A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with \(VMO\) coefficients. (English) Zbl 1107.35030
In recent years, there has been considerable interest in extending the well-known Calderon-Zygmund estimates for the Laplacian to more general equations, in particular equations with highest order coefficients lying in the Sarason space VMO. In addition, the analogous estimates with Morrey spaces replacing Lebesgue spaces have been considered. These Morrey space estimates have been proved by refining the proofs for the \(L^p\) estimates. The author shows that the Morrey space estimates can be derived from the \(L^p\) estimates by the Campanato technique. In fact the proof the discussed estimates is almost identical to the proof used by Campanato to derive Hölder estimates (by first estimating suitable \(L^{p,\mu}\) norms). This approach could be applied also to the study of any equation with non-VMO coefficients which have \(L^p\) estimates.
Reviewer: Lubomira Softova (Bari)
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35J15 | Second-order elliptic equations |
| 35K10 | Second-order parabolic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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