Sobolev Space

Since its definition involves generalized derivatives rather than ordinary ones it is complete that is it is a banach space.

Sobolev space. The space w 3 p represented by a blue dot at the point 1 p 3 embeds into the spaces indicated by red dots all lying on a line with slope n. Sobolev see so1 so2. Sobolev embedding theorem graphical representation of the embedding conditions. With the above motivation we can de ne sobolev spaces as follows.

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. For k2n 0 and 1 p 1 the sobolev space of order kis de ned as wk p ff2l p. The space w l p omega was defined and first applied in the theory of boundary value problems of mathematical physics by s l.

Note that wk p is a subspace of the banach space lp. The derivatives are understood in a suitable weak sense to make the space complete i e. Here we collect a few basic. 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.