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  • 1.
    Alvarado, Ryan
    et al.
    University of Missouri.
    Brigham, Dan
    University of Missouri.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mitrea, Marius
    University of Missouri.
    Ziade, Elia
    University of Missouri.
    Sharp Geometric Maximum Principles for Semi-Elliptic Operators with Singular Drift2011In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 18, no 4, p. 613-620Article in journal (Refereed)
    Abstract [en]

    We discuss a sharp generalization of the Hopf-Oleinik boundary point principle (BPP) for domains satisfying an interior pseudo-ball condition, for non-divergence form, semi-elliptic operators with singular drift. In turn, this result is used to derive a version of the strong maximum principle under optimal pointwise blow-up conditions for the coefficients of the differential operator involved. We also explain how a uniform two-sided pseudo-ball condition may be used to provide a purely geometric characterization of Lyapunov domains, and clarify the role this class of domains plays vis-a-vis to the BPP.

  • 2.
    Alvino, Angelo
    et al.
    University Napoli Federico II.
    Cianchi, Andrea
    University Florence.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mercaldo, Anna
    University Napoli Federico II.
    Well-posed elliptic Neumann problems involving irregular data and domains2010In: ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, ISSN 0294-1449, Vol. 27, no 4, p. 1017-1054Article in journal (Refereed)
    Abstract [en]

    Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented. (C) 2010 Elsevier Masson SAS. All rights reserved.

  • 3.
    Björn, Jana
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form2000In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 12, no 1, p. 81-113Article in journal (Refereed)
    Abstract [en]

    We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.

  • 4.
    Carbery, Anthony
    et al.
    Edinburgh University, UK.
    Maz'ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Mitrea, Marius
    University of Missouri, USA.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The integrability of negative powers of the solution of the Saint Venant problem2014In: 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)
    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 .

  • 5.
    Cianchi, A.
    et al.
    Dipartimento di Matematica e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Neumann problems and isocapacitary inequalities2008In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 89, no 1, p. 71-105Article in journal (Refereed)
    Abstract [en]

    A priori bounds for solutions to (nonlinear) elliptic Neumann problems in open subsets O of Rn are established via inequalities relating the Lebesgue measure of subsets of O to their relative capacity. Both norm and capacitary estimates for solutions, and norm estimates for their gradients are derived which improve classical results even in the case of the Laplace equation. © 2007 Elsevier Masson SAS. All rights reserved.

  • 6.
    Cianchi, Andrea
    et al.
    University of Florence, Italy .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Boundedness of solutions to the Schrodinger equation under Neumann boundary conditions2012In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 98, no 6, p. 654-688Article in journal (Refereed)
    Abstract [en]

    We deal with Neumann problems for Schrodinger type equations, with non-necessarily bounded potentials, in possibly irregular domains in R-n. Sharp balance conditions between the regularity of the domain and the integrability of the potential for any solution to be bounded are established. The regularity of the domain is described either through its isoperimetric function or its isocapacitary function. The integrability of the sole negative part of the potential plays a role, and is prescribed via its distribution function. The relevant conditions amount to the membership of the negative part of the potential to a Lorentz type space defined either in terms of the isoperimetric function, or of the isocapacitary function of the domain. (c) 2012 Elsevier Masson SAS. All rights reserved.

  • 7.
    Cianchi, Andrea
    et al.
    University of Florence, Italy .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds2013In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 135, no 3, p. 579-635Article in journal (Refereed)
    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.

  • 8.
    Cianchi, Andrea
    et al.
    University of Firenze, Italy .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems2014In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 212, no 1, p. 129-177Article in journal (Refereed)
    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.

  • 9.
    Cianchi, Andrea
    et al.
    University of Florence, Italy.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. University of Liverpool, England.
    GLOBAL GRADIENT ESTIMATES IN ELLIPTIC PROBLEMS UNDER MINIMAL DATA AND DOMAIN REGULARITY2015In: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 14, no 1, p. 285-311Article in journal (Refereed)
    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.

  • 10.
    Cianchi, Andrea
    et al.
    University of Florence, Italy .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems2014In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 16, no 3, p. 571-595Article in journal (Refereed)
    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.

  • 11.
    Cianchi, Andrea
    et al.
    University of Florence.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds2011In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 87, no 3, p. 469-491Article in journal (Refereed)
    Abstract [en]

    Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold M are established in terms of the isocapacitary function of M. The relevant capacity takes a different form according to whether M has finite or infinite volume. Conditions involving the more standard isoperimetric function of M can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.

  • 12.
    Cianchi, Andrea
    et al.
    University of Florence, Italy.
    Mazya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. University of Liverpool, England.
    Sobolev inequalities in arbitrary domains2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 293, p. 644-696Article in journal (Refereed)
    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.

  • 13.
    Cianchi, Andrea
    et al.
    University of Florence.
    Mazya, Vladimir G.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations2011In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 36, no 1, p. 100-133Article in journal (Refereed)
    Abstract [en]

    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.

  • 14.
    Costea, Serban
    et al.
    n/a.
    Maz'ya, Vladimir
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities.2009In: Sobolev spaces in mathematics. II, Springer , 2009, p. 103-121Chapter in book (Other (popular science, discussion, etc.))
  • 15.
    Costin, Ovidio
    et al.
    Department of Mathematics Ohio State University.
    Maz´ya, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Sharp Hardy-Leray inequality for axisymmetric divergence-free fields2008In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 32, no 4, p. 523-532Article in journal (Refereed)
    Abstract [en]

    We show that the sharp constant in the classical n-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for n = 2 without the axisymmetry assumption.

  • 16. Filippas, S
    et al.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Tertikas, A
    Sharp Hardy-Sobolev inequalities2004In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 339, no 7, p. 483-486Article in journal (Refereed)
    Abstract [en]

    Let Omega be a smooth bounded domain in R-N, N greater than or equal to 3. We show that Hardy's inequality involving the distance to the boundary, with best constant (1/4), may still be improved by adding a multiple of the critical Sobolev norm.

  • 17.
    Filippas, Staffas
    et al.
    University of Creta.
    Maz´ya, Vladimir G.
    University of Liverpool, Liverpool, UK.
    Tertikas, Ahilles
    University of Creta.
    Critical Hardy-Sobolev inequalities2007In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 87, no 1, p. 37-56Article in journal (Refereed)
  • 18.
    Filippas, Stasis
    et al.
    University of Crete.
    Maz´ya, Vladimir G.
    Department of Mathematics, Ohio State University Columbus, OH 43210, USA.
    Tertikas, Ahilles
    University of Crete.
    On a question of Brezis and Marcus2006In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 25, no 4, p. 491-501Article in journal (Refereed)
    Abstract [en]

    Motivated by a question of Brezis and Marcus, we show that the Lp–Hardy inequality involving the distance to the boundary of a convex domain, can be improved by adding an Lq norm qp, with a constant depending on the interior diameter of Ω.

  • 19.
    G. Maz'ya , Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain2009In: Comptes Rendus Mathematique, ISSN 1631-073X, Vol. 347, no 9-10, p. 517-520Article in journal (Refereed)
    Abstract [en]

    It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Mazya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

  • 20.
    G. Maz'ya, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Movchan , A
    University of Liverpool.
    Uniform asymptotic approximations of Greens functions in a long rod2008In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 31, no 17, p. 2055-2068Article in journal (Refereed)
    Abstract [en]

    Asymptotic approximations for Greens function for the operator -Delta in a long rod are derived. These approximations are uniformly valid over the whole domain including the end regions of the rod. Connections are established between the asymptotic approximations in a long rod and the asymptotics in thin domains. Overview of asymptotic approximations of Greens kernels in a domain with a small hole and domains with singularly perturbed smooth or conical boundaries is also given.

  • 21.
    G. Maz'ya, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Rossmann , J
    University of Rostock.
    A maximum modulus estimate for solutions of the Navier-Stokes system in domains of polyhedral type2009In: MATHEMATISCHE NACHRICHTEN, ISSN 0025-584X , Vol. 282, no 3, p. 459-469Article in journal (Refereed)
    Abstract [en]

    The authors prove a maximum modulus estimate for solutions of the stationary Navier-Stokes system in bounded domains of polyhedral type.

  • 22.
    Galaktionov, V A
    et al.
    University of Bath, England .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Boundary characteristic point regularity for Navier-Stokes equations: Blow-up scaling and Petrovskii-type criterion (a formal approach)2012In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 75, no 12, p. 4534-4559Article in journal (Refereed)
    Abstract [en]

    The three-dimensional (3D) Navier-Stokes equations less thanbrgreater than less thanbrgreater thanu(t) + (u . del)u = -del p + Delta u, divu = 0 in Q(0), (0.1) less thanbrgreater than less thanbrgreater thanwhere u = [u, v, w](T) is the vector field and p is the pressure, are considered. Here, Q(0) subset of R-3 x [-1, 0) is a smooth domain of a typical backward paraboloid shape, with the vertex (0, 0) being its only characteristic point: the plane {t = 0} is tangent to. partial derivative Q(0) at the origin, and other characteristics for t is an element of [0,-1) intersect. partial derivative Q(0) transversely. Dirichlet boundary conditions on the lateral boundary. partial derivative Q(0) and smooth initial data are prescribed: less thanbrgreater than less thanbrgreater thanu = 0 on. partial derivative Q(0), and u(x, -1) = u(0)(x) in less thanbrgreater than less thanbrgreater thanQ(0) boolean AND {t = -1} (div u(0) = 0). (0.2) less thanbrgreater than less thanbrgreater thanExistence, uniqueness, and regularity studies of (0.1) in non-cylindrical domains were initiated in the 1960s in pioneering works by Lions, Sather, Ladyzhenskaya, and Fujita-Sauer. However, the problem of a characteristic vertex regularity remained open. less thanbrgreater than less thanbrgreater thanIn this paper, the classic problem of regularity (in Wieners sense) of the vertex (0, 0) for (0.1), (0.2) is considered. Petrovskiis famous "2 root log log-criterion of boundary regularity for the heat equation (1934) is shown to apply. Namely, after a blow-up scaling and a special matching with a boundary layer near. partial derivative Q(0), the regularity problem reduces to a 3D perturbed nonlinear dynamical system for the first Fourier-type coefficients of the solutions expanded using solenoidal Hermite polynomials. Finally, this confirms that the nonlinear convection term gets an exponentially decaying factor and is then negligible. Therefore, the regularity of the vertex is entirely dependent on the linear terms and hence remains the same for Stokes and purely parabolic problems. less thanbrgreater than less thanbrgreater thanWell-posed Burnett equations with the minus bi-Laplacian in (0.1) are also discussed.

  • 23. Gibiansky, Ekaterina
    et al.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Movchan, Alexander
    Three-scale asymptotics for a diffusion problem coupled with the wave equation2005In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 84, no 6, p. 585-600Article in journal (Refereed)
  • 24.
    Hellsten, H.
    et al.
    Ericsson Microwave Systems AB, Mölndal, SE 431 84, Sweden.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Vainberg, B.
    Department of Mathematics, UNCC, Charlotte, NC 28223, United States.
    The spectrum of water waves produced by moving point sources, and a related inverse problem2003In: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 38, no 4, p. 345-354Article in journal (Refereed)
    Abstract [en]

    Parameters of moving sources on the sea surface are recovered by spectral analysis of the induced surface waves. The method can be an alternative to the standard way of seeing the ship directly, in particular, when the direct observation is impossible. © 2003 Elsevier B.V. All rights reserved.

  • 25.
    Jaye, Benjamin J
    et al.
    Kent State University, OH 44240 USA .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Verbitsky, Igor E
    University of Missouri, MO 65211 USA .
    Quasilinear elliptic equations and weighted Sobolev-Poincare inequalities with distributional weights2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 232, no 1, p. 513-542Article in journal (Refereed)
    Abstract [en]

    We introduce a class of weak solutions to the quasilinear equation -Delta(p)u = sigma vertical bar u vertical bar(P-2)u in an open set Omega subset of R-n with p andgt; 1, where Delta(p)u = del. (vertical bar del u vertical bar(p-2)del u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients sigma which satisfy the inequality less thanbrgreater than less thanbrgreater than-Lambda integral(Omega) vertical bar del h vertical bar(p)dx andlt;= andlt;vertical bar h vertical bar(p), sigma andgt; andlt;= lambda integral(Omega) vertical bar del h vertical bar(p)dx, less thanbrgreater than less thanbrgreater thanfor all h is an element of C-0(infinity)(Omega). Here 0 andlt; Lambda andlt; +infinity-, and less thanbrgreater than less thanbrgreater than0 andlt; lambda andlt; (p - 1)(2-p) if p andgt;= 2, or 0 andlt; lambda andlt; 1 if 0 andlt; p andlt; 2. less thanbrgreater than less thanbrgreater thanAs we shall demonstrate, these conditions on lambda are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on sigma. less thanbrgreater than less thanbrgreater thanWe also study weak solutions of the closely related equation -Delta p nu = (p - 1)vertical bar del nu vertical bar(p) +sigma, under the same conditions on . Our results for this latter equation will allow us to characterize the class of sigma satisfying the above inequality for positive lambda and Lambda. thereby extending earlier results on the form boundedness problem for the Schrodinger operator to p not equal 2.

  • 26.
    Jaye, B.J.
    et al.
    Department of Mathematics, University of Missouri, Columbia, MO, 65211, United States, Department of Mathematics, Kent State University, Kent, OH, 44240, United States.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Verbitsky, I.E.
    Department of Mathematics, University of Missouri, Columbia, MO, 65211, United States.
    Existence and regularity of positive solutions of elliptic equations of Schrödinger type2012In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 118, no 2, p. 577-621Article in journal (Refereed)
    Abstract [en]

    We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schrödinger type, for an arbitrary open Ω ⊆ ℝn under only a form-boundedness assumption on σ ∈ D′(Ω) and ellipticity assumption on A ∈ L∞(Ω)n×n. We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient, As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schrödinger type operator H = -div(A∇·)-σ with arbitrary distributional potential σ ∈ D′(Ω), and give examples clarifying the relationship between these two properties. © 2012 Hebrew University Magnes Press.

  • 27.
    Jin, Tianling
    et al.
    Rutgers State University.
    Mazya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Van Schaftingen, Jean
    Univ Catholique Louvain.
    Pathological solutions to elliptic problems in divergence form with continuous coefficients2009In: COMPTES RENDUS MATHEMATIQUE, ISSN 1631-073X, Vol. 347, no 13-14, p. 773-778Article in journal (Refereed)
    Abstract [en]

    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).

  • 28.
    Kamotski, I.
    et al.
    UCL, England .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Estimate for a solution to the water wave problem in the presence of a submerged body2013In: Russian journal of mathematical physics, ISSN 1061-9208, E-ISSN 1555-6638, Vol. 20, no 4, p. 453-467Article in journal (Refereed)
    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.

  • 29.
    Kamotski, I. V.
    et al.
    UCL, London, England .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    On the Linear Water Wave Problem in the Presence of a Critically Submerged Body2012In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 44, no 6, p. 4222-4249Article in journal (Refereed)
    Abstract [en]

    We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Mazya in 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g., H-loc(2) boolean AND L-infinity) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

  • 30.
    Karlin, V
    et al.
    Univ Cent Lancashire, Ctr Res Fire & Explos Studies, Preston PR1 2HE, Lancs, England Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Liverpool, Dept Math Sci, Liverpool L69 3BX, Merseyside, England Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge CB3 9EW, England.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Movchan, AB
    Univ Cent Lancashire, Ctr Res Fire & Explos Studies, Preston PR1 2HE, Lancs, England Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Liverpool, Dept Math Sci, Liverpool L69 3BX, Merseyside, England Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge CB3 9EW, England.
    Willis, JR
    Univ Cent Lancashire, Ctr Res Fire & Explos Studies, Preston PR1 2HE, Lancs, England Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Liverpool, Dept Math Sci, Liverpool L69 3BX, Merseyside, England Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge CB3 9EW, England.
    Bullough, R
    Univ Cent Lancashire, Ctr Res Fire & Explos Studies, Preston PR1 2HE, Lancs, England Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Liverpool, Dept Math Sci, Liverpool L69 3BX, Merseyside, England Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge CB3 9EW, England.
    Numerical solution of nonlinear hypersingular integral equations of the Peierls type in dislocation theory2000In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 60, no 2, p. 664-678Article in journal (Refereed)
    Abstract [en]

    In this paper we analyze hypersingular integral equations of the Peierls type specified on the whole real axis. The method of "approximate approximations" [Maz'ya, Proceedings of the Eighth Conference on the Mathematics of Finite Elements and Applications, VIII, MAFELAP 1993, Brunel University, 1994, pp. 107-104] is employed in order to produce numerical solutions of high accuracy, and the results are applied to problems in the theory of dislocations. The critical Peierls stress is evaluated for different configurations of dislocations and for different types of the interatomic force law.

  • 31. Karlin, V.
    et al.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Schmidt, G.
    High accuracy periodic solutions to the Sivashinsky equation2003Article in journal (Refereed)
    Abstract [en]

    The aim of this work is the accurate calculation of periodic solutions to the Sivashinsky equation, which models dynamics of the long wave instability of laminar premixed flame. A highly accurate computational algorithm was developed in both one and two spatial dimensions and its crucial implementation details are presented. The algorithm is based on the concept of saturated asymptotic approximations and can be straightforwardly extended to a wide variety of nonlinear integro-differential equations. The development of such an algorithm was motivated by difficulties in interpretation of the results of numerical experiments with the Sivashinsky equation using spectral methods. The computations carried out by the algorithm in question are in good agreement with the results obtained earlier by spectral methods. Analysis of the accuracy of obtained numerical solutions and of their stabilization to steady states supports the idea of the instability of the steady coalescent pole solutions (with maximal possible number of poles) to the Sivashinsky equation in large domains through huge linear transient amplification of nonmodal perturbations of small but finite amplitudes.

  • 32. Kondratiev, V
    et al.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Shubin, M
    Discreteness of spectrum and strict positivity criteria for magnetic Schrodinger operators2004In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 29, no 03-Apr, p. 489-521Article in journal (Refereed)
    Abstract [en]

    We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrodinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local energy of the magnetic field. The conditiions for the discreteness of spectrum depend, in particular, on a functional parameter which is a decreasing function of one variable whose argument is the normalized local energy of the magnetic field. This function enters the negligibility condition of sets for the scalar potential. We give a description for the range of all admissible functions which is precise in a certain sense. In case when there is no magnetic field, our results extend the discreteness of spectrum and positivity criteria by Molchanov [Molchanov, A. M. (1953). On the discreteness of the spectrum conditions for self-adjoint differential equations of the second order (Russian). Trudy Mosk. Matem. Obshchestva (Proc. Moscow Math. Society) 2:169-199] and Maz'ya [Maz'ya, V. G. (1973). On (p, l)-capacity, imbedding theorems and the spectrum of a self-adjoint elliptic operator. Math. USSR Izv. 7:357-387].

  • 33.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Differential Equations with Operator Coefficients: with Applications to Boundary Value Problems for Partial Differential Equations1999Book (Refereed)
  • 34.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Theory of a higher-order Sturm-Liouville equation.1997Book (Refereed)
  • 35.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    An asymptotic theory of higher order operator differential equations with nonsmooth nonlinearities2004In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 217, p. 448-488Article in journal (Refereed)
  • 36.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary2005In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 184, no 2, p. 185-213Article in journal (Refereed)
    Abstract [en]

    [No abstract available]

  • 37.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary2003In: Annali della Scuola normale superiore di Pisa.. Classe di scienze, Vol. 2, no 3, p. 551-600Article in journal (Refereed)
  • 38.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotics of a singular solution to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary2003In: Function spaces, differential operators and nonlinear analysis.  .: The Hans Triebel anniversary volume / [ed] Dorothee Haroske,Thomas Runst ,Hans-Jürgen Schmeisser, Basel: Birkhäuser Verlag , 2003, p. 75-115Chapter in book (Other academic)
    Abstract [en]

    Contains collection of papers that deal with the symbiotic relationship between the theory of function spaces, harmonic analysis, linear and nonlinear partial differential equations, spectral theory and inverse problems. This book is a tribute to Hans Triebel's work and shows his influence in development of the modern theory of function spaces.

  • 39.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary behavior of solutions to linear and nonlinear elliptic equations in plane convex domains2001In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 8, no 1-2, p. 189-193Article in journal (Refereed)
  • 40.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Sharp conditions for the classical asymptotic behaviour near a point for solutions to quasilinear elliptic systems2004In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 38, no 2, p. 143-165Article in journal (Refereed)
    Abstract [en]

    It is well known that distributional solutions of an elliptic equation with constant coefficients behave asymptotically near an interior point as sums of polynomials and linear combinations of derivatives of a fundamental solution. We consider a class of quasilinear elliptic systems and give mild conditions ensuring the same asymptotic behaviour. The sharpness of our conditions is illustrated by examples. The results are obtained as corollaries of a general theorem on the asymptotics of solutions to nonlinear ordinary differential equations in Banach spaces.

  • 41.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Movchan, A.B.
    Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom.
    Fields in non-degenerate 1D-3D elastic multi-structures2001In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 54, no 2, p. 177-212Article in journal (Refereed)
    Abstract [en]

    We analyse 1D-3D elastic multi-structures defined as solids involving finite-size three-dimensional elastic regions connected with thin rods. In the limit, when the thickness of the thin rods tends to zero, one has a union of a three-dimensional region and a set of thin rods. Classes of degenerate and non-degenerate multi-structures are specified, and asymptotic expansions of solutions of mixed boundary-value problems of linear elasticity are constructed. Asymptotic analysis, given in this work, provides rigorous justification of the existing engineering pile-structure models, and it also enables one to construct new models of high accuracy.

  • 42.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Movchan, Alexander
    Department of Mathematical Sciences, University of Liverpool, Liverpool, UK.
    Asymptotic analysis of fields in multi-structures1999Book (Refereed)
  • 43.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Rossmann, Jürgen
    FB Mathematik, Rostock University, Rostock, Germany.
    Elliptic boundary value problems in domains with point singularities1997Book (Refereed)
  • 44.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maz´ya, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Rossmann, Jürgen
    FB Mathematik Rostock University, Rostock, Germany.
    Spectral problems associated with corner singularities of solutions of elliptic equations2001Book (Refereed)
    Abstract [en]

    This book focuses on the analysis of eigenvalues and eigenfunctions that describe singularities of solutions to elliptic boundary value problems in domains with corners and edges. The authors treat both classical problems of mathematical physics and general elliptic boundary value problems

  • 45. Kresin, G. I.
    et al.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Best constants in the Miranda-Agmon inequalities for solutions of elliptic systems and the classical maximum modulus principle for fluid and elastic half-spaces2003In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 82, no 2, p. 157-185Article in journal (Refereed)
  • 46.
    Kresin, Gershon
    et al.
    Department of Computer Science and Mathematics, Ariel University Center of Samaria.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Maximum principles and sharp constants for solutions of elliptic and parabolic systems2012Book (Other academic)
  • 47.
    Kresin, Gershon
    et al.
    Ariel University Centre Samaria.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    OPTIMAL ESTIMATES FOR THE GRADIENT OF HARMONIC FUNCTIONS IN THE MULTIDIMENSIONAL HALF-SPACE2010In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, ISSN 1078-0947, Vol. 28, no 2, p. 425-440Article in journal (Refereed)
    Abstract [en]

    A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a multidimensional half-space is obtained under the assumption that functions boundary values belong to L-p. This representation is concretized for the cases p = 1,2, and infinity.

  • 48.
    Kresin, Gershon
    et al.
    Ariel University Center of Samaria, Ariel, Israel.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Sharp real-part theorems for high order derivatives2012In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 181, no 2, p. 107-125Article in journal (Refereed)
    Abstract [en]

    We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane C + . It is assumed that the boundary value of the real part of the function on ∂C + belongs to Lp. This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to Lp. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in C + . As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary.

  • 49. Kresin, Gershon
    et al.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Sharp pointwise estimates for solutions of strongly elliptic second order systems with boundary data from L-P2007In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 86, no 7, p. 783-805Article in journal (Refereed)
    Abstract [en]

    The strongly elliptic system Aij partial derivative(2)u/partial derivative x(i)partial derivative x(j) = 0 with constant m x m matrix-valued coefficients A(ij) = A(ji) for a vector-valued functions u = (u(1),...,u(m)) in the half-space R-+(n) = {x = (x(1),..., x(n)) : x(n) > 0} as well as in a domain Omega subset of R-n with smooth boundary partial derivative Omega and compact closure Omega is considered. A representation for the sharp constant C-p in the inequality vertical bar u(x)vertical bar <= C(p)x(n)((1-n)/p) parallel to u vertical bar x(n)=0 parallel to p is obtained, where vertical bar center dot vertical bar is the length of a vector in the m-dimensional Euclidean space, x epsilon R-+(n), and parallel to center dot parallel to(p) is the L-p-norm of the modulus of an m-component vector-valued function, 1 <= p <= infinity. It is shown that lim vertical bar x - O-x vertical bar((n-1)/p) sup{vertical bar u(x)vertical bar : parallel to u vertical bar partial derivative Omega parallel to p <= 1} =C-p(O-x), x -> O-x where O-x is a point at partial derivative Omega nearest to x epsilon Omega, u is the solution of Dirichlet problem in Omega for the strongly elliptic system A(ij)partial derivative(2)u/partial derivative x(i)partial derivative x(j) = 0 with boundary data from [L-p(partial derivative Omega)](m), and C-p(O-x) is the sharp constant in the aforementioned inequality for u in the tangent space R-+(n) (O-x) to partial derivative Omega at O-x. As examples, Lame ' and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula C-p = 2 Gamma((n + 2)/2)/pi((n+p-1)/(2p)) {Gamma((2p + n - 1)/(2p - 2))/Gamma((n + 1)p/(2p - 2))}((p-1)/p) is derived, where 1 < p < infinity.

  • 50.
    Kresin, Gershon
    et al.
    Department of Computer Science and Mathematics, College of Judea and Samaria, Ariel, Israel.
    Maz´ya, Vladimir G.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Sharp real-part theorems: A unified approach2007Book (Refereed)
    Abstract [en]

    This volume contains a coherent point of view on various sharp pointwise inequalities for analytic functions in a disk in terms of the real part of the function on the boundary circle or in the disk itself. Inequalities of this type are frequently used in the theory of entire functions and in the analytic number theory.

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