Moduli of continuity, functional spaces,\break andelliptic boundary value problems. The full regularity spacesC α 0,λ(Ω̅)

Media type: E-Article

Title: Moduli of continuity, functional spaces,\break andelliptic boundary value problems. The full regularity spacesC α 0,λ(Ω̅)

Contributor: Beirão da Veiga, Hugo

Published: Walter de Gruyter GmbH, 2018

Language: English

DOI: 10.1515/anona-2016-0041

ISSN: 2191-950X; 2191-9496

Keywords: Analysis

Origination:

Footnote:

Description: Abstract Let 𝑳 {\boldsymbol{L}} be a second order uniformly elliptic operator, andconsider the equation 𝑳 ⁢ u = f {\boldsymbol{L}u=f} under the boundarycondition u = 0 {u=0} . We assume data f in genericalsubspaces of continuous functions D ω ¯ {D_{\overline{\omega}}} characterized by agiven modulus of continuity ω ¯ ⁢ ( r ) {\overline{\omega}(r)} , and show that thesecond order derivatives of the solution u belong tofunctional spaces D ω ^ {D_{\widehat{\omega}}} , characterized by a modulus ofcontinuity ω ^ ⁢ ( r ) {\widehat{\omega}(r)} expressed in terms of ω ¯ ⁢ ( r ) {\overline{\omega}(r)} .Results are optimal. In some cases, as for Hölder spaces, D ω ^ = D ω ¯ {D_{\widehat{\omega}}=D_{\overline{\omega}}} . In this case we say that full regularityoccurs. In particular, full regularity occurs for the new class offunctional spaces C α 0 , λ ⁢ ( Ω ¯ ) {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as aparticular case, the classical Hölder spaces C 0 , λ ⁢ ( Ω ¯ ) = C 0 0 , λ ⁢ ( Ω ¯ ) {C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})} .Few words, concerning the possibility of generalizations andapplications to non-linear problems, are expended at the end of theintroduction and also in the last section.

Access State: Open Access

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