On classical solutions to elliptic boundary value problems. The full regularity spaces $C^{0,\,\la}_\al(\Ov)$

Abstract

Let L be a second order, uniformly elliptic operator, and consider the equation L u=f under the homogeneous boundary condition u=0. It is well known that f in C(Om) (Om= Omega) does not guarantee second order derivatives D^2 u in C(Om). This gap led to look for functional spaces C_*(Om), contained in C(Om), as large as possible, for which f in C_*(Om) merely guarantees the continuity of D^2 u (but nothing more, say). Hölder continuity is too restrictive to fulfill this minimal requirement since in this case D^2 u inherits the whole regularity enjoyed by f (we say that "full regularity" occurs). This two opposite situations led us to look for significant cases in which intermediate regularity (i.e., between "mere continuity" and "full regularity") occurs. This holds for data in Log spaces D^{0,a}(Om) (a= alpha) where 0< a < infty, simply obtained by replacing in the modulus of continuity of Hölder spaces the quantity 1/|x-y| by log(1/|x-y|). If f in D^{0,a} for some fixed a > 1, then D^2 u in D^{0,a-1}. This regularity is optimal. The above picture opened the way to further investigation. Below we study the more general problem of data f in subspaces of continuous functions D_{om}, characterized by a given modulus of continuity om(r). Hölder and Log spaces are particular cases. A significant new, lets say curious, case is shown by the family of functional spaces C^{0,l}_a (l= lambda), where 0 <= l < 1, and a in R. In particular, C^{0,l}_0= C^{0,l}, is a Hölder space, and C^{0,0}_a = D^{0,a}, is a Log space. Main point is that full regularity occurs for l > 0, and arbitrary a in R. If f in C^{0,l}_a then D^2 u in C^{0,l}_a.