Interior mollification of higher weak derivatives #
For a function in W^{k+2,p}(Ω), the Fréchet derivative of the mollification of its
order-k+1 weak derivative is the mollification of its order-k+2 weak derivative.
The convolution uses zero extensions, but the identity holds only where the translated
kernel support is contained in Ω; no regularity of the boundary is assumed.
The result supplies the successive derivative identities needed to construct smooth local approximations in the iterated weak Sobolev spaces. It uses the weak-derivative identity recorded at each graph step and the interior convolution theorem.
The classical argument is in L. C. Evans, Partial Differential Equations, §5.3.1.
The first derivative identity for the mollified Sobolev jet. The gradient stored by
Wkp is converted to a linear functional using the real inner product.
Pointwise derivative form of hasFDerivAt_indicator_convolution_value.
The classical derivative of the mollified kth iterated weak-gradient field
is the mollified (k+1)st field in the interior of the domain.
Pointwise derivative form of hasFDerivAt_indicator_convolution_iteratedGradient.
With a normalized smooth bump, the support condition is supplied by a closed ball contained in the domain. This version can be applied at each stage of a Sobolev jet without choosing a separate support bound for the bump.
The derivative of a normalized-bump mollification of the value of a first-order Sobolev function, at points whose bump stays in the domain.
Pointwise derivative form of hasFDerivAt_indicator_convolution_normed_value.
Pointwise derivative form of
hasFDerivAt_indicator_convolution_normed_iteratedGradient.