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Vol 224, No 3 (2017)

Article

On Monotonicity of Some Functionals Under Monotone Rearrangement with Respect To One Variable

Bankevich S.V.

Abstract

The Pólya–Szegö inequality is considered for a monotone rearrangement with integrand depending on the rearrangement variable. The inequality is proved for integrands having polynomial growth.

Journal of Mathematical Sciences. 2017;224(3):385-390
pages 385-390 views

Local Boundary Regularity for the Navier–Stokes Equations in Non-Endpoint Borderline Lorentz Spaces

Barker T.

Abstract

Local regularity up to the flat part of the boundary is proved for certain classes of distributional solutions that are LL3,q with q finite. The corresponding result for the interior case was recently proved by Wang and Zhang, see also Phuc’s paper. For local regularity up to the flat part of the boundary, q = 3 was established by G. A. Seregin. Our result can be viewed as an extension of it to L3,q with q finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new ϵ-regularity criterion, are central themes in providing this extension.

Journal of Mathematical Sciences. 2017;224(3):391-413
pages 391-413 views

An Alternative Approach Towards the Higher Order Denoising of Images. Analytical Aspects

Bildhauer M., Fuchs M., Weickert J.

Abstract

Theoretical aspects of a variational model for the denoising of images which can be interpreted as a substitute for a higher order approach, are investigated. In this model, the smoothness term that usually involves the highest derivatives is replaced by a mixed expression for a second unknown function in which only derivatives of lower order occur. The main results concern the existence and uniqueness as well as the regularity properties of the solutions to this variational problem. They are established under various assumptions imposed on the growth rates of the different parts of the energy functional.

Journal of Mathematical Sciences. 2017;224(3):414-441
pages 414-441 views

Stabilization Technique Applied To Curve Shortening Flow in the Plane

Mikayelyan H.

Abstract

The method proposed by T. I. Zelenjak is applied to the mean curvature flow in the plane. A new type of monotonicity formula for star-shaped curves is obtained.

Journal of Mathematical Sciences. 2017;224(3):442-447
pages 442-447 views

The Multiplicity of Positive Solutions to A Quasilinear Equation Generated By The Il′in–Caffarelli–Cohn–Nirenberg Inequality

Nazarov A.I., Neterebskii B.O.

Abstract

The Euler–Lagrange equation for the functional related to the V. P. Il’in inequality also known as the Caffarelli–Kohn–Nirenberg inequality is considered. We prove that if the space dimension is even, then changing some parameters, one can obtain arbitrary many different positive solutions for this equation.

Journal of Mathematical Sciences. 2017;224(3):448-455
pages 448-455 views

On Variational Representations of the Constant in the Inf-Sup Condition for the Stokes Problem

Repin S.

Abstract

Variational representations of the constant cΩ in the inf-sup condition for the Stokes problem in a bounded Lipschitz domain in ℝd, d ≥ 2, are deduced. For any pair of admissible functions, the respective variational functional provides an upper bound of cΩ and the exact infimum of it is equal to cΩ. Minimization of the functionals over suitable finite dimensional subspaces generates monotonically decreasing sequences of numbers converging to cΩ and, therefore, they can be used for numerical evaluation of the constant.

Journal of Mathematical Sciences. 2017;224(3):456-467
pages 456-467 views

Remark on Wolf’s Condition for Boundary Regularity of the Navier–Stokes Equations

Seregin G.

Abstract

The Wolf regularity condition is proved up to the boundary for solutions to the Navier–Stokes equations satisfying nonslip boundary condition.

Journal of Mathematical Sciences. 2017;224(3):468-474
pages 468-474 views

Proof of Schauder Estimates for Parabolic Initial-Boundary Value Model Problems VIA O. A. Ladyzhenskaya’s Fourier Multipliers Theorem

Solonnikov V.A.

Abstract

The paper is concerned with estimates of the Hölder norms of solutions of model parabolic initialboundary value problems in a half-space. The proof is based on O. A. Ladyzhenskaya’s theorem on the Fourier multipliers in anisotropic Hölder spaces and on K. K. Golovkin’s theorem on equivalent norms in these spaces.

Journal of Mathematical Sciences. 2017;224(3):475-491
pages 475-491 views