R Boundedness Fourier Multipliers And Problems Of Elliptic And Parabolic Type

$\mathcal {R}$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type

The property of maximal $L_p$-regularity for parabolic evolution equations is investigated via the concept of $\mathcal R$-sectorial operators and operator-valued Fourier multipliers. As application, we consider the $L_q$-realization of an elliptic boundary value problem of order $2m$ with operator-valued coefficients subject to general boundary conditions. We show that there is maximal $L_p$-$L_q$-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.

R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type

Introduction Notations and conventions $\mathcal R$-Boundedness and Sectorial Operators: Sectorial operators The classes ${\mathcal{BIP}}(X)$ and $\mathcal H^\infty(X)$ $\mathcal R$-bounded families of operators $\mathcal R$-sectorial operators and maximal $L_p$-regularity Elliptic and Parabolic Boundary Value Problems: Elliptic differential operators in $L_p(\mathbb{R}^n;E)$ Elliptic problems in a half space: General Banach spaces Elliptic problems in a half space: Banach spaces of class $\mathcal{HT}$ Elliptic and parabolic problems in domains Notes References.

R-boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type