\(L^p (I,C^{\alpha} (\Omega))\) regularity for diffusion equations with non-smooth data. (English) Zbl 1514.35075
Summary: We prove an \(L^p (I,C^{\alpha} (\Omega))\) regularity result for a diffusion equation with mixed boundary conditions, \(L^{\infty}\) coefficients and an \(L^q\) initial condition. We provide explicit control of the \(L^p (I,C^{\alpha} (\Omega))\) norm with respect to the data. To prove our result, we first establish \(C^{\alpha} (\Omega)\) control of the stationary equation, extending a result by R. Haller-Dintelmann et al. [Appl. Math. Optim. 60, No. 3, 397–428 (2009; Zbl 1179.49041)].
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
Citations:
Zbl 1179.49041References:
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