### Abstract

The property of maximal L_{p}-regularity for parabolic evolution equations is investigated via the concept of 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.

Original language | English |
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Journal | Memoirs of the American Mathematical Society |

Issue number | 788 |

Publication status | Published - 2003 Nov |

Externally published | Yes |

### Fingerprint

### Keywords

- Differential operators with operator-valued coefficients
- L-theory for boundary value problems of general type
- Lopatinskii-Shapiro condition
- Maximal L-regularity
- Operator-valued Fourier multipliers
- R-boundedness

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Memoirs of the American Mathematical Society*, (788).

**R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type.** / Denk, Robert; Hieber, Matthias Georg; Prüss, Jan.

Research output: Contribution to journal › Article

*Memoirs of the American Mathematical Society*, no. 788.

}

TY - JOUR

T1 - R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type

AU - Denk, Robert

AU - Hieber, Matthias Georg

AU - Prüss, Jan

PY - 2003/11

Y1 - 2003/11

N2 - The property of maximal Lp-regularity for parabolic evolution equations is investigated via the concept of 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 Lp-Lq-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.

AB - The property of maximal Lp-regularity for parabolic evolution equations is investigated via the concept of 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 Lp-Lq-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.

KW - Differential operators with operator-valued coefficients

KW - L-theory for boundary value problems of general type

KW - Lopatinskii-Shapiro condition

KW - Maximal L-regularity

KW - Operator-valued Fourier multipliers

KW - R-boundedness

UR - http://www.scopus.com/inward/record.url?scp=0242287337&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242287337&partnerID=8YFLogxK

M3 - Article

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 788

ER -