Estimates in the Generalized Morrey Spaces for Linear Parabolic Systems

Details Estimates in the Generalized Morrey Spaces for Linear Parabolic Systems Estimates in the Generalized Morrey Spaces for Linear Parabolic Systems

2008-11-20

arxiv.org/abs/0811.3360v2

Mathematics

Analysis of PDEs

Scientific paper

The purpose of this paper is to study the parabolic system $u_t^i-D_\alpha(a_{ij}^{\alpha\beta}D_\beta u^j)=-div f^i$ in the generalized Morrey Space $L_{\phi}^{2,\lambda}$ . We would like to understand the regularity of the solutions of this system. It will be shown that 1: if $a_{ij}^{\alpha\beta}\in C(\bar{Q_{T}})$ then $Du\in L_\phi^{2,\lambda}$, and 2: if $a_{ij}^{\alpha\beta}\in VMO(Q_{T})$ then $Du\in L_\phi^{2,\lambda}$. Moreover we will be able to obtain estimates on the gradient of the solutions to the system, which will tell us about the regularity of the solutions.

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