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A fast approximation method to three dimensional equations in quasi-static uncoupled thermoelasticity is pro-posed. We approximate the density via Gaussian approximating functions introduced in the method approximate approximations. In this way the action of the integral operators on such functions is presented in a simple analytical form. If the density has separated representation, the problem is reduced to the computation of one-dimensional integrals which admit efficient cubature procedures. The comparison of the numerical and exact solution shows that these formulas are accurate and provide the predicted approximation rate 2, 4, 6 and 8., Funding Agencies|Universita degli Studi di Roma La Sapienza within the CRUI-CARE Agreement

Sobolev embeddings into Orlicz spaces on domains in the Eu-clidean space or, more generally, on Riemannian manifolds are considered. Highly irregular domains where the optimal degree of integrability of a func-tion may be lower than the one of its gradient are focused. A necessary and sufficient condition for the validity of the relevant embeddings is established in terms of the isocapacitary function of the domain. Compact embeddings are discussed as well. Sufficient conditions involving the isoperimetric function of the domain are derived as a by-product., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) Prin 2017 "Direct and inverse problems for partial differential equations: theoretical aspects and applications"; GNAMPA of the Italian INdAM -National Institute of High Mathematics [201758MTR2]

Various notions of dissipativity for partial differential operators and their applications are surveyed. We deal with functional dissipativity and its particular case L-p-dissipativity. Most of the results are due to the authors., Funding Agencies|RUDN University Strategic Academic Leadership Program

Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain., Funding Agencies|Italian Ministry of Education, University and Research (MIUR) [2017AYM8XW, 201758MTR2]; GNAMPA of the Italian INdAM -National Institute of High Mathematics

The paper discusses new cubature formulas for the Riesz potential and the fractional Laplacian (-Delta)(alpha/2), 0 < alpha < 2, in the framework of the method approximate approximations. This approach, combined with separated representations, makes the method successful also in high dimensions. We prove error estimates and report on numerical results illustrating that our formulas are accurate and provide the predicted convergence rate 2, 4, 6, 8 up to dimension 10(4)., Funding Agencies|RUDN University Strategic Academic Leadership Program

We discuss conditions for Lipschitz and C-1 regularity of solutions for a uniformly elliptic equation in divergence form. We focus on coefficients having the form that was introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz or C-1 regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous.

We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x = 0. In particular, we treat the case of solutions that are not Lipschitz continuous at x = 0. We show that our estimate is sharp., Funding Agency: RUDN University Strategic Academic Leadership Program

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second-order equations with real constant coefficients in the layer , where and . The homogeneous equation is considered with initial data in , . For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to , pamp;gt;n + 2 and . Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained., Funding Agencies|RUDN University Program 5-100

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 derived for domains containing lower-dimensional clusters of inclusions., Funding Agencies|RUDN University Program 5-100; EPSRC (UK)UK Research & Innovation (UKRI)Engineering & Physical Sciences Research Council (EPSRC) [EP/L024926/1]; EUEuropean Commission [MSCA-IF-2016-747334-CAT-FFLAP]

In the present paper we consider the Dirichlet problem for the second order differential operator E = del(A del), where A is a matrix with complex valued L-infinity entries. We introduce the concept of dissipativity of E with respect to a given function phi : R+ -> R+. Under the assumption that the ImA is symmetric, we prove that the condition vertical bar s phi (s)vertical bar vertical bar < ImA (x)xi, xi >vertical bar <= 2 root phi(s)[s phi(s)] < ReA (x)xi, xi > (for almost every x is an element of Omega subset of R-N and for any s > 0, xi is an element of R-N) is necessary and sufficient for the functional dissipativity of E. (c) 2020 Elsevier Ltd. All rights reserved., Funding Agencies|RUDN University Program [5-100]

We deal with m-component vector-valued solutions to the Cauchy problem for a linear both homogeneous and nonhomogeneous weakly coupled second-order parabolic system in the layer R-T(n+1) = R-n x (0, T). We assume that coefficients of the system are real and depending only on t, n >= 1 and T < infinity. The homogeneous system is considered with initial data in [L-p(R-n)](m), 1 <= p <= infinity. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to [L-p(R-T(n+1) )](m) boolean AND [C-alpha<((R-T(n+1)))over bar>](m), alpha is an element of (0, 1). Explicit formulas for the sharp coefficients in pointwise estimates for solutions to these problems and their directional derivative are obtained., Funding Agencies|RUDN University Program 5-100

We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order O(h(2M)), M = 1, 2, 3, 4, are provided., Funding Agencies|Universita degli Studi di Roma La Sapienza within the CRUICARE Agreement

Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space L-P. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.

Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in Rn. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a pointwise estimate for the functions in question via a Riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace. (C) 2019 Elsevier Ltd. All rights reserved., Funding Agencies|Research Project of the Italian Ministry of University and Research (MIUR) Prin 2015 "Partial differential equations and related analytic-geometric inequalities" [2015HY8JCC]; GNAMPA of the Italian INdAM -National Institute of High Mathematics; RUDN University Program 5-100, Russia

Second-order estimates are established for solutions to the p-Laplace system with right-hand side in L-2. The nonlinear expression of the gradient under the divergence operator is shown to belong to W-1,W-2, and hence to enjoy the best possible degree of regularity. Moreover, its norm in 1471,2 is proved to be equivalent to the norm of the right-hand side in L-2. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well. (C) 2019 Elsevier Masson SAS. All rights reserved., Funding Agencies|Italian Ministry of University and Research (MIUR)Ministero dell Istruzione, dell Universita e della Ricerca (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM -National Institute of High Mathematics; RUDN University Program [5-100]

For the general second order linear differential operator with complex-valued distributional coefficients a(j,k), b(j), and c in an open set (n) (n 1), we present conditions which ensure that -L0 is accretive, i.e., Re -L0 phi,phi 0 for all phi C-0(infinity) (Omega).

Necessary and sufficient conditions are found for the unique solvability of the Neumann problem for the p-Laplace operator. They characterize both the domain and measures on the right-hand sides.

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrodinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200., Funding Agencies|Ministry of Education and Science of the Russian Federation [02, a03.21.0008]

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on f(n-1) is obtained under the assumption that f belongs to L-p. It is assumed that the kernel of the integral depends on the parameters alpha and beta. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of alpha, beta in the case p = infinity. Conditions ensuring the validity of some analogues of the Khavinsons conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space., Funding Agencies|Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

Best possible second-order regularity is established for solutions to p-Laplacian type equations with and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L (2)-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all., Funding Agencies|Italian Ministry of University and Research (MIUR) [2012TC7588]; GNAMPA of the Italian INdAM - National Institute of High Mathematics; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

We find necessary and sufficient conditions for the L-p-dissipatiyity of the Dirichlet problem for systems of partial differential operators of the first order with complex locally integrable coefficients. As a by product we obtain sufficient conditions for a certain class of systems of the second order., Funding Agencies|Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems, no a priori restrictions on the coefficient matrices are imposed.

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function f on R-n is obtained under the assumption that f belongs to L-p. The explicit value of the coefficient is found for the cases p = 1 and p = 2.

This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations.

In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Mazya (Invent Math 175(2):287-334, 2009). As a preliminary stage of this work, in Mayboroda and Mazya (Invent Math 196(1):168, 2014) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order., Funding Agencies|Alfred P. Sloan Fellowship; NSF [DMS 1344235, DMS 1220089]; UMN MRSEC Seed grant [DMR 0212302]; Simons Fellowship in Mathematical Sciences; Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

A short survey of a series of results by the author, partly obtained in collaboration with Yu. Burago.

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplaces operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterized by a small parameter which is much larger when compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.

Boundary value problems for a class of quasilinear elliptic equations, with an Orlicz type growth and L-1 right-hand side are considered. Both Dirichlet and Neumann problems are contemplated. Existence and uniqueness of generalized solutions, as well as their regularity, are established. The case of measure right-hand sides is also analyzed. (C) 2017 Published by Elsevier Ltd., Funding Agencies|Research Project of the Italian Ministry of University and Research (MIUR) Prin [2015HY8JCC]; GNAMPA of the Italian INdAM (National Institute of High Mathematics); Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

The existence of solutions to a class of quasilinear elliptic problems on noncompact Riemannian manifolds, with finite volume, is investigated. Boundary value problems, with homogeneous Neumann conditions, in possibly irregular Euclidean domains are included as a special instance. A nontrivial solution is shown to exist under an unconventional growth condition on the right-hand side, which depends on the geometry of the underlying manifold. The identification of the critical growth is a crucial step in our analysis, and entails the use of the isocapacitary function of the manifold. A condition involving its isoperimetric function is also provided. (C) 2017 Elsevier Inc. All rights reserved., Funding Agencies|MIUR (Italian Ministry of Education, University and Research) [2012TC7588]; GNAMPA of the Italian INdAM (National Institute of High Mathematics); Ministry of Education and Science of the Russian Federation [02.a03.21.0008]

For highly perforated domains the paper addresses a novel approach to study mixed boundary value problems for the equations of linear elasticity in the framework of mesoscale approximations. There are no assumptions of periodicity involved in the description of the geometry of the domain. The size of the perforations is small compared to the minimal separation between neigh-boring defects and here we discuss a class of problems in perforated domains, which are not covered by the homogenization approximations. The mesoscale approximations presented here are uniform. Explicit asymptotic formulas are supplied with the remainder estimates. Numerical illustrations, demonstrating the efficiency of the asymptotic approach developed here, are also given.

A theory of Sobolev inequalities in arbitrary open sets in R-n is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities exhibit the same critical exponents as in the classical framework. Moreover, they involve constants independent of the geometry of the domain, and hence yield genuinely new results even in the case when just smooth domains are considered. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. (C) 2016 Elsevier Inc. All rights reserved., Funding Agencies|Research Project of Italian Ministry of University and Research (MIUR) Prin [2012TC7588]; GNAMPA of the Italian INdAM (National Institute of High Mathematics)

We propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions. We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions. (C) 2015 Elsevier Inc. All rights reserved.

A Brezis–Gallouet–Wainger logarithmic interpolation-embedding inequality is proved for various classes of irregular domains, in particular, for power cusps and λ-John domains.

The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.

This is a survey of results mostly relating elliptic equations and systems of arbitrary even order with rough coefficients in Lipschitz graph domains. Asymptotic properties of solutions at a point of a Lipschitz boundary are also discussed.

The authors consider boundary value problems for the Navier-Stokes system in a polyhedral domain, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are arbitrarily combined on the faces of the polyhedron. They prove existence and regularity theorems for weak solutions in weighted (and nonweighted) L (p) Sobolev and Holder spaces with sharp integrability and smoothness parameters.

We construct a function u is an element of W-loc(1.1) (B(0, 1)) which is a solution to div(A del u) = 0 in the sense of distributions, where A is continuous and u is not an element of W-loc(1.p) (B(0, 1)) for p greater than 1. We also give a function u is an element of W-loc(1.1)(B(0, 1)) such that u is an element of W-loc(1.p) (B(0, 1)) for every p less than infinity, u satisfies div(A del u) = 0 with A continuous but u is not an element of W-loc(1.infinity) (B(0, 1)). This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335-338).