Weak solutions and Sobolev spaces

For a locally integrable function u, a locally integrable function v is its weak derivative in direction i when ∫u ∂_iφ=−∫vφ for every smooth compactly supported test function φ. Compact support removes boundary terms, so this definition initially describes interior differentiation. If a classical derivative is continuous, integration by parts shows it is also the weak derivative. The converse fails: absolute value has a weak derivative even though its classical derivative is undefined at zero. Weak derivatives represented by functions are unique up to equality almost everywhere.

Every locally integrable function has distributional derivatives, but those derivatives need not be locally integrable functions. A step function has a delta distribution as its derivative, which cannot be represented by an Lp function on an interval containing the jump. It therefore fails the first-order Sobolev condition there. By contrast, the first derivative of |x| is sign(x), an essentially bounded function. Changing the assigned value of sign at zero changes no integrals and no Sobolev class. The second distributional derivative of |x| is 2δ₀, so membership at one derivative order does not imply membership at two.

For p=2, W1,2 is written H1 and has norm squared equal to the L2 norm squared of the function plus that of its gradient. It is a Hilbert space, allowing orthogonality and bounded functional arguments. Convergence in H1 controls both quantities, while L2 convergence alone can hide increasingly rapid oscillation in derivatives. Sobolev embedding theorems provide stronger conclusions only with the dimension, derivative order, exponent, and domain hypotheses stated. In particular, an arbitrary H1 function in several dimensions need not have a continuous representative.

Boundary conditions need a comparable level of care. Define H1_0 as the closure of smooth compactly supported functions in the H1 norm. On a bounded Lipschitz domain, this agrees with H1 functions having zero trace. A trace is a continuous boundary restriction operator in the appropriate spaces, rather than a license to choose arbitrary point values for an equivalence class. Nonzero Dirichlet data can be handled by a suitable extension and subtraction. These definitions keep a boundary condition meaningful even when the solution is not continuous up to every boundary point.

For −Δu=f and homogeneous Dirichlet data, multiply by a zero-boundary test function and integrate by parts. The weak problem asks for u in H1_0 such that ∫∇u·∇v=∫fv for every v in H1_0, assuming for example f in L2 and a bounded Lipschitz domain. Only one derivative of u appears. This enlarges the admissible solution class without changing the sign of the original operator. A smooth weak solution recovers the classical equation in the interior by testing against compactly supported probes. Whether a particular weak solution is smooth requires a separate regularity theorem.

In the spectral lab, fix a modest decay p and increase mode count N. The added amplitudes may look small while their short wavelengths noticeably sharpen the profile. Each spatial derivative multiplies the nth sine coefficient by n, so visual amplitude alone misses the quantities entering an H1 norm. The lab displays a finite sum and every such sum is smooth; it neither creates a genuine jump nor proves an infinite series is in a Sobolev space. Use it to motivate the derivative-weighted estimate, then apply an actual convergence calculation when passing to infinitely many modes.