Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)

Abstract

Let 𝑳{\boldsymbol{L}} be a second order uniformly elliptic operator, and consider the equation 𝑳⁢u=f{\boldsymbol{L}u=f} under the boundary condition u=0{u=0}. We assume data f in generical subspaces of continuous functions Dω¯{D_{\overline{\omega}}} characterized by a given modulus of continuityω¯⁢(r){\overline{\omega}(r)}, and show that the second order derivatives of the solution u belong to functional spaces Dω^{D_{\widehat{\omega}}}, characterized by a modulus of continuityω^⁢(r){\widehat{\omega}(r)} expressed in terms of ω¯⁢(r){\overline{\omega}(r)}. Results are optimal. In some cases, as for Hölder spaces, Dω^=Dω¯{D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity occurs. In particular, full regularity occurs for the new class of functional spaces Cα0,λ⁢(Ω¯){C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular case, the classical Hölder spaces C0,λ⁢(Ω¯)=C00,λ⁢(Ω¯){C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}. Few words, concerning the possibility of generalizations and applications to non-linear problems, are expended at the end of the introduction and also in the last section.

Keywords

• linear elliptic boundary value problems
• data spaces of uniformly continuous functions
• uniform continuity properties of higher order derivatives
• intermediate and full regularity
• new functional spaces
• 35a09
• 35b65
• 35j25