\({\mathcal R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. (English) Zbl 1274.35002
Mem. Am. Math. Soc. 788, 114 p. (2003).
Summary: 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 that the top order coefficients are bounded and uniformly continuous.
MSC:
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 35K25 | Higher-order parabolic equations |
| 47A60 | Functional calculus for linear operators |
| 47B38 | Linear operators on function spaces (general) |
