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BETA
G. Maz'ya, Vladimir
Alternative names
Publications (10 of 167) Show all publications
Mazya, V., Movchan, A. B. & Nieves, M. J. (2016). MESOSCALE MODELS AND APPROXIMATE SOLUTIONS FOR SOLIDS CONTAINING CLOUDS OF VOIDS. Multiscale Modeling & simulation, 14(1), 138-172
Open this publication in new window or tab >>MESOSCALE MODELS AND APPROXIMATE SOLUTIONS FOR SOLIDS CONTAINING CLOUDS OF VOIDS
2016 (English)In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 14, no 1, p. 138-172Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
SIAM PUBLICATIONS, 2016
Keyword
mesoscale approximations; singularly perturbed problems; elasticity; multiply perforated domains; asymptotic analysis
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-127592 (URN)10.1137/151006068 (DOI)000373366500006 ()
Available from: 2016-05-03 Created: 2016-05-03 Last updated: 2017-11-30
Cianchi, A. & Mazya, V. (2016). Sobolev inequalities in arbitrary domains. Advances in Mathematics, 293, 644-696
Open this publication in new window or tab >>Sobolev inequalities in arbitrary domains
2016 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 293, p. 644-696Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2016
Keyword
Sobolev inequalities; Irregular domains; Boundary traces; Optimal norms; Representation formulas
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-127410 (URN)10.1016/j.aim.2016.02.012 (DOI)000373093200014 ()
Note

Funding Agencies|Research Project of Italian Ministry of University and Research (MIUR) Prin [2012TC7588]; GNAMPA of the Italian INdAM (National Institute of High Mathematics)

Available from: 2016-05-02 Created: 2016-04-26 Last updated: 2017-11-30
Cianchi, A. & Maz´ya, V. (2015). GLOBAL GRADIENT ESTIMATES IN ELLIPTIC PROBLEMS UNDER MINIMAL DATA AND DOMAIN REGULARITY. Communications on Pure and Applied Analysis, 14(1), 285-311
Open this publication in new window or tab >>GLOBAL GRADIENT ESTIMATES IN ELLIPTIC PROBLEMS UNDER MINIMAL DATA AND DOMAIN REGULARITY
2015 (English)In: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 14, no 1, p. 285-311Article in journal (Refereed) Published
Abstract [en]

This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued.

Place, publisher, year, edition, pages
American Institute of Mathematical Sciences (AIMS), 2015
Keyword
Elliptic boundary value problems; gradient estimates; capacity; perimeter; rearrangements
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-114244 (URN)10.3934/cpaa.2015.14.285 (DOI)000347920400019 ()
Note

Funding Agencies|MIUR (Ministero dellIstruzione e dellUniversita); GNAMPA (Gruppo Nazionale per l Analisi Matematica, la Probabilita e le loro Applicazioni) of the Italian INdAM (Istituto Nazionale di Alta Matematica)

Available from: 2015-02-16 Created: 2015-02-16 Last updated: 2017-12-04
Lanzara, F., Maz´ya, V. & Schmidt, G. (2014). Fast cubature of volume potentials over rectangular domains by approximate approximations. Applied and Computational Harmonic Analysis, 36(1), 167-182
Open this publication in new window or tab >>Fast cubature of volume potentials over rectangular domains by approximate approximations
2014 (English)In: Applied and Computational Harmonic Analysis, ISSN 1063-5203, E-ISSN 1096-603X, Vol. 36, no 1, p. 167-182Article in journal (Refereed) Published
Abstract [en]

In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order O(h(6)) up to dimension 10(8).

Place, publisher, year, edition, pages
Elsevier, 2014
Keyword
Multi-dimensional convolution, Advection-diffusion potential, Separated representation, Higher dimensions
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-102767 (URN)10.1016/j.acha.2013.06.003 (DOI)000327920000009 ()
Available from: 2014-01-07 Created: 2013-12-26 Last updated: 2017-12-06
Cianchi, A. & Maz´ya, V. (2014). Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems. Archive for Rational Mechanics and Analysis, 212(1), 129-177
Open this publication in new window or tab >>Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems
2014 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 212, no 1, p. 129-177Article in journal (Refereed) Published
Abstract [en]

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2014
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-105404 (URN)10.1007/s00205-013-0705-x (DOI)000330986200003 ()
Available from: 2014-03-21 Created: 2014-03-21 Last updated: 2017-12-05
Cianchi, A. & Maz´ya, V. (2014). Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems. Journal of the European Mathematical Society (Print), 16(3), 571-595
Open this publication in new window or tab >>Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems
2014 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 16, no 3, p. 571-595Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
European Mathematical Society, 2014
Keyword
Nonlinear elliptic equations; Dirichlet problems; Neumann problems; gradient estimates; rearrangements; Lorentz spaces; Orlicz spaces
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-105429 (URN)10.4171/JEMS/440 (DOI)000331328800004 ()
Available from: 2014-03-21 Created: 2014-03-21 Last updated: 2017-12-05
Mayboroda, S. & Maz´ya, V. (2014). Regularity of solutions to the polyharmonic equation in general domains. Inventiones Mathematicae, 196(1), 0464
Open this publication in new window or tab >>Regularity of solutions to the polyharmonic equation in general domains
2014 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 196, no 1, p. 0464-Article in journal (Refereed) Published
Abstract [en]

The present paper establishes boundedness of derivatives for the solutions to the polyharmonic equation of order 2m in arbitrary bounded open sets of , 2a parts per thousand currency signna parts per thousand currency sign2m+1, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2014
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-106126 (URN)10.1007/s00222-013-0464-1 (DOI)000333160600001 ()
Available from: 2014-04-25 Created: 2014-04-24 Last updated: 2017-12-05
Carbery, A., Maz'ya, V., Mitrea, M. & Rule, D. (2014). The integrability of negative powers of the solution of the Saint Venant problem. Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, XIII(2), 465-531
Open this publication in new window or tab >>The integrability of negative powers of the solution of the Saint Venant problem
2014 (English)In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. XIII, no 2, p. 465-531Article in journal (Refereed) Published
Abstract [en]

We initiate the study of the finiteness condition∫ Ω u(x) −β dx≤C(Ω,β)<+∞ whereΩ⊆R n is an open set and u is the solution of the Saint Venant problem Δu=−1 in Ω , u=0 on ∂Ω . The central issue which we address is that of determining the range of values of the parameter β>0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C(Ω,β) . Classes of domains for which our analysis applies include bounded piecewise C 1 domains in R n , n≥2 , with conical singularities (in particular polygonal domains in the plane), polyhedra in R 3 , and bounded domains which are locally of classC 2 and which have (finitely many) outwardly pointing cusps. For example, we show that if u N is the solution of the Saint Venant problem in the regular polygon Ω N with N sides circumscribed by the unit disc in the plane, then for each β∈(0,1) the following asymptotic formula holds: % {eqnarray*} \int_{\Omega_N}u_N(x)^{-\beta}\,dx=\frac{4^\beta\pi}{1-\beta} +{\mathcal{O}}(N^{\beta-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v(0)=0 , ∇v(0)=0 and Δv≥c>0 .

Place, publisher, year, edition, pages
Scuola Normale Superiore, 2014
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-108526 (URN)000339985500008 ()2-s2.0-84908458320 (Scopus ID)
Available from: 2014-06-29 Created: 2014-06-29 Last updated: 2017-12-05Bibliographically approved
Cianchi, A. & Maz´ya, V. (2013). Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds. American Journal of Mathematics, 135(3), 579-635
Open this publication in new window or tab >>Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds
2013 (English)In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 135, no 3, p. 579-635Article in journal (Refereed) Published
Abstract [en]

We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds M of finite volume. Sharp conditions ensuring L-q(M) and L-infinity(M) bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of M.

Place, publisher, year, edition, pages
Johns Hopkins University Press, 2013
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-96118 (URN)10.1353/ajm.2013.0028 (DOI)000320010700001 ()
Available from: 2013-08-14 Created: 2013-08-14 Last updated: 2017-12-06Bibliographically approved
Kamotski, I. & Maz´ya, V. (2013). Estimate for a solution to the water wave problem in the presence of a submerged body. Russian journal of mathematical physics, 20(4), 453-467
Open this publication in new window or tab >>Estimate for a solution to the water wave problem in the presence of a submerged body
2013 (English)In: Russian journal of mathematical physics, ISSN 1061-9208, E-ISSN 1555-6638, Vol. 20, no 4, p. 453-467Article in journal (Refereed) Published
Abstract [en]

We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a submerged body. Under some geometrical requirements, we derive an explicit bound for the solution depending on the domain and the functions on the right-hand side.

Place, publisher, year, edition, pages
MAIK Nauka/Interperiodica, 2013
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-103310 (URN)10.1134/S1061920813040079 (DOI)000328853300007 ()
Available from: 2014-01-16 Created: 2014-01-16 Last updated: 2017-12-06
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