Interior mollification of domain Sobolev functions #
A function and its weak derivative in Lᵖ(Ω), for 1 ≤ p ≤ ∞, can be extended by zero
and convolved with a smooth compactly supported kernel. Wherever the translated kernel
support lies inside Ω, the classical derivative of the mollification is the mollification
of the weak derivative. Zero extension need not preserve weak differentiability at the boundary;
the support condition ensures that no boundary term enters the identity.
HasWeakFDerivOn.hasFDerivAt_indicator_convolution_right states this for arbitrary kernels.
HasWeakFDerivOn.hasFDerivAt_indicator_convolution_normed specializes to normalized smooth bumps,
with the geometric condition that the closed ball of the outer radius lies inside the domain.
The vector-valued formulation applies to successive weak derivative fields when constructing
smooth local approximations in higher-order Sobolev spaces.
References #
L. C. Evans, Partial Differential Equations, Chapter 5, §5.3.1.
If U is a weak derivative field of u on Omega, with u ∈ Lᵖ(Omega) and
U ∈ L^q(Omega) for 1 ≤ p, q ≤ ∞, the convolution of the zero extension of u with a
smooth compactly supported kernel rho has derivative at x the convolution of the zero
extension of U with rho, whenever x - tsupport rho ⊆ Omega.
Where x - tsupport rho ⊆ Omega, the Fréchet derivative of the convolution of the zero
extension of u with rho equals the convolution of the zero extension of its weak derivative
field U with rho. The fields may have different integrability exponents 1 ≤ p, q ≤ ∞.
At any point x whose closed ball of radius phi.rOut lies in Omega, the mollification
of the zero extension of u by the normalized bump has derivative the mollification of the
zero extension of its weak derivative field U.
At a point x with closedBall x phi.rOut ⊆ Omega, the Fréchet derivative of the
mollification of the zero extension of u by the normalized bump equals the mollification
of the zero extension of its weak derivative field U.