Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)
| Autor: | Hugo Beirão da Veiga |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
35b65
QA299.6-433 Topological tensor product 010102 general mathematics Mathematical analysis linear elliptic boundary value problems 35a09 new functional spaces 01 natural sciences Moduli Moduli of algebraic curves uniform continuity properties of higher order derivatives 35j25 intermediate and full regularity 0103 physical sciences 010307 mathematical physics Boundary value problem 0101 mathematics data spaces of uniformly continuous functions Analysis Mathematics |
| Zdroj: | Advances in Nonlinear Analysis, Vol 7, Iss 1, Pp 15-34 (2018) |
| ISSN: | 2191-9496 |
| Popis: | 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 D ω ¯ {D_{\overline{\omega}}} characterized by a given modulus of continuity ω ¯ ( r ) {\overline{\omega}(r)} , and show that the second order derivatives of the solution u belong to functional spaces D ω ^ {D_{\widehat{\omega}}} , characterized by a modulus of continuity 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 regularity occurs. In particular, full regularity occurs for the new class of functional spaces C α 0 , λ ( Ω ¯ ) {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular 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 and applications to non-linear problems, are expended at the end of the introduction and also in the last section. |
| Databáze: | OpenAIRE |
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