Morrey-type regularity of solutions to parabolic problems with discontinuous data. (English) Zbl 1270.35153
Summary: We consider regular oblique derivative problem in cylinder \(Q_{T} = \Omega \times (0, T), \, {\Omega\subset \mathbb R^n}\) for uniformly parabolic operator \({\mathfrak P = D_t - \sum_{i,j=1}^n a^{ij}(x)D_{ij}}\) with VMO principal coefficients. Its unique strong solvability is proved in [the author, Manuscr. Math. 103, No. 2, 203–220 (2000; Zbl 0963.35032)], when \({\mathfrak P u \in L^p(Q_T)}, \, {p \in (1,\infty)}\). Our aim is to show that the solution belongs to the generalized Sobolev-Morrey space \({W^{2,1}_{p,\omega}(Q_T)}\), when \({\mathfrak P u\in L^{p,\omega} (Q_T)}\), \({p \in (1,\infty)}\), \({\omega(x,r): \, {\mathbb R}^{n+1}_+ \to {\mathbb R}_+}\). For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy-Dirichlet problem.
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35D35 | Strong solutions to PDEs |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
non-divergence type; regular oblique derivative problem; VMO principal coefficients; Dirichlet problemCitations:
Zbl 0963.35032References:
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