Maximal regularity for the heat equation with various boundary conditions in an infinite layer

Abstract

This paper treats resolvent $L_q$ estimate and maximal $L_p-L_q$ regularity for the heat equation with various boundary conditions in an infinite layer. We need to consider two boundary conditions on upper boundary and lower boundary. We are able to choose any pair of Dirichlet, Neumann and Robin boundary conditions. We construct the solutions of Fourier multiplier operators and we use a theorem for an integral operator, which derives $L_q$-boundedness and $L_p-L_q$ boundedness. The key is that the holomorphic symbols can be properly estimated from above.