chapter  2
34 Pages

Variational Formulation of the Dirichlet and Neumann Problems

Let fc G R. We denote by Hk(R2) the standard Sobolev space (see [2], [11], [14] and [16]) consisting of three-component distributions u E 5'(IR2) whose Fourier transforms u are regular distributions [12] generated by functions £&(£), and such that

For a domain S C R2, we denote by Hk(S) the space of the restric­ tions to S of all the elements of Hk(R2). The norm in H k(S) is defined by

The norm on H k(S) for nonnegative integers k is equivalent to

where a = (oq, a ^ ,<23) 7^ 0 is a multiindex with nonnegative com­ ponents. From now on we do not distinguish between equivalent norms.