Electron. J. Diff. Equ., Vol. 2016 (2016), No. 204, pp. 1-8.

Poincare inequality and Campanato estimates for weak solutions of parabolic equations

Junichi Aramaki

We shall show that the Poincare type inequality holds for the weak solution of a parabolic equation. The key is to control the $L^p$ norm of the first derivative of the weak solution with respect to the time variable. The inequality is necessary to get an estimate in the Campanato space $\mathcal{L}^{p,\mu }$ for general parabolic equations.

Submitted December 24, 2015. Published July 28, 2016.
Math Subject Classifications: 35A09, 35K10, 35D35.
Key Words: Poincare type inequality; weak solution; parabolic equation.

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Junichi Aramaki
Division of Science
Faculty of Science and Engineering
Tokyo Denki University
Hatoyama-machi, Saitama 350-0394, Japan
email: aramaki@mail.dendai.ac.jp

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