We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $\Omega$ subject to homogeneous Dirichlet boundary conditions. prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying condition $\lvert 1 / p - 2 \rvert < 4 + \varepsilon$ some $\varepsilon > 0$. further show that operator admits property of maximal regularity and its $\mathrm{H}^{\infty}$-calculus is bo...

We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers.

We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel e −(u(θ)+u(θ)) cosh θ−θ ′ 2 .

The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Several conditions for the separability and Fredholmness in Banach-valued Lp spaces are given. Sharp estimates for resolvent, discreetness of spectrum, and completeness of root elements of the corresponding differential operators are obtained. In the last section, some ...

In this paper, we study the problem of controllability of impulsive neutral evolution integrodifferential equations with state dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii’s fixed point theorem, theory of resolvent operators, and an abstract phase space. An example is given to illustrate the theory.

Structure– phenomenological hereditary model for prediction of viscoelastic properties of layered polymer composites based on hereditary mechanics and theory of laminated plates relationships was elaborated. Interrelated creep and relaxation constitutive equations of layered composite was derived on the basis of algebra of resolvent operators and matrix algorithms. On example of cross-ply carbo...

We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the non...

Nonlinear vector integral equations are considered. Solution estimates and solvability conditions are derived. Applications to the periodic boundary value problem are also discussed. Under some restrictions our results improve the well-known ones. The main tool in the paper is the recent estimates for the resolvent of Hilbert-Schmidt operators.