Regularity theorems for the solutions of general elliptic systems and boundary value problems

Abstract

AbstractIn the preceding chapter, we developed rather completely the regularity properties of weak and strong solutions of certain first order variational problems and second order differential equations involving only one unknown function. Complete results concerning the solutions of the corresponding problems involving several unknown functions were obtained only in the case where v = 2; these results were obtained by the writer before the war and were described in Chapter 1. In the preceding chapter some first differentiability results were obtained for certain systems of equations but these results did not imply the continuity of the first derivatives. In 1952, the writer presented a paper (Morrey [10]) at the Arden House Conference on Partial Differential Equations, in which it was proved that any vector solution of class C1 of a regular variational problem of class C n μ , n ≥ 2, was also of class C n μ . These results still leave a gap in the theory for systems, which can only be filled by an extension of the De Giorgi-Nash results, developed in § 5–3, to systems or some entirely new device. But, in proving the C n gm differentiability results for systems of equations, the writer was forced to use the very important formulas of F. John ([1], [3]) for fundamental solutions and to use methods which are appropriate for the discussion of elliptic systems of higher order. Hence the inclusion of this chapter in this book.KeywordsWeak SolutionElliptic SystemGeneral BoundaryPrincipal PartRegularity TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.