality
 * $$\|\partial_\alpha f - f_\alpha\| \ge |\|\partial_\alpha f\| - \|f_\alpha\||$$

, the sequence $$(\varphi \partial_\alpha f_l)_{l \in \mathbb N}$$ is, for large enough $$l$$, dominated by the function $$2 \|\varphi\|_\infty f_\alpha$$, and the sequence $$(\partial_\alpha \varphi f_l)_{l \in \mathbb N}$$ is dominated by the function $$2 \|\partial_\alpha \varphi\|_\infty f$$. incomplete: Why are the dominating functions L1? Therefore
 * $$\begin{align}

\int_{\R^d} \partial_\alpha \varphi(x) f(x) dx =& \lim_{l \to \infty} \int_{\R^d} \partial_\alpha \varphi(x) f_l(x) dx & \text{ dominated convergence} \\ & = \lim_{l \to \infty} (-1)^{|\alpha|} \int_{\R^d} \varphi(x) \partial_\alpha f_l(x) dx & \\ &= (-1)^{|\alpha|} \int_{\R^d} \varphi(x) f_\alpha(x) dx & \text{ dominated convergence} \end{align}$$ , which is why $$f_\alpha$$ is the $$\alpha$$th-weak derivative of $$f$$ for all $$d$$-dimensional multiindices $$\alpha \in \mathbb N_0^d$$ with $$|\alpha| \le n$$.

Approximation by smooth functions
We shall now prove that for any $$L^p$$ function, we can find a sequence of bump functions converging to that function in $$L^p$$ norm.

approximation by simple functions and lemma 12.1, ||f_eps-f|| le ||f_eps - g_eps|| + ||g_eps - g|| + ||g - f||

Let $$\Omega \subset \R^d$$ be a domain, let $$r > 0$$, and $$U \subset \Omega$$, such that $$U + B_r(0) \subseteq \Omega$$. Let furthermore $$u \in \mathcal W^{m, p}(U)$$. Then $$\mu_\epsilon * f$$ is in $$C^\infty(U)$$ for $$\epsilon < r$$ and $$\lim_{\epsilon \to 0} \|\mu_\epsilon * f - f\|_{W^{m, p}(U)} = 0$$.

Proof: The first claim, that $$\mu_\epsilon * f \in C^\infty(U)$$, follows from the fact that if we choose
 * $$\tilde f(x) = \begin{cases}

f(x) & x \in U \\ 0 & x \notin U \end{cases}$$ Then, due to the above section about mollifying $$L^p$$-functions, we know that the first claim is true.

The second claim follows from the following calculation, using the one-dimensional chain rule:
 * $$\frac{\partial^\alpha}{\partial x^\alpha} (\mu_\epsilon * f) (y)= \int_{\R^d} \frac{\partial^\alpha}{\partial x^\alpha}\mu_\epsilon(y -x) f(x) dx = (-1)^{|\alpha|} \int_{\R^d} \frac{\partial^\alpha}{\partial y^\alpha}\mu_\epsilon(y -x) f(x) dx$$
 * $$=\int_{\R^d} \mu_\epsilon(y -x) \frac{\partial^\alpha}{\partial y^\alpha} f(x) dx = (\mu_\epsilon * \frac{\partial^\alpha}{\partial y^\alpha}f) (y)$$

Due to the above secion about mollifying $$L^p$$-functions, we immediately know that $$\lim_{\epsilon \to 0} \|\mu_\epsilon * \frac{\partial^\alpha}{\partial y^\alpha}f - f\| = 0$$, and the second statement therefore follows from the definition of the $$W^{m, p}(U)$$-norm.

Let $$\Omega \subseteq \R^d$$ be an open set. Then for all functions $$v \in W^{m, p}(\Omega)$$, there exists a sequence of functions in $$C^\infty(\Omega) \cap W^{m, p}(\Omega)$$ approximating it.

Proof:

Let's choose
 * $$U_i := \{x \in \Omega : \text{dist}(\partial \Omega, x) > \frac{1}{i} \wedge \|x\| < i\}$$

and
 * $$V_i =\begin{cases}

U_3 & i = 0 \\ U_{i+3} \setminus \overline{U_{i+1}} & i > 0 \end{cases}$$

One sees that the $$V_i$$ are an open cover of $$\Omega$$. Therefore, we can choose a sequence of functions $$(\tilde \eta_i)_{i \in \N}$$ (partition of the unity) such that By defining $$\Eta_i := \{\tilde \eta_j \in \{\tilde \eta_m\}_{m \in \N} : \text{supp } \tilde \eta_j \subseteq V_i\}$$ and
 * 1) $$\forall i \in \N : \forall x \in \Omega : 0 \le \tilde \eta_i(x) \le 1$$
 * 2) $$\forall x \in \Omega : \exists \text{ only finitely many } i \in \N : \tilde \eta_i(x) \neq 0$$
 * 3) $$\forall i \in \N : \exists j \in \N : \text{supp } \tilde \eta_i \subseteq V_j$$
 * 4) $$\forall x \in \Omega : \sum_{i=0}^\infty \tilde \eta_i(x) = 1$$
 * $$\eta_i(x) := \sum_{\eta \in \Eta_i} \eta(x)$$, we even obtain the properties

where the properties are the same as before except the third property, which changed. Let $$|\alpha| = 1$$, $$\varphi$$ be a bump function and $$(v_j)_{j \in \N}$$ be a sequence which approximates $$v$$ in the $$L^p(\Omega)$$-norm. The calculation
 * 1) $$\forall i \in \N : \forall x \in \Omega : 0 \le \eta_i(x) \le 1$$
 * 2) $$\forall x \in \Omega : \exists \text{ only finitely many } i \in \N : \eta_i(x) \neq 0$$
 * 3) $$\forall i \in \N: \text{supp } \eta_i \subseteq V_i$$
 * 4) $$\forall x \in \Omega : \sum_{i=0}^\infty \tilde \eta_i(x) = 1$$
 * $$\int_\Omega \eta_i(x) v_j(x) \frac{\partial^\alpha}{\partial x^\alpha} \varphi(x) dx = - \int_\Omega \left(\frac{\partial^\alpha}{\partial x^\alpha}\eta_i(x) v_j(x) + \eta_i(x) \frac{\partial^\alpha}{\partial x^\alpha} v_j(x)\right) \varphi(x) dx$$

reveals that, by taking the limit $$j \to \infty$$ on both sides, $$v \in W^{m, p}(\Omega)$$ implies $$\eta_i v \in W^{m, p}(\Omega)$$, since the limit of $$\eta_i(x) \frac{\partial^\alpha}{\partial x^\alpha} v_j(x)$$ must be in $$L^p(\Omega)$$ since we may choose a sequence of bump functions $$\varphi_k$$ converging to 1.

Let's choose now
 * $$W_i = \begin{cases}

U_{i+4} \setminus \overline{U_i} & i \ge 1 \\ U_4 & i = 0 \end{cases}$$ We may choose now an arbitrary $$\delta > 0$$ and $$\epsilon_i$$ so small, that Let's now define
 * $$\|\eta_{\epsilon_i} * (\eta_i v) - \eta_i v\|_{W^{m, p}(\Omega)} < \delta \cdot 2^{-(j+1)}$$
 * 1) $$\text{supp } (\eta_{\epsilon_i} * (\eta_i v)) \subset W_i$$
 * $$w(x) := \sum_{i=0}^\infty \eta_{\epsilon_i} * (\eta_i v)(x)$$

This function is infinitely often differentiable, since by construction there are only finitely many elements of the sum which do not vanish on each $$W_i$$, and also since the elements of the sum are infinitely differentiable due to the Leibniz rule of differentiation under the integral sign. But we also have:
 * $$\|w - v\|_{W^{m, p}(\Omega)} = \left\|\sum_{i=0}^\infty \eta_{\epsilon_i} * (\eta_i v) -\sum_{i=0}^\infty (\eta_i v)\right\|_{W^{m, p}(\Omega)} \le \sum_{i=0}^\infty \|\eta_{\epsilon_i} * (\eta_i v) - \eta_i v\|_{W^{m, p}(\Omega)} < \delta \sum_{i=0}^\infty 2^{-(j+1)} = \delta$$

Since $$\delta$$ was arbitrary, this finishes the proof.

Let $$\Omega$$ be a bounded domain, and let $$\partial \Omega$$ have the property, that for every point $$x \in \partial \Omega$$, there is a neighbourhood $$\mathcal U_x$$ such that
 * $$\Omega \cap \mathcal U_x = \{(x_1, \ldots, x_d) \in \R^d : x_i < f(x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{d-1}) \}$$

for a continuous function $$f$$. Then every function in $$W^{m, p}(\Omega)$$ can be approximated by $$C^\infty(\overline{\Omega})$$-functions in the $$W^{m, p}(\Omega)$$-norm.

Proof: to follow