Abstract:The paper contains $L_p$-estimates and a theorem on the local in time solvability of the problem arising as a result of linearization of the free boundary problem for two viscous fluids, compressible and incompressible, contained in a bounded vessel, separated by a free interface, and subject to mass and capillary forces. This result is known for the case of $p=2$; it serves as an analytical

Abstract:The objective of the note is to prove a regularity result for weak solutions to the Navier–Stokes equations that are locally in $L_\infty (L^{3,\infty })$. It reads that, in a sense, the number of singular points at each time is at most finite. This note is inspired by a recent paper of H. J. Choe, J. Wolf, M. Yang.

Abstract:Formal asymptotic algorithms are considered for a class of meso-scale approximations for problems of vibration of elastic membranes that contain clusters of small inertial inclusions distributed along contours of predefined smooth shapes. Effective transmission conditions have been identified for inertial structured interfaces, and approximations to solutions of eigenvalue problems have been

Abstract:A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system

Abstract:A class of weighted Hardy inequalities is treated. The sharp constants depend on the lowest eigenvalues of auxiliary Schrödinger operators on a sphere. In particular, for some block radial weights these sharp constants are given in terms of the lowest eigenvalue of a Legendre type equation.

Abstract:Our goal in this work is to explain an unexpected feature of the expanding level sets of the solutions of a system where a half-plane in which reaction-diffusion phenomena occur exchanges mass with a line having a large diffusion of its own. The system was proposed by H. Berestycki, L. Rossi, and the second author as a model of enhancement of biological invasions by a line of fast diffusion

Abstract:An existence theory is developed for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoration of missing data in an image as the steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. The existence of a weak solution is established by

Abstract:This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $(\partial _t - \Delta _x)^s$ for $s \in (0,1)$. The regularity estimates are completely local in nature. This aspect is of

Abstract:The existence of generalized Lagrange multipliers is proved for a class of evolution problems for linear differential operators of various types subject to constraints on the derivatives. Those Lagrange multipliers and the respective solutions are stable for the vanishing of the coercive parameter and are naturally associated with evolution variational inequalities with time-dependent convex

Abstract:Let $u$ be a solution to the normalized $p$-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $20\}} \ \text { in } \ B_1(0) \end{equation*} where $u(x)\ge 0$ and $\chi _A$ is the characteristic function of the set $A$. The main result is that for almost every free boundary point with respect to the $(n-1)$-Hausdorff measure, there is a neighborhood where the free boundary

Abstract:This is a brief historical overview of the Sobolev mixed norm theory of linear elliptic and parabolic equations and the recent development in this theory based on the Rubio de Francia extrapolation theorem. A self-contained proof of this theorem along with other relevant tools of Real Analysis are also presented as well as an application to mixed norm estimates for fully nonlinear equations

Abstract:The dyadic model , , is considered. It is shown that in the case of nontrivial right-hand side the system may have two different Leray-Hopf solutions.

Abstract:Let be the classical Lebesgue spaces of -periodic functions and the best approximation of  by the space  in . For , , the symbol stands for the space of functions  of the form In this paper, all spaces  are described that provide a sharp constant in several inequalities for approximation of classes of convolutions with a kernel . In particular, necessary and sufficient conditions are obtained

Abstract:Two Kirchhoff plates, which are described by Neumann problems for biharmonic equations, overlap along a thin strip. In the interior of the strip, the plates are connected by rivets, which are modeled by the Sobolev point transmission conditions. By taking the boundary layer phenomenon into account, homogenization with respect to a small parameter (the relative period of the distribution of

Abstract:In this paper, the topological classification of nonseparating (respectively, separating) real algebraic nonsingular -curves of bidegree on the quadric ellipsoid is completed. In particular, it is shown that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. The strategy is to reduce

Abstract:We construct a sequence such that the trigonometric series converges to zero everywhere on a subsequence . We show, for any such series, that the must be very sparse, and that the support of the related distribution must be quite large.

Abstract:Let , and let . The paper is devoted to approximation of classes of functions  for every representable as where is an entire function of type not exceeding  , , and . The approximating space  consists of functions of the form Under some conditions on and , linear operators with values in  are constructed for which . For the constant (it is an analog of the well-known Favard constant) cannot

Abstract:The method of phase-amplitude transformations is used for extraction of harmonics of a given order from trigonometric polynomials Such transformations take polynomials to similar polynomials by using two simplest operations: multiplication by a real constant and shift by a real phase , i.e., . The harmonic is extracted by addition of similar polynomials: where the and are defined by explicit

Abstract:Some observations about original Welschinger invariants defined in the paper Invariants of real symplectic -manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195-234, are collected. None of their proofs is difficult, nevertheless these remarks do not seem to have been made before. The main result is that when is a real rational algebraic surface, Welschinger

Abstract:The 0th cohomology group is computed for a complex of groups of cobordism-framed correspondences. In the case of ordinary framed correspondences, an analogous computation was completed by A. Neshitov in his paper ``Framed correspondences and the Milnor-Witt -theory''. Neshitov's result is, at the same time, a computation of the homotopy groups , and the present work might be used subsequently

Abstract:Stability is proved for the rest state in the problem of evolution of two viscous fluids, compressible and incompressible, contained in a bounded vessel and separated by a free interface. The fluids are subject to mass and capillary forces. The proof of stability is based on ``maximal regularity'' estimates for the solution in the anisotropic Sobolev-Slobodetskiĭ spaces with an exponential

Abstract:A polynomial-time algorithm is constructed that, given a graph , finds the full set of nonequivalent Cayley representations of over the group , where and . This result implies that the recognition and isomorphism problems for Cayley graphs over can be solved in polynomial time.

Abstract:It is shown that the Orlicz-Lorentz spaces , , with Orlicz function and weight sequence are uniformly isomorphic to subspaces of if the norm satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces . The approach is based on combinatorial averaging techniques, and a new result of independent interest is proved, which relates suitable averages with Orlicz-Lorentz

Abstract:The classical balayages of measures and subharmonic functions are extended to a system of rays with common origin on the complex plane . For an arbitrary subharmonic function of finite order on , this allows one to build a -subharmonic function on that is harmonic outside of , coincides with on outside of a polar set, and has the same growth order as . Applications are given to the investigation

Abstract:The Maxwell operator is studied in a three-dimensional cylinder whose cross-section is a simply connected bounded domain with Lipschitz boundary. It is assumed that the coefficients of the operator are scalar functions depending on the longitudinal variable only. We show that the square of such an operator is unitarily equivalent to the orthogonal sum of four scalar elliptic operators of second

Abstract:A posteriori estimates are proved for the accuracy of approximations of solutions to variational problems with nonstandard power functionals. More precisely, these are integral functionals with power type integrands having a variable exponent . It is assumed that is bounded away from one and infinity. Estimates in the energy norm are obtained for the difference of the approximate and exact

Abstract:The paper is devoted to classification of irreducible real linear representations of noncommutative compact connected Lie groups whose Morse matrix coefficients have the minimal number of critical points permitted by the topology of .

Abstract:The periodic Schrödinger operator on a discrete periodic graph is treated. The discrete spectrum is estimated for the perturbed operator , , where is a decaying potential. In the case when the potential has a power asymptotics at infinity, an asymptotics is obtained for the discrete spectrum of the operator for a large coupling constant.

Abstract:On polyhedral cones, a new family of valuations is introduced; they are valued in the space of bounded polyhedra.

Abstract:All (finite connected) spatial graphs are supplied with an additional structure -- the replenished skeleton and its disk framing, -- in such a way that the problem of isotopic classification of spatial graphs endowed with this structure admits reduction to two problems: the (classical) problem of isotopic classification of tangles and the (close to classical) problem of isotopic classification

Abstract:In , consider a selfadjoint matrix elliptic second order differential operator , , with periodic coefficients depending on . The principal part of the operator is given in a factorized form, the operator involves first and zero order terms. Approximation is found for the operator exponential , , for small in the ( )-operator norm with a suitable . The results are applied to study the behavior

Abstract:In the present paper, sandwich classification is established for the overgroups of the subsystem subgroup of the Chevalley group for the three types of the pair (the root system and its subsystem) listed below such that the group is (up to a torus) a Levi subgroup of the parabolic subgroup with Abelian unipotent radical. Namely, it is shown that for any overgroup of this sort, there exists

Abstract:Quadratic forms on bimodules are defined and the sandwich classification theorem is proved for subgroups of the general linear group normalized by the elementary unitary group if is a regular bimodule with sufficiently large hyperbolic part.

Abstract:Realizations are studied for a particular block of 5-dimensional Lie algebras (within the well-known Mubarakzyanov classification) in the form of algebras of holomorphic vector fields on homogeneous real hypersurfaces of the 3-dimensional complex space. All (locally) holomorphically homogeneous and Levi nondegenerate real hypersurfaces associated with algebras in the block in question are

Abstract:The explicit upper Bellman function is found for the natural dyadic maximal operator acting from into . As a consequence, it is shown that the norm of the natural operator equals for all , and so does the norm of the classical dyadic maximal operator. The main result is a partial consequence of a theorem for the so-called -trees, which generalize dyadic lattices. The Bellman function in this

Abstract:The classical Grauert and Ramspott theorems constitute the foundation of the Oka principle on Stein spaces. In this paper, similar results are established on the maximal ideal space of the Banach algebra  of bounded holomorphic functions on the open unit disk . The results are illustrated by some examples and applications to the theory of operator-valued functions.

Abstract:A complete series of trace formulas for the one-dimensional Stark operator on the entire axis with a rapidly decreasing potential is constructed. The Stark equation is related to the cylindrical Korteweg-de Vries equation by the pair (-). For this equation, infinite series of integrals of motion is constructed that corresponds to the trace formulas for the Stark operator.

Abstract:In the author's earlier paper [Revyakov M., J. Multivariate Anal. 116 (2013) 25-34] concerning mathematical statistics, a need arose to employ functions called ``Schur-convex functions of the 2nd order with respect to two variables''. In the present paper, the class of Schur-convex functions of the 2nd order in variables is introduced. Necessary and sufficient conditions (in the form of analogs

Abstract:Cylindrical acoustic waveguides with constant cross-section  are considered, specifically, a straight waveguide and a locally curved waveguide that depends on a parameter . For , in two different settings ( and ), the task is to find an eigenvalue that is embedded in the continuous spectrum of the waveguide and, hence, is inherently unstable. In other words, a solution of the Neumann problem

Abstract:The SRA-free condition for metric spaces (that is, spaces without Small Rough Angles) was introduced by Zolotov to study rectifiability for self-contracted curves in various metric spaces. A Möbius invariant version of this notion is introduced, which allows one to show that the zz-distance associated with the respective Möbius structure on the circle is nondegenerate. This result is an important

Abstract:A special version of the corona theorem in several variables, valid when all but one of the data functions are smooth, is used to generalize, to the polydisk and to the ball, the results obtained by El Fallah, Kellay, and Seip about the cyclicity of nonvanishing bounded holomorphic functions in sufficiently large Banach spaces of analytic functions determined either by weighted sums of powers

Abstract:The rigidity theorem for homotopy invariant presheaves with Witt-transfers on the category of smooth schemes over a field of characteristic different form two is proved. Namely, for any such sheaf , isomorphism is established, where is an essentially smooth local Henselian scheme with a separable residue field over . As a consequence, the rigidity theorem for the presheaves for any smooth

Abstract:We consider a third-order nonselfadjoint differential operator with square summable coefficients whose domain is defined by quasiperiodic boundary conditions. For this operator, using the method of similar operators, we obtain an asymptotic behavior of eigenvalues, estimates of the deviations of spectral projections, and the equiconvergence of spectral decompositions.

Abstract:Let be a semilocal Dedekind domain with fraction field . It is shown that two hereditary -orders in central simple -algebras that become isomorphic after tensoring with and with some faithfully flat étale -algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement

Abstract:This paper is devoted to almost subnormal subgroups of the general linear group of degree over a division ring that satisfy a generalized power central group identity.

Abstract:This is a continuation of the study of subgroups of the Chevalley group over a ring with root system and weight lattice that contain the elementary subgroup over a subring of . A. Bak and A. V. Stepanov considered recently the case of the symplectic group (simply connected group with root system ) in characteristic 2. In the current article, that result is extended to the case of and for the

Abstract:In , consider a selfadjoint matrix second order elliptic differential operator , . The principal part of the operator is given in a factorized form, the operator contains first and zero order terms. The operator is positive definite, its coefficients are periodic and depend on . The behavior in the small period limit is studied for the operator exponential , . The approximation in the -operator

Abstract:The paper is devoted to a noncommutative holomorphic functional calculus and its application to noncommutative algebraic geometry. A description is given for the noncommutative (infinite-dimensional) affine spaces , , , and for the projective spaces within Kapranov's model of noncommutative algebraic geometry based on the sheaf of formally-radical holomorphic functions of elements of a nilpotent