Multidimensional Sobolev Spaces

We have introduced one-dimensional Sobolev spaces in Chap. 2 . The fundamental concept that is required is that of weak derivative. In the case of higher dimension, we need to talk about weak partial derivatives. The basic identity that allows to introduce such fundamental concept is again the integration-by-parts formula which in the multidimensional framework is a consequence of the classical divergence theorem and the product rule. Though many concepts are similar to the one-dimensional situation, technical issues are much more involved.