Mollification and density

Choose a nonnegative even smooth kernel $\rho$ supported in the unit ball with integral one, and define $\rho_\varepsilon(x)=\varepsilon^{-n}\rho(x/\varepsilon)$. For a locally integrable whole-space function, convolution is smooth wherever the compact averaging ball stays in a region of integrability. On an open domain without an extension, the same formula is defined only where the averaging ball lies inside the domain. Smoothing does not create boundary values that the original function never supplied. Its length scale is the kernel radius, not a promise of uniform pointwise error for arbitrary data.

If $u\in W^{k,p}(\mathbb R^n)$ with finite $p$, weak differentiation commutes with convolution: differentiate the kernel first and then use the weak identity to transfer derivatives back onto $u$. Translation continuity in $L^p$ implies that convolving each derivative with this approximate identity converges in $L^p$. Summing the finitely many component errors gives convergence in $W^{k,p}$. The same reasoning works locally inside a domain. It does not require the original derivative to be continuous, only integrable in the specified exponent.

For any open $\Omega$ and finite $p$, smooth functions on $\Omega$ that belong to $W^{k,p}(\Omega)$ are dense in that space. A locally finite partition of unity lets different neighborhoods use different smoothing radii, avoiding a false assumption that one global radius stays inside an arbitrary domain. This is the Meyers–Serrin density theorem. Smoothness on the open set does not mean smooth extension across its boundary, and these approximants need not have compact support. The theorem deliberately makes no regularity demand on the boundary for this interior formulation.

On a bounded Lipschitz domain, a bounded extension operator followed by whole-space smoothing gives smooth restrictions dense in $W^{1,p}(\Omega)$ for finite $p$. By contrast, $C_c^\infty(\Omega)$ is dense only in the subspace defined as $W^{1,p}_0(\Omega)$, not in all of $W^{1,p}$. A cutoff used to force compact support creates a derivative term involving its rapidly changing boundary layer. That extra term can prevent convergence even if the removed strip has very small volume. A constant function with nonzero boundary trace is the simplest warning.

The finite-p assumption matters. A corner such as $|x|$ lies in $W^{1,\infty}(-1,1)$, but its derivative is the sign function. Derivatives of smooth approximants are continuous, and cannot approach this jump in essential-supremum norm: an error smaller than one would force incompatible one-sided values at zero. Thus uniform approximation of the function is weaker than approximation in the full $W^{1,\infty}$ norm. Mollified derivatives can converge at almost every point while their largest essential error remains bounded away from zero.

On the whole space, a unit-mass kernel preserves the integral of an integrable function and nonnegativity of nonnegative data. Compactly supported kernels also enlarge support only by their radius. A heat kernel is another approximate identity with noncompact tails, so it smooths immediately but does not retain compact support. Derivative identities and norm convergence remain available under the appropriate integrability assumptions. When using heat flow as an illustration, distinguish its exact multiplier from a finite convolution stencil or a numerical solution of a PDE.