Partial Differential Equations/Sobolev spaces

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In this chapter, it will be defined, what Sobolev spaces are. Also, some properties of Sobolev spaces will be shown.

Weak derivatives and Sobolev spaces[edit]

Weak derivatives[edit]

Let \Omega \subseteq \R^d be a domain, \alpha \in \N_0^d a multiindex, and f \in L^p(\Omega), where p \in [1, \infty]. We call g \in L^p(\Omega) the weak derivative of order \alpha of f \in L^p(\Omega), if and only if:

\frac{\partial^\alpha}{\partial x^\alpha} T_f = T_g

We denote the weak derivative of f with the notation \frac{\partial^\alpha}{\partial x^\alpha} f, if it exists.

Properties[edit]

It can be easily observed that the weak derivatives are linear, i. e. \frac{\partial^\alpha}{\partial x^\alpha} (\lambda f + g) = \lambda \frac{\partial^\alpha}{\partial x^\alpha} f + \frac{\partial^\alpha}{\partial x^\alpha} g. Furthermore, they are equal to the actual derivatives if these exist. This follows due to integration by parts.

Weak gradient[edit]

The weak gradient of f \in L^p is defined as follows:

\nabla f := \left( \begin{smallmatrix}
\frac{\partial^{(1, 0, \ldots, 0)}}{\partial x^{(1, 0, \ldots, 0)}} f \\
\vdots \\
\frac{\partial^{(0, \ldots, 0, 1)}}{\partial x^{(0, \ldots, 0, 1)}}f \\
\end{smallmatrix} \right)

, i. e. it is the sorted d-dimensional vector of all weak derivatives with |\alpha| = 1.

Sobolev spaces[edit]

Let m \in \N_0, p \in [1, \infty], and \Omega be a domain. Then the Sobolev space \mathcal W^{m, p}(\Omega) is defined as follows:

\mathcal W^{m, p}(\Omega) := \{f \in L^p : \frac{\partial^\alpha}{\partial x^\alpha} f \text{ exists and } \frac{\partial^\alpha}{\partial x^\alpha} f \in L^p\}

The Sobolev spaces also have a norm: \|f\|_{W^{m, p}(\Omega)} := \sum_{|\alpha| \le m} \left\| \frac{\partial^\alpha}{\partial x^\alpha} f \right\|_{L^p}

Theorem 7.1[edit]

This expression is really a norm.

Proof: To show that the expression is a norm, we have to check the three defining properties for a norm:

  • Definiteness, i. e. \|x\| = 0 \Leftrightarrow x = 0
  • Homogeneity, i. e. \|\alpha x\| = |\alpha|\|x\| for every scalar \alpha
  • and the triangle inequality, i. e. \|x + y\| \le \|x\| + \|y\|.

We start with definiteness: If f = 0 (i. e. f is the function constantly zero), then obviously \|f\|_{W^{m, p}} = 0, since the derivatives of zero (which is a constant function) are zero. Furthermore, if \|f\|_{W^{m, p}} = 0, then in particular it follows that \|f\|_{L^p} = 0, which implies that f=0 since \|f\|_{L^p} is a norm.

Homogeneity follows directly from the linearity of the derivative and the homogeneity of the \| \cdot \|_{L^p}-norm.

The triangle inequality follows from the linearity of the derivative, the triangle inequality for the \| \cdot \|_{L^p}-norm and the commutation law for addition:

\|f + g\|_{W^{m, p}} := \sum_{|\alpha| \le k} \left\| \frac{\partial^\alpha}{\partial x^\alpha} f + \frac{\partial^\alpha}{\partial x^\alpha} g \right\|_{L^p} \le \sum_{|\alpha| \le k} \left\| \frac{\partial^\alpha}{\partial x^\alpha} f \right\|_{L^p} + \left\| \frac{\partial^\alpha}{\partial x^\alpha} g \right\|_{L^p} =: \|f\|_{W^{m, p}} + \|g\|_{W^{m, p}}

Theorem 7.2[edit]

All Sobolev spaces are Banach spaces.

Proof: We use the fact that L^p-spaces are Banach spaces. Let (u_i)_{i \in \N} be a Cauchy sequence in \mathcal W^{m, p}(\Omega). Then, since for all multi-indices \alpha \in \N_0^d, we find that:

\left\|\frac{\partial^\alpha}{\partial x^\alpha} u_i - \frac{\partial^\alpha}{\partial x^\alpha} u_j\right\| \le \sum_{|\alpha| \le k} \left\| \frac{\partial^\alpha}{\partial x^\alpha} (u_i - u_j) \right\|_{L^p}

Therefore, the sequences \left( \frac{\partial^\alpha}{\partial x^\alpha} u_i \right)_{i \in \N} are all Cauchy sequences in L^p and therefore have a limit u_\alpha. We choose u := u_{(0, \ldots, 0)} and show that u is the limit of (u_i)_{i \in \N} in \mathcal W^{m, p}(\Omega): Let \varphi \in \mathcal D(\Omega) (and therefore also \varphi \in \mathcal S). Then:

\int_{\R^d} \frac{\partial^\alpha}{\partial x^\alpha} \varphi(x) u(x) dx = \lim_{i \to \infty} \int_{\R^d} \frac{\partial^\alpha}{\partial x^\alpha} \varphi(x) u_i(x) dx = \lim_{i \to \infty} (-1)^{|\alpha|} \int_{\R^d} \varphi(x) \frac{\partial^\alpha}{\partial x^\alpha} u_i(x) dx

, and due to the dominated convergence theorem,

\lim_{i \to \infty} (-1)^{|\alpha|} \int_{\R^d} \varphi(x) \frac{\partial^\alpha}{\partial x^\alpha} u_i(x) dx = (-1)^{|\alpha|} \int_{\R^d} \varphi(x) u_\alpha(x) dx

, which is why u_\alpha is the weak derivative of order \alpha of u. This finishes the proof, since it shows that u_i \to u, i \to \infty with respect to the norm of \mathcal W^{m, p}(\Omega).

Approximation by smooth functions[edit]

Approximation of continuous functions[edit]

Recall the definition of the multi-dimensional mollifiers. Let \Omega \subseteq \R^d be a domain and u \in C_0(\Omega), i. e. u is a continuous function with compact support.

Then u * \eta_\epsilon \in C^\infty(\Omega) and u * \eta_\epsilon \to u uniformly as \epsilon \to 0.

Proof: First, that u * \eta_\epsilon \in C^\infty(\Omega) follows by the definition of the convolution and the Leibniz integral rule (depending on how you define the convolution, you also need to prove that u * \eta_\epsilon \in C^\infty(\Omega) = \eta_\epsilon \in C^\infty(\Omega) * u, so that you apply the derivative to \eta_\epsilon). This shows the first part.

We further know that u is a continuous function. Since u has compact support, it is also uniformly continuous. Furthermore:

(u * \eta_\epsilon) (y) = \int_{\R^d} u(x) \eta_\epsilon(y - x) dx

Since u is uniformly continuous, we may choose \epsilon > 0 such that \|x - y\| < 2 \epsilon \Rightarrow |u(x) - u(y)| < \delta. Therefore, we find, since \int_{\R^d} \eta_\epsilon(x) dx = 1, the following:

\left| \int_{\R^d} u(x) \eta_\epsilon(y - x) dx - u(y) \right| \le \int_{\R^d} |u(x) - u(y)| \eta_\epsilon(y - x) dx < \delta

Since the smallest possible \delta decreases with decreasing \epsilon, and is arbitrary, the claim is proven.

Approximation of Lp-functions[edit]

Let \Omega \subseteq \R^d be a domain, 1 \le p < \infty and u \in L^p(\Omega), and let furthermore \text{supp } u + B_\epsilon(0) \subseteq \Omega.

Then u * \eta_\epsilon \in C^\infty(\Omega) and u * \eta_\epsilon \to u uniformly as \epsilon \to 0.

Proof: The first claim, namely u * \eta_\epsilon \in C^\infty(\Omega), follows exactly the same way as in the subsubchapter about mollifying continuous functions, by the Leibnis rule for integrating under the integral sign.

We will use that C_0(\Omega) is a dense subset of L^p(\Omega). This means by definition that

\forall f \in L^p(\Omega): \forall \epsilon > 0: \exists g \in C_0(\Omega) : \|f - g\|_{L^p} < \epsilon

i. e. one can approximate functions in L^p(\Omega) arbitrarily well with functions in C_0(\Omega).

This is proven by approximating the elementary functions, which are used in the definition of the Lebesgue integral, with continuous functions. It is OK that they have only compact support, because all L^p-functions h have the property that for arbitrary \epsilon > 0 there exists a bounded set M such that

\left| \int_{\R^d \setminus M} h(x) dx \right| < \epsilon

.

Let' choose an arbitrary \epsilon > 0. We choose now first g \in C_0(\Omega) such that \|g - f\|_{L^p} < \frac{\epsilon}{4}.

Then we choose \delta_1 > 0 such that \|g - g * \eta_{\delta_1} \| < \frac{\epsilon}{4}.

Third, we notice that f * \eta_\delta - g * \eta_\delta = (f - g) * \eta_\delta. With the triangle inequality and the reversed triangle inequality, we furthermore obtain:

|\|(f - g) * \eta_\delta\|_{L^p} - \|f - g\|_{L^p}| \le \|((f - g) * \eta_\delta) - (f - g)\|_{L^p}

If we choose now \delta_2 > 0 such that \|((f - g) * \eta_{\delta_2}) - (f - g)\| < \frac{\epsilon}{4}, we obtain, together with the fact that \|f - g\|_{L^p} < \frac{\epsilon}{4}, that

\|f * \eta_{\delta_2} - g * \eta_{\delta_2}\|_{L^p} < \frac{\epsilon}{2}

We now choose \delta = \min\{\delta_1, \delta_2\} and obtain with the triangle inequality:

\|f - f * \eta_\delta\|_{L^p} \le \|f - g\|_{L^p} + \|g - g * \eta_\delta\|_{L^p} + \|f * \eta_\delta - g * \eta_\delta\|_{L^p} < \frac{\epsilon}{4} + \frac{\epsilon}{4} + \frac{\epsilon}{2} = \epsilon

Since \epsilon > 0 was arbitrary, this finishes the proof.

Lemma 7.3[edit]

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.

Partition of unity[edit]

Let \Omega \subseteq \R^d be an open set, and \bigcup_{\alpha \in A} M_\alpha be an open cover of \Omega. Then there exists a sequence of functions \eta_i such that the following conditions are satisfied:

  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 : \exists \alpha \in A : \text{supp } \eta_i \subseteq M_\alpha
  4. \forall x \in \Omega : \sum_{i=0}^\infty \eta_i(x) = 1

Proof: We will proof this by explicitly constructing such a sequence of functions. First, we construct a sequence of closed balls B_i with the properties

  1. \forall i \in \N : \exists \alpha \in A : B_i \subseteq M_\alpha
  2. \forall x \in \Omega : \exists \text{ only finitely many } i \in \N : x \in B_i
  3. \bigcup_{i \in \N} \text{interior}( B_i) = \Omega.

In order to do this, we first start with the definition of a sequence compact sets:

K_i := \{x \in \Omega : \text{dist}(\partial \Omega, x) \ge \frac{1}{i}, \|x\| \le i\}

This sequence obviously has the properties

  • \bigcup_{j \in \N} K_j = \Omega
  • K_j \subset \text{interior}(K_{j+1})

We now construct our sequence of closed balls such that

K_1 \subset \bigcup_{1 \le i \le k_1} \text{interior}(B_i) \subseteq \text{interior}(K_2)
and K_{j+1} \setminus \text{interior}(K_j) \subset \bigcup_{k_j < i \le k_{j+1}} B_i \subseteq \text{interior}(K_{j+2}) \setminus K_{j-1}

We do this in the following way: To meet the first condition, we first cover K_1 with balls by choosing for every x \in K_1 a ball B_x such that B_x \subseteq ( M_\alpha \cap \text{interior}(K_2)). Since these balls cover K_1, and K_1 is compact, we may choose a finite subcover B_1, \ldots B_{k_1}.

To meet the second condition, we do just the same thing, noticing that K_{j+1} \setminus \text{interior}(K_j) is compact and \text{interior}(K_{j+2}) \setminus K_{j-1} is open.

This sequence of open balls fulfills the conditions which we wanted.

We recall now the Definition of the multi-dimensional mollifier and note that we can recenter it to an arbitrary x_0 \in \R^d by using the formula

\eta_{\epsilon, x_0} (x) := \eta_\epsilon (x - x_0)

What we do now, is that for each B_i, we choose \tilde \eta_i as the mollifier which is adjusted to the ball, i. e. the ball is it's support. Then we choose

\eta(x) := \sum_{i=0}^\infty \tilde \eta_i(x)

and

\eta_i := \frac{\tilde \eta_i}{\eta},

and we have found a sequence of functions which satisfies the conditions we wanted.

Approximation of Wm, p-functions with smooth functions on open sets[edit]

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

  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

By defining \Eta_i := \{\tilde \eta_j \in \{\tilde \eta_m\}_{m \in \N} : \text{supp } \tilde \eta_j \subseteq V_i\} and

\eta_i(x) := \sum_{\eta \in \Eta_i} \eta(x), we even obtain the properties
  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

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

\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

  1. \|\eta_{\epsilon_i} * (\eta_i v) - \eta_i v\|_{W^{m, p}(\Omega)} < \delta \cdot 2^{-(j+1)}
  2. \text{supp } (\eta_{\epsilon_i} * (\eta_i v)) \subset W_i

Let's now define

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.

Approximation of Wm, p-functions with smooth functions on compact sets[edit]

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