2015/10/16 by Hugo Beirao da Veiga, da Veiga, Hugo Beirao
Mathematics · #35A09 #35B65 #35J25 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A09 #msc:35B65 #msc:35J25
paper · pdf · doi:10.48550/arxiv.1510.04926
arxiv created 2015/10/16 · arxiv updated 2015/10/19
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 D2 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 D2 u (but nothing more, say). Hölder continuity is too restrictive to fulfill this minimal requirement since in this case D2 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 D0,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 D0,a for some fixed a > 1, then D2 u in D0,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 Dom, 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 C0,la (l= lambda), where 0 <= l < 1, and a in R. In particular, C0,l0= C0,l, is a Hölder space, and C0,0a = D0,a, is a Log space. Main point is that full regularity occurs for l > 0, and arbitrary a in R. If f in C0,la then D2 u in C0,la.