## Characterizations of Sobolev Spaces

I will present here a few useful characterizations of Sobolev spaces. The source is *Functional Analysis, Sobolev Spaces and Partial Differential Equations* by Haim Brezis. Consider with . We denote by the conjugate of , i.e. . Then the following properties are equivalent:

- (i) ;
- (ii) there exists a constant such that
;

- (iii) there exists a constant such that for all and all with we have
(note that makes sense for and . Furthermore, we can take in (ii) and (iii). If we have

.

## One dimensional trace of a Sobolev function

Check that the mapping from to is a continuous linear functional on . Deduce that there exists a unique such that

Show that is the solution of some differential equation with appropriate boundary conditions and compute explicitly.

*H. Brezis, Functional Analysis, Ex 8.18*

## A lemma of J. L. Lions

Let and be three Banach spaces with norms and . Assume that with *compact *injection and that with *continuous *injection. Prove that

satisfying .

*Applications:*

- Prove that for every there exists satisfying.
- Pick . Prove that for every there exists such that .

*Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011*

## Sobolev space impossible extension

Let . And for (**!!! Correction: **) consider . Prove that:

1) does not have Lipschitz boundary (i.e. its boundary is not locally the graph of Lipschitz functions).

2) .

3) For every ball which contains , there is no function in which extends .

This problem gives a counter example which states that if doesn’t have Lipschitz boundary, then there may be no extension to some functions in to greater Sobolev spaces.

## Generalized Poincare Inequality

Consider a bounded domain with Lipschitz boundary. If is a non-zero, closed subspace in , which does not contain the non-zero constant functions, then there is a constant , depending on , such that .

Note that this generalizes the usual Poincare inequality, which says that the above inequality holds for some on the space , a space which does not contain the non-zero constant functions.

## Function which vanishes at infinity

Let with . Show that ,

PHD Cincinnati (6103)