Maximal Regularity of Second Order Delay Equations in Banach Spaces

Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈Rto have maximal regularity in HSlder continuous function spaces C^α （R, X）, where X is a Banach space, A is a closed operator in X, F, G ∈L（C（[-r, 0], X）, X） are delay operators for some fixed r 〉 0.

Analyticity of the Sine Family with L^P-Maximal Regularity for Second Order Cauchy Problems

If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on R^N, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. This is used to give a sufficient condition of the maximal regularity in the sense of Triebel spaces for vector-valued Cauchy problems with Dirichlet boundary conditions.

Existence theory for Rosseland equation and its homogenized equation

The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.

Embedding and Maximal Regular Differential Operators in Sobolev-Lions Spaces

This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.

Operator-valued Fourier Multipliers on Periodic Triebel Spaces

We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions.