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On «power-logarithmic» solutions of the Dirichlet problem for elliptic systems in K d × R n - d , where K d is a d-dimensional cone

Vladimir A. Kozlov;  Vladimir G. Maz'ya — 1996

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A description of all «power-logarithmic» solutions to the homogeneous Dirichlet problem for strongly elliptic systems in a n -dimensional cone K = K d × R n - d is given, where K d is an arbitrary open cone in R d and n > d > 1 .

Gradient regularity via rearrangements for p -Laplacian type elliptic boundary value problems

Andrea Cianchi;  Vladimir G. Maz'ya — 2014

Journal of the European Mathematical Society

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.

On pointwise interpolation inequalities for derivatives

Vladimir G. Maz'ya;  Tatjana Olegovna Shaposhnikova — 1999

Mathematica Bohemica

Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where ∇ k is the gradient of order k , ℳ is the Hardy-Littlewood maximal operator, and I z is the Riesz potential of order z , are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space M ( W p m ( ℝ n ) → W p l ( ℝ n ) ) is described.

Water-wave problem for a vertical shell

Nikolai G. Kuznecov;  Vladimir G. Maz'ya — 2001

Mathematica Bohemica

The uniqueness theorem is proved for the linearized problem describing radiation and scattering of time-harmonic water waves by a vertical shell having an arbitrary horizontal cross-section. The uniqueness holds for all frequencies, and various locations of the shell are possible: surface-piercing, totally immersed and bottom-standing. A version of integral equation technique is outlined for finding a solution.

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