Sobolev Spaces

From weak derivatives to function spaces¶

In the previous sections we defined the weak derivative of a locally integrable function: $v = D^\alpha u$ weakly if

This makes sense for any $u \in L^1_{\mathrm{loc}}(\Omega)$, but the derivative $v$ might not be in $L^p$, or might not exist as an $L^p$ function at all. Sobolev spaces are defined by requiring that weak derivatives exist and have controlled size.

The completeness proof is simple because it reduces to the completeness of $L^p$: each derivative converges separately, and the weak derivative structure is preserved under $L^p$ limits. This is a common pattern: Sobolev spaces inherit their analytical properties from $L^p$.

The notation $H^k$ reflects the Hilbert space structure. For PDE applications, $H^1(\Omega)$ is the most important space: it controls the function and its first derivatives in $L^2$.

What does the Sobolev norm measure?¶

In the introduction (Remark 1), we asked “what do $L^p$ norms measure?” and found that they capture two features of a function:

• Height (amplitude): the peak value, controlled by $L^\infty$.

• Width (spatial extent): the size of the support, contributing to $L^p$ for $p < \infty$.

But these two features miss something fundamental. Consider the sequence $u_n(t) = \sin(n\pi t)$ on $[0,1]$:

Every function in this sequence has the same height, the same support, and the same $L^2$ norm, yet they are clearly different: $u_{100}$ oscillates 100 times faster than $u_1$. The $L^2$ norm is completely blind to this.

This is the same sequence that appeared in the weak convergence chapter (Example 2): $\sin(n\pi t) \rightharpoonup 0$ weakly in $L^2$, but $\|\sin(n\pi t)\|_{L^2} = 1/\sqrt{2}$ for all $n$. The $L^2$ norm cannot distinguish rapid oscillation from no oscillation at all.

Source

import numpy as np
import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 3, figsize=(15, 4.5))
t = np.linspace(0, 1, 1000)

# --- Panel 1: sin(nπt) for increasing n ---
ax = axes[0]
for n, color in [(1, 'C0'), (3, 'C1'), (10, 'C2'), (30, 'C4')]:
    ax.plot(t, np.sin(n * np.pi * t), color=color, lw=1.2, alpha=0.8,
            label=rf'$\sin({n}\pi t)$')
ax.set_xlabel(r'$t$', fontsize=11)
ax.set_ylabel(r'$u_n(t)$', fontsize=11)
ax.set_title('Same $L^2$ norm, different frequencies', fontsize=11)
ax.legend(fontsize=8, loc='upper right')
ax.tick_params(labelsize=9)

# --- Panel 2: L2 norm vs H1 norm ---
ax = axes[1]
ns = np.arange(1, 31)
l2_norms = np.ones_like(ns, dtype=float) / np.sqrt(2)
h1_norms = np.sqrt(0.5 + 0.5 * (ns * np.pi)**2)
ax.plot(ns, l2_norms, 'C0o-', ms=4, lw=1.5, label=r'$\|u_n\|_{L^2}$')
ax.plot(ns, h1_norms, 'C3s-', ms=4, lw=1.5, label=r'$\|u_n\|_{H^1}$')
ax.set_xlabel(r'$n$', fontsize=11)
ax.set_ylabel('Norm', fontsize=11)
ax.set_title(r'$L^2$ blind to frequency; $H^1$ sees it', fontsize=11)
ax.legend(fontsize=9)
ax.set_yscale('log')
ax.tick_params(labelsize=9)

# --- Panel 3: Fourier weights ---
ax = axes[2]
freqs = np.arange(0, 20)
for s, color, label in [(0, 'C0', '$s=0$ ($L^2$)'),
                         (1, 'C1', '$s=1$ ($H^1$)'),
                         (2, 'C3', '$s=2$ ($H^2$)')]:
    weights = (1 + freqs**2)**s
    ax.plot(freqs, weights, 'o-', color=color, ms=5, lw=1.5, label=label)
ax.set_xlabel(r'Frequency $|n|$', fontsize=11)
ax.set_ylabel(r'Weight $(1 + n^2)^s$', fontsize=11)
ax.set_title(r'Sobolev weights penalize high frequencies', fontsize=11)
ax.legend(fontsize=9)
ax.set_yscale('log')
ax.tick_params(labelsize=9)

plt.tight_layout()
plt.show()

Left: the functions $\sin(n\pi t)$ oscillate faster but have constant $L^2$ norm. Center: the $L^2$ norm (blue) stays flat while the $H^1$ norm (red) grows linearly in $n$, showing that the derivative detects oscillation. Right: the Fourier weights $(1 + n^2)^s$ that define $H^s$; higher $s$ penalizes high frequencies more aggressively.

The third feature: frequency¶

The Sobolev norm adds the missing dimension. Computing the derivative:

So the $H^1$ norm sees the oscillation:

The derivative term $\|u_n'\|_{L^2}$ blows up like $n\pi$: differentiation converts oscillation frequency into amplitude. The $H^1$ norm detects the frequency because derivatives penalize oscillation.

The bump function scaling heuristic¶

Following Tao, we can make this precise with a model function that independently controls all three features. Fix a bump function $\phi \in C_c^\infty(\mathbb{R})$ with $\operatorname{supp}(\phi) \subset [-1, 1]$, and consider

where:

• $A > 0$ controls the height (amplitude),

• $R > 0$ controls the width (spatial extent: support $\approx [-R, R]$),

• $N > 0$ controls the frequency (oscillation rate).

The $L^p$ norm scales as (by the change of variables $y = x/R$):

so $\|f_{A,R,N}\|_{L^p} \approx A \, R^{1/p}$. The frequency $N$ does not appear: the $L^p$ norm is blind to oscillation.

The $W^{s,p}$ norm behaves differently. Each derivative of $\sin(Nx)$ pulls down a factor of $N$:

(the lower-order terms involve derivatives of $\phi$, which contribute factors of $R^{-1}$ instead of $N$, and are negligible when $N \gg 1/R$). Therefore:

The three parameters are now visible:

The extra factor $N^s$ is the whole point: the Sobolev norm penalizes high-frequency oscillation, with the penalty growing as $N^s$. Higher Sobolev index $s$ means stronger frequency penalty.

Regularity as frequency decay¶

The bump function heuristic shows that the Sobolev norm penalizes high frequency. But how many degrees of regularity does a function have? To make this precise, consider the one-parameter family

where $\phi \in C_c^\infty(\mathbb{R})$ is a fixed bump. The prefactor $N^{-s}$ is chosen so that $\|G_{s,N}\|_{W^{s,p}}$ remains bounded as $N \to \infty$ (from the scaling $\|f_{A,R,N}\|_{W^{s,p}} \approx A R^{1/p} N^s$ with $A = N^{-s}$ and $R = 1$).

Computing derivatives (keeping only the dominant term when $N \gg 1$):

The behavior splits sharply at $k = s$:

The following plot shows $G_{s,N}$ for $s = 2$ and increasing $N$. The function itself shrinks (because of the $N^{-s}$ prefactor), but its second derivative stays $O(1)$ and its third derivative grows.

Source

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-1.5, 1.5, 4000)
dx = x[1] - x[0]

def bump(x):
    """Smooth bump function supported on [-1, 1]."""
    result = np.zeros_like(x)
    mask = np.abs(x) < 1
    result[mask] = np.exp(-1 / (1 - x[mask]**2))
    return result / np.max(result)

s = 2  # regularity index
Ns = [5, 15, 50]
phi = bump(x)

fig, axes = plt.subplots(1, 3, figsize=(12, 3.5))
titles = [
    '$G_{2,N}(x)$  (k=0 < s)',
    "$G_{2,N}''(x)$  (k=2 = s)",
    "$G_{2,N}'''(x)$  (k=3 > s)",
]
deriv_orders = [0, 2, 3]

for col, (k, title) in enumerate(zip(deriv_orders, titles)):
    ax = axes[col]
    for N in Ns:
        g = N**(-s) * phi * np.sin(N * x)
        # Compute k-th derivative numerically
        gk = g.copy()
        for _ in range(k):
            gk = np.gradient(gk, dx)
        lp_norm = np.sqrt(np.trapezoid(gk**2, x))
        ax.plot(x, gk, lw=1.0, label=f'$N={N}$, $\\|\\cdot\\|_2={lp_norm:.2f}$')
    ax.set_title(title, fontsize=10)
    ax.set_xlabel('$x$')
    ax.legend(fontsize=7)
    ax.axhline(0, color='gray', lw=0.5, zorder=0)
    ax.grid(True, alpha=0.3)

fig.suptitle('Regularity threshold for $G_{s,N}$ with $s=2$: '
             '$L^2$ norms decay ($k<s$), stay bounded ($k=s$), or blow up ($k>s$)',
             fontsize=10, y=1.02)
plt.tight_layout()
plt.show()

The Fourier picture¶

For $H^s = W^{s,2}$ on the torus $[0, 2\pi]$ (or $\mathbb{R}^d$ via the Fourier transform), the connection to frequency is exact. If $u(x) = \sum_n \hat{u}_n \, e^{inx}$, then $D^s u$ has Fourier coefficients $(in)^s \hat{u}_n$, and Parseval gives:

The weight $(1 + |n|^2)^s$ penalizes high frequencies $|n|$ by a factor of $|n|^{2s}$. This makes the frequency interpretation precise:

• $L^2$ ($s = 0$): all frequencies weighted equally. The norm is $\sum |\hat{u}_n|^2$: total energy, regardless of frequency.

• $H^1$ ($s = 1$): weight $(1 + n^2)$. High frequencies are penalized quadratically.

• $H^k$ ($s = k$): weight $(1 + n^2)^k$. Higher $k$ means stronger suppression of high frequencies.

A function is in $H^s$ if and only if its Fourier coefficients decay fast enough to compensate for the weight $(1 + |n|^2)^s$. Smoother functions have faster-decaying Fourier coefficients, so they live in higher Sobolev spaces.

Examples: what is and isn’t in $W^{1,2}$¶