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For such solutions, the theorem on uniqueness “in the small” was also established a s well a s the fact that their differentiability properties depend only on the differentiability properties of F. However, the existence theorem, of which we spoke above, yielded, for the general case, only generalized solutions, functions with considerably poorer properties. 3) was devoted. This work proceeded basically along the line of imposing various supplementary restrictions on F under which it was possible either to prove the existence theorem immediately in the class of sufficiently broad functions (as was done by Bernstein, Tonelli, Hopf, and others) or to improve gradually the differentiability properties of the generalized solutions of the problem (Morrey, Sigalov, Plotnikov, and others).
U . p ) and a , the distance from Q1 to S, where Q, is a strictly interior subregion, the distance from 8,to S \ S', and any norms S' and u I s , , where Q, i s adjacent to the portion S'of the boundary of S, then we need to impose certain restrictions regarding the growth of u , ) and a and their partial derivatives a s J p I+a. The following example shows that one of the necessary conditions is the 23 ADMISSIBLE EXTENSIONS O F T H E CONCEPT if a ( x . u. p ) does not have unbounded singularities for x E and for finite u and p .
19), we can see that this is impossible. 20), the constant c will increase without bound a s R -+ 0. Lu=O, u J s = O has only the zero solution in the class W i ( Q ) . In this case, the operator L has a bounded inverse in L, (Q). Therefore, )Iu 1) L l ( e ) is bounded in terms of ! J L U I I ~ ,for ( ~ ) an arbitrary function u in Wg(&)that vanishes on S. 19). 5) with the same 'Y and p. Let u s average both members of Eq. 19) with the aid of some sort of nonnegative infinitely differentiable kernel wp ( Ix -- Y I 1 (see 11, 21).
