(Finite differences). For a locally summable function , we introduce the notations Here the is the -th coordinate vector, .

(Cutoff function). Let . The cutoff function is any function that satisfies the following conditions:

The importance of the cutoff function is evident from the following propositions. Note, that we have to restrict the original set to a subset .

(Finite difference in Sobolev space). Let and . Then for any subset for such that dist .

Due to the proposition ( Local approximation by smooth functions ) it is enough to prove the statement for a smooth . We have Hence, for

(Finite difference basics). Let be a bounded set and are locally summable functions. Then If, in addition, then

We verify the statements as follows:

(Existence of derivative via finite difference). Let , , and for some constant and any such that dist . Then

According to the proposition ( Weak compactness of bounded set ) there exists a sequence , such that for every . Then we pass the formula ( Integration by part for finite differences ) to the limit and use the definition ( Weak derivative ) to establish that in the weak sense. Then follows from .