Sobolev Space

The space w l p omega was defined and first applied in the theory of boundary value problems of mathematical physics by s l.

Sobolev space. More about the spaces wk p will be discussed a bit later. We could start with c functions with compact support on rd and complete it in the norm u k p defined by u p k p n1 nd 0 n n1 nd k dn1 x1 dn d x d. In mathematics a sobolev space is a vector space of functions equipped with a norm that is a combination of lp norms of the function together with its derivatives up to a given order.

Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning par tial differential equations. With the above motivation we can de ne sobolev spaces as follows. The sobolev spaces w k p rd are defined as the space of functions u on rd such that u and all its partial derivatives dn1 x1 dn d x d u of order n n 1 n d k are in l p.

The derivatives are understood in a suitable weak sense to make the space complete i e. Note that wk p is a subspace of the banach space lp. Sobolev see so1 so2.

The white circle at 0 0 indicates the impossibility of optimal embeddings into l. Since its definition involves generalized derivatives rather than ordinary ones it is complete that is it is a banach space. For k2n 0 and 1 p 1 the sobolev space of order kis de ned as wk p ff2l p.