1957/02/15 by R. Friedberg · 5 citations
Mathematics · Psychology · #Meromorphic and Entire Functions #History and Theory of Mathematics #Behavioral and Psychological Studies
paper · doi:10.1073/pnas.43.2.236
openalex publication_date 1957/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
If we have shown that for 1nf < jal, Vhu for sufficiently small h have their L (N2-IaI +1)-norms uniformly bounded in h and since the same is true for Vh ¶/i and Vh c#I, we may apply relation (3) to v = Vhau and conclude that IIVhaUjI(N -Ia I iX) is uniformly bounded in h for all sufficiently small h.But as h -O-0, Vhau con- verges in the distribution sense to (b/lx)au.If we consider the sequence V2-nhU in L2(N2-1Ja ,), then, by the uniform boundedness of the norms, we can find a subsequence converging weakly to an element of L2(N2-aI) which must equal (6/bx)au in that neighborhood.Since all the distribution derivatives of u are lo- cally in L2, our conclusion follows from a well-known theorem of Sobolev.6Remark 2: By a refinement of the argument, we can remove the restriction that u E L2(G) and prove regularity for any distribution solution.Similarly, bounds may be obtained for the L2-norms of u and its derivatives on a compact G1 in terms of L2-norms of 4 , t and its derivatives and any negative Dirichlet norm of u on any G2 containing 6, in its interior.1 For a survey of recent results in the elliptic case cf.