DOCSLIB.ORG

Advances in Nonlinear Pdes ! Book$Of$Abstracts$

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

! ! Advances in Nonlinear PDEs ! in#honor#of##Nina#N.#Uraltseva## Book$of$abstracts$ St.!Petersburg,!!2014!!! ! ! Advances in Nonlinear PDEs September 3-5, 2014 Advances in Nonlinear PDEs ! Interna2onal#Conference# in#honor#of#the#outstanding#mathema2cian# Nina#N.#Uraltseva## and#on#the#occasion#of#her#80th#anniversary# St.!Petersburg,!!September!3@5,!2014!!! Supported!by:! St.!Petersburg!Steklov!Ins7tute! of!Mathema7cs! 1 Advances in Nonlinear PDEs September 3-5, 2014 ! ! Organizing(Commi,ee:( Darya%Apushkinskaya% Alexander%Mikhailov% Alexander%Nazarov% Sergey%Repin% Ta;ana%Vinogradova% Nadya%Zalesskaya% 2 Advances in Nonlinear PDEs September 3-5, 2014 Cauchy-Dirichlet problem for quasilinear parabolic systems with a nonsmooth in time principal matrix Arina Arkhipova St. Petersburg State University, Russia [email protected] We consider quasilinear parabolic systems with nonsmooth in time principal matrices. Only boundedness of these matrices is assumed in time variable. We prove partial regularity up to the parabolic boundary of a cylinder. To prove the result, we apply the method of modified A-caloric approximation assuming that the matrix A is not the constant one but depends on the time variable. The talk reports on results obtained jointly with J.Stara and O.John. 3 Advances in Nonlinear PDEs September 3-5, 2014 Semilinear elliptic problems with a Hardy potential Catherine Bandle University of Basel, Switzerland [email protected] We consider problems of the type p Δu +V (x)u = u in a bounded domain in n where µ V is a Hardy potential and δ(x) is the distance δ 2(x) from a point x to the boundary of the domain. € We are interested in the € existence of positive solutions, and the interplay between the nonlinearity € € and the boundary € singularity. If 0 < p <1 the nonlinearity € gives rise to dead cores and if p >1 to boundary blowup. We give a fairly complete picture of the radial solutions and use those solutions as upper and lower solutions for general domains. € The talk reports on results € obtained in collaboration with V.Moroz (Swansea), W.Reichel (KIT Karslruhe) and M.A.Pozio (La Sapienza Rome) 4 Advances in Nonlinear PDEs September 3-5, 2014 Stochastic counterparts of the Cauchy problem for quasilinear systems of parabolic equations Yana Belopolskaya St. Petersburg State University for Architecture and Civil Engineering, Russia [email protected] We consider the Cauchy problem for two types of quasilinear parabolic systems, namely systems with diagonal principal parts and systems with nondiagonal principal parts. For systems of the first type, investigated in the famous monograph [1] we construct probabilistic representations for classical, generalized and viscosity solutions of the Cauchy problem and use them to investigate the PDE system solution. The probabilistic counterpart is constructed in terms of stochastic differential equations for corresponding Markov processes and their multiplicative operator functionals. For systems of the second type studied in a number of papers started from [2] we construct a probabilistic representation of a generalized solution of the Cauchy problem for a PDE system of the second type in terms of time reversed stochastic flows, generated by solutions of corresponding stochastic differential equations. Partial financial support of RFBR Grant 12-01- 00457-a and Minoobrnauki project 1.370.2011 is gratefully acknowledged. References [1] O. Ladyzenskaya, V. Solonnikov, N. Uraltzeva Linear and quasilinear equations of parabolic type 1967, Nauka. [2] H. Amann Dynamic theory of quasilinear parabolic systems, Mathematische Zeitschrift (1989), Vol. 202, Issue 2, pp 219- 250. 5 Advances in Nonlinear PDEs September 3-5, 2014 Non-uniqueness of hydrodynamic equations from molecular dynamics Stamatis Dostoglou University of Missouri-Columbia, USA [email protected] Hydrodynamic equations can be obtained at the limit of microscopic Newtonian equations as the number of molecules increases while the length scale changes accordingly. For some examples of molecule systems, with reasonable initial conditions and interaction potential, the resulting hydrodynamic equations for certain scaling turn out to be quite easy to analyze and show how microscopic fluctuations can lead to apparent macroscopic non-uniqueness with respect to their macroscopic initial conditions. The contents of the talk are part of an ongoing project to obtain rigorous Reynolds equations and are based on work currently done with Jianfei Xue. 6 Advances in Nonlinear PDEs September 3-5, 2014 Normal equations and nonlocal stabilization by feedback control for equations of Navier- Stokes type Andrey Fursikov Moscow State University, Russia [email protected] We study so-called parabolic equations of normal type to understand better properties of equations of Navier-Stokes type. By definition semilinear parabolic equation is normal parabolic equation (NPE) if its nonlinear term defined by operator Bsatisfies the condition: ∀v ∈ H1 vector B(v) is collinear to v. In other words solutions of NPE does not satisfies energy estimate “in the most degree". For € Burgers and 3D Helmholtz € equations we € derive normal € parabolic equations (NPE), which nonlinear terms B(v) are orthogonal projections of nonlinear terms for corresponding original equations on the straight line generated by the vector . The structure of dynamical flow corresponding to these NPE v will be described. € For NPE corresponding to Burgers equation we construct nonlocal stabilization to zero of solutions by starting, impulse, or distributed feedback € controls supported in an arbitrary fixed sub domain of the spatial domain. The last result is applied to nonlocal stabilization of solutions for Burgers equations. 7 Advances in Nonlinear PDEs September 3-5, 2014 On attractors of m-Hessian evolutions Nina Ivochkina St. Petersburg State University, Russia [email protected] Let Ω be a bounded domain in n, 2,1 Q = Ω × (0;∞), u ∈C (Q ), uxx be the Hesse matrix of u in space variables. We denote by Tp[u] = Tp (uxx ), 1 p n $1 the p-trace of u and introduce p- ≤€ ≤ xx € Hessian evolution operator by € E [u]€:= u T [u]+€T [u]. € p t p −1 p We investigate € asymptotic behavior of € solutions € of the following € initial boundary € value problems: E [u] = f , u = φ, 1≤ m ≤ n, (1) € m ∂ 'QT where . In ∂'QT = {Ω × {0}} ∪{∂Ω × [0;T]} particular, we have proved € 2,1 Theorem 1. Let f ≥ν > 0, f ∈C (QT ) for all € 2,1 T ∈[0;∞), φ ∈C (QT ), φ = 0 on ∂Ω × [0;∞), 2 ∂Ω ∈C . Assume that limt → ∞ f (x,t) = f (x) and 2 there exists € a solution € u ∈C (Ω ) to the Dirichlet problem € € € € T [u] = f , u = 0. € € m ∂Ω Then all solutions u ∈C 2,1(Ω × [0;∞)) to the problem (1) tend uniformly € in C to the function u (x) , when t →∞ . € It is of interest the following non existence theorem. € € € € 8 Advances in Nonlinear PDEs September 3-5, 2014 Theorem 2. Assume that there are points x0, x1 ∈Ω such that φxx (x0,0) is (m −1)-positive matrix, while φxx (x1,0) is not (m −1)-positive. Then there are no solutions in C 2,1(Q ) to the problem (1), whatever T € f > 0, ∂Ω, T > 0, φ had been. € € € €Eventually, we formulate the existence theorem assuming sufficiently smooth€ data in (1). € € € € Theorem 3. Let f ≥ν > 0, ∂Ω is ( m −1)-convex hypersurface, φ(x,0) ∈K m −1(Ω ). Assume that compatibility conditions are satisfied. Then there exists a unique in C 2,1(Q ) solution to the problem (1). T € € € € The work is supported by the RFBR grant No. 12-01-00439 and by the grants Sci. Schools RF 1771.2014.1,€ and by the St. Petersburg State University grant 6.38.670.2013. 9 Advances in Nonlinear PDEs September 3-5, 2014 Analyticity of the free boundary in the thin obstacle problem Herbert Koch University of Bonn, Germany [email protected] We prove analyticity of the regular part of the boundary of the contact set for the thin obstacle problem. A key step is a change of variables which reduces the problem to a Monge-Ampere type equation, which in the relevant regime becomes a fully nonlinear variant of a Grushin-type operator. We prove analyticity of solutions to this fully nonlinear equation under non degeneracy assumptions. This talk is based on results obtained in collaboration with A.Petrosyan and W.Shi. 10 Advances in Nonlinear PDEs September 3-5, 2014 On uniqueness in the water wave theory Vladimir Kozlov Linköping University, Sweden [email protected] This talk concerns the classical free-boundary problem that describes two-dimensional steady gravity waves on water of finite depth. I will discuss some uniqueness results. 11 Advances in Nonlinear PDEs September 3-5, 2014 Common features of homogenization and of large scale limits in statistical mechanics Stephan Luckhaus Leipzig University, Germany [email protected] We try to present large scale limits in equilibrium statistical mechanics from an analytic- functional analytic point of view. The results presented are joint with Roman Kotecky. We also try to point out the analogues with results in homogenization. The setting for both types of result is that of quasiconvex functionals. 12 Advances in Nonlinear PDEs September 3-5, 2014 Criteria for the Poincare-Hardy inequalities Vladimir Maz’ya University of Liverpool, UK and Linköping University, Sweden [email protected] This is a survey of necessary and sufficient conditions for validity of various integral inequalities containing arbitrary weights (measures and distributions). These results have direct applications to the spectral theory of elliptic partial differential operators. 13 Advances in Nonlinear PDEs September 3-5, 2014 From Uraltseva to Zhikov, forty years of degenerate operators in Russia Giuseppe Mingione University of Parma, Italy [email protected] In 1967 a seminal paper of Nina Ural'tseva appeared, featuring the proof of the C1,β -nature of solutions to the degenerate equation p-Laplacian equations.
Recommended publications
  • Futureacademy.Org.UK

    Futureacademy.Org.UK

    The European Proceedings of Social & Behavioural Sciences EpSBS Future Academy ISSN: 2357-1330 http://dx.doi.org/10.15405/epsbs.2017.08.55 EEIA-2017 2017 International conference "Education Environment for the Information Age" PROBLEMS OF AN INTERDISCIPLINARITY IN COMPARATIVE EDUCATION IN THE INFORMATION AGE Vladimir Myasnikov (a), Irina Naydenova (b), Natalia Naydenova (c)*, Tatyana Shaposhnikova (d) *Corresponding author (a) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia (b) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia (c) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia, [email protected]* (d) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia Abstract The article is devoted to the topical issues for the methodology of modern comparative research in education, which at the moment in the information age should be interdisciplinary in describing and explaining the educational phenomena and practices; build on the results of research in other areas of knowledge and carried out by specialists from different scientific fields. In modern conditions, the interdisciplinary of comparative research requires an expansion of the framework of this approach, their orientations to results of new disciplines taking into account electronic means of the analysis and assessment of data with the prospect of moving from interdisciplinarity to transdisciplinarity. The interdisciplinarity in comparative research is studied in the works of foreign and domestic scientists.
  • Friedrichs Type Inequalities in Arbitrary Domains

    Friedrichs Type Inequalities in Arbitrary Domains

    Friedrichs type inequalities in arbitrary domains Andrea Cianchi Vladimir Maz’ya To Ari Laptev on the occasion of his 70th birthday 1 Introduction Almost one century ago, in his paper [8] on boundary value and eigenvalue problems in the elasticity theory of plates, K.Friedrichs showed that, given a bounded open set Ω R2 satisfying a very mild regularity condition on the boundary ∂Ω, there exists a constant⊂ C = C(Ω) such that u 2 Ω C u 2 Ω + u 2 Ω (1.1) k kL ( ) ≤ k∇ kL ( ) k kL (∂ ) for every sufficiently smooth function u : Ω R. Here, 2 Ω stands for the → k · kL (∂ ) L2-norm on ∂Ω with respect to the (n 1)-dimensional Hausdorff measure. As noticed in the same paper, an application of inequality− (1.1) to the first-order derivatives of u yields 2 u 2 Ω C u 2 Ω + u 2 Ω (1.2) k∇ kL ( ) ≤ k∇ kL ( ) k∇ kL (∂ ) for some constant C = C(Ω). Inequality (1.1) can be regarded as a forerunner of inequalities in Sobolev type spaces involving trace norms over the boundary of the underlying domain. The latter play a role in a variety of problems in the theory of partial differential equations, including the analysis of solutions to elliptic equations subject to Robin boundary conditions. Criteria for the validity of inequalities of the form u Lq (Ω) C u L p (Ω) + C u Lr (∂Ω), (1.3) k k ≤ 1 k∇ k 2 k k A. Cianchi: Dipartimento di Matematica e Informatica “U.Dini", Università di Firenze, Viale Mor- gagni 67/a 50134, Firenze (Italy); email: andrea.ciamchi@unifi.it V.
  • EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: R.J.Wilson@Open.Ac.Uk

    EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: [email protected]

    CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania e-mail: [email protected] NEWSLETTER No. 47 KRZYSZTOF CIESIELSKI Mathematics Institute March 2003 Jagiellonian University Reymonta 4 EMS Agenda ................................................................................................. 2 30-059 Kraków, Poland e-mail: [email protected] Editorial by Sir John Kingman .................................................................... 3 STEEN MARKVORSEN Department of Mathematics Executive Committee Meeting ....................................................................... 4 Technical University of Denmark Building 303 Introducing the Committee ............................................................................ 7 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] An Answer to the Growth of Mathematical Knowledge? ............................... 9 SPECIALIST EDITORS Interview with Vagn Lundsgaard Hansen .................................................. 15 INTERVIEWS Steen Markvorsen [address as above] Interview with D V Anosov .......................................................................... 20 SOCIETIES Krzysztof Ciesielski [address as above] Israel Mathematical Union ......................................................................... 25 EDUCATION Tony Gardiner
  • Layer Potentials for General Linear Elliptic Systems 1

    Layer Potentials for General Linear Elliptic Systems 1

    Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 309, pp. 1{23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LAYER POTENTIALS FOR GENERAL LINEAR ELLIPTIC SYSTEMS ARIEL BARTON Abstract. In this article we construct layer potentials for elliptic differential operators using the Babuˇska-Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be constructed in very general settings. We then generalize several well known properties of layer potentials for harmonic and second order equations, in particular the Green's formula, jump relations, adjoint relations, and Verchota's equivalence between well-posedness of boundary value problems and invertibility of layer potentials. 1. Introduction There is by now a very rich theory of boundary value problems for the Laplace operator, and more generally for second order divergence form operators − div Ar. The Dirichlet problem − div Aru = 0 in Ω; u = f on @Ω; kukX ≤ CkfkD and the Neumann problem − div Aru = 0 in Ω; ν · Aru = g on @Ω; kukX ≤ CkgkN are known to be well-posed for many classes of coefficients A and domains Ω, and with solutions in many spaces X and boundary data in many boundary spaces D and N. A great deal of current research consists in extending these well posedness results to more general situations, such as operators of order 2m ≥ 4 (for example, [19, 25, 45, 47, 53, 54]; see also the survey paper [22]), operators with lower order terms (for example, [24, 30, 34, 55, 62]) and operators acting on functions defined on manifolds (for example, [46, 50, 51]).
  • $ L^ P $ Solvability of the Stationary Stokes Problem on Domains With

    $ L^ P $ Solvability of the Stationary Stokes Problem on Domains With

    Abstract The Dirichlet boundary value problem for the Stokes operator with Lp data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all p ∈ (2 − ε, ∞] and also its solvability in C(D) for the data in C(∂D). arXiv:1007.5495v1 [math.AP] 30 Jul 2010 1 Lp solvability of the Stationary Stokes problem on domains with conical singularity in any dimension ∗ Martin Dindos The University of Edinburgh and Maxwell Institute of Mathematical Sciences JCMB, The King’s Buildings Mayfield Road, Edinburgh EH9 3JZ, Scotland [email protected] Vladimir Maz’ya Department of Mathematical Sciences, M&O Building, University of Liverpool, Liverpool L69 7ZL, UK and Department of Mathematics, Link¨oping University, SE-581 83 Link¨oping, Sweden [email protected] In memory of Michael Sh. Birman Keywords: Lam´eand Stokes systems, Lp solvability, Dirichlet problem 1 Introduction In this paper we study the Stokes system (which is the linearized version of the stationary Navier-Stokes system) on a fixed domain D ⊂ Rn, for n ≥ 3. In fact, we establish our result for a both Lam´esystem (ν < 1/2) and the Stokes system (ν = 1/2). We want to consider a classical question of the solvability of the Lp Dirichlet problem on the domain D. Let us recall that the Dirichlet problem for the system (1.1) is Lp solvable on the domain D if for all vector fields f ∈ Lp(∂D) there is a pair of (u, p) (here u : D → Rn, p : D → R) such that − ∆u + ∇p = 0, div u + (1 − 2ν)p =0 in D, (1.1) u ∂D = f almost everywhere, ∗ p u ∈ L (∂D), ∗Mathematics Subject Classifications: 35J57, 35J47 2 and moreover for some C > 0 independent of f the estimate ∗ ku kLp(∂D) ≤ CkfkLp(∂D) holds.
  • Curriculum Vitae

    Curriculum Vitae

    Umberto Mosco WPI Harold J. Gay Professor of Mathematics May 18, 2021 Department of Mathematical Sciences Phone: (508) 831-5074, Worcester Polytechnic Institute Fax: (508) 831-5824, Worcester, MA 01609 Email: [email protected] Curriculum Vitae Current position: Harold J. Gay Professor of Mathematics, Worcester Polytechnic Institute, Worcester MA, U.S.A. Languages: English, French, German, Italian (mother language) Specialization: Applied Mathematics Research Interests:: Fractal and Partial Differential Equations, Homog- enization, Finite Elements Methods, Stochastic Optimal Control, Variational Inequalities, Potential Theory, Convex Analysis, Functional Convergence. Twelve Most Relevant Research Articles 1. Time, Space, Similarity. Chapter of the book "New Trends in Differential Equations, Control Theory and Optimization, pp. 261-276, WSPC-World Scientific Publishing Company, Hackenseck, NJ, 2016. 2. Layered fractal fibers and potentials (with M.A.Vivaldi). J. Math. Pures Appl. 103 (2015) pp. 1198-1227. (Received 10.21.2013, Available online 11.4.2014). 3. Vanishing viscosity for fractal sets (with M.A.Vivaldi). Discrete and Con- tinuous Dynamical Systems - Special Volume dedicated to Louis Niren- berg, 28, N. 3, (2010) pp. 1207-1235. 4. Fractal reinforcement of elastic membranes (with M.A.Vivaldi). Arch. Rational Mech. Anal. 194, (2009) pp. 49-74. 5. Gauged Sobolev Inequalities. Applicable Analysis, 86, no. 3 (2007), 367- 402. 6. Invariant field metrics and dynamic scaling on fractals. Phys. Rev. Let- ters, 79, no. 21, Nov. (1997), pp. 4067-4070. 7. Variational fractals. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) No. 3-4, pp. 683-712. 8. A Saint-Venant type principle for Dirichlet forms on discontinuous media (with M.
  • International Congress of Mathematics 2002 Satellite

    International Congress of Mathematics 2002 Satellite

    中 央 研 究 院 數 學 研 究 所 INSTITUTE OF MATHEMATICS ACADEMIA SINICA TAIPEI 10617, TAIWAN TEL : 886-2-23685999 FAX : 886-2-23689771 International Congress of Mathematics 2002 Satellite Conference – Taipei 2002 International Conference on Nonlinear Analysis 13 August – 17 August 2002 Institute of Mathematics, Academia Sinica Organized by Nai-Heng Chang 張乃珩 (Academia Sinica) Kin-Ming Hui 許健明 (Academia Sinica) Fon-Che Liu 劉豐哲 (Academia Sinica) Tai-Ping Liu 劉太平 (Academia Sinica) Organized by National Science Council 中 央 研 究 院 數 學 研 究 所 INSTITUTE OF MATHEMATICS ACADEMIA SINICA TAIPEI 10617, TAIWAN TEL : 886-2-23685999 FAX : 886-2-23689771 Table of Contents August 13, Tuesday, 2002 13:30 – 14:30 Kyuya Masuda (Himeji Institute of Technology) Analyticity of solutions for some evolution equations .......................................................................... 1 14:40 – 15:40 Vladimir Maz'ya (Linkoping University) The Schrödinger and the relativistic Schrödinger operators on the energy space: boundedness and compactness criteria ....................................................................................................................... 2 16:00 – 17:00 Mariano Giaquinta (Scuola Normale Superiore, Pisa) Graph currents and applications ............................................................................................................ 3 August 14, Wednesday, 2002 09:00 – 10:00 Claude Bardos (University of Paris 7) Mean field dynamics of N quantum particles, Hartree, Hartree Fock approximations and Coulomb Potential ................................................................................................................................
  • Abstract and Applied Analysis, Volume 2011, Article ID 545264, 19 Pages, 2011

    Abstract and Applied Analysis, Volume 2011, Article ID 545264, 19 Pages, 2011

    Nikolai Azbelev −−− the Giant of Causal Mathematics Efim A. Galperin Universite du Quebec a Montreal, Canada One of the greatest mathematicians of all times, Nikolai Viktorovich Azbelev has died. He is one of those great scholars who rise to lasting prominence after leaving this world of fast market values in science. He left immortal ideas that are changing mathematics. I met him 20 years ago at the first world Congress of Nonlinear Analysts (1992) in Tampa, USA, and later in Ariel and Athens. We talked a lot about differential equations with deviating arguments, the branch of mathematics not duly recognized at the time and, I am afraid, not clearly understood right now. The classical theories of ODEs and PDEs well presented in all textbooks, monographs and high level articles in mathematics are physically invalid, and Nikolai Azbelev felt it strongly although he could not tell it from the podium of a congress. Even now, it is not common to talk about it. To put things straight, consider the 2nd law of Newton usually formulated in textbooks as follows: ma = mx’’ = F(t, x, v), v = x’ = lim[x(t+ dt) – x(t)]/dt as dt→ 0. Here the mass m is presumed to be constant. For m ≠ const, Georg Buquoy proposed (1812) another formula: mdv + (v – w)dm = F(t, x, v)dt where v – w is the relative velocity with which dm is ejected from a moving body. Both formulas are however non-causal, thus physically invalid, since at any current moment t, the value x(t + dt), dt > 0, does not exist and cannot be known (measured) at a future moment t + dt > t not yet realized.
  • Sobolev Inequalities for the Symmetric Gradient in Arbitrary Domains

    Sobolev Inequalities for the Symmetric Gradient in Arbitrary Domains

    Sobolev inequalities for the symmetric gradient in arbitrary domains Andrea Cianchi Dipartimento di Matematica e Informatica \U. Dini", Universit`adi Firenze Viale Morgagni 67/A, 50134 Firenze, Italy e-mail: cianchi@unifi.it Vladimir G. Maz'ya Department of Mathematics, Link¨opingUniversity, SE-581 83 Link¨oping,Sweden and RUDN University 6 Miklukho-Maklay St, Moscow, 117198, Russia e-mail: [email protected] Abstract 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. 1 Introduction Diverse mathematical models for physical phenomena, described by some vector-valued func- arXiv:1901.09897v1 [math.FA] 27 Jan 2019 n tion u :Ω ! R , depend on the derivatives of u just through the symmetric part Eu of its n distributional gradient ru. Here, Ω is an open set in R , n ≥ 2, and 1 T Eu = 2 ru + (ru) ; where (ru)T stands for the transpose matrix of ru. Instances in this connection are provided by the theory of non-Newtonian fluids, where u represents the velocity of a fluid [2, DL, 22, 28, 39, 40, 41], and the theories of plasticity and nonlinear elasticity, where u stands for the displacement of a body [28, 32, 51].
  • Recent Progress in Elliptic Equations and Systems of Arbitrary Order with Rough Coefficients in Lipschitz Domains

    Recent Progress in Elliptic Equations and Systems of Arbitrary Order with Rough Coefficients in Lipschitz Domains

    Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains Vladimir Maz'ya Department of Mathematical Sciences University of Liverpool, M&O Building Liverpool, L69 3BX, UK Department of Mathematics Link¨opingUniversity SE-58183 Link¨oping, Sweden Tatyana Shaposhnikova Department of Mathematics Link¨opingUniversity SE-58183 Link¨oping, Sweden Abstract. This is a survey of results mostly relating elliptic equations and systems of arbitrary even order with rough coefficients in Lipschitz graph domains. Asymp- totic properties of solutions at a point of a Lipschitz boundary are also discussed. 2010 MSC. Primary: 35G15, 35J55; Secondary: 35J67, 35E05 Keywords: higher order elliptic equations, higher order elliptic systems, Besov spaces, BMO, VMO, Lipschitz graph domains, Green's function, asymptotic be- haviour of solutions Introduction Results on the regularity of solutions up to the boundary of a domain play a very important role in the theory of linear elliptic equations and systems. The following classical example serves as an illustration. Consider the Dirichlet problem ( ∆u = 0 in Ω; Tr u = f on @Ω; where Ω is a bounded domain with smooth boundary in Rn and Tr u stands for the 1 boundary value (trace) of u. Let u 2 Wp (Ω), 1 < p < 1, that is Z 1=p p p kuk 1 := jruj + juj dx < 1; Wp (Ω) Ω 1−1=p and let f belong to the Besov space Bp (@Ω) with the norm Z Z p Z 1=p jf(x) − f(y)j p n+p−2 dσxdσy + jf(x)j dσx : @Ω @Ω jx − yj @Ω 1 It is well-known that 1 1−1=p Tr Wp (Ω) = Bp (@Ω): 1−1=p 1 Moreover, the harmonic extension u of f 2 Bp (@Ω) belongs to Wp (Ω) and the 1 1−1=p norm of u in Wp (Ω) is equivalent to the above norm in Bp (@Ω).
  • Unsolved Problems in Analysis and Partial Differential Equations

    Unsolved Problems in Analysis and Partial Differential Equations

    Integr. Equ. Oper. Theory (2018) 90:25 https://doi.org/10.1007/s00020-018-2460-8 Published online April 26, 2018 Integral Equations c The Author(s) 2018 and Operator Theory Open Problems Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations Vladimir Maz’ya Abstract. This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations. Mathematics Subject Classification. Primary: 46E35, 42B37, 26D10 31A10, 35J25, 35J30 35K10, 35L10, 35R05 74J15, 76D, 74B05. Keywords. Sobolev spaces, Capacitary inequality, Hardy inequality, Riesz potential space, Singular integral operator, Fredholm radius, Maximal operator, Boundary value problems for elliptic operators, Laplace equa- tion, Cauchy problem, Lipschitz domains, Jordan domains, Green func- tion, Capacity potential, Polyharmonic potential, Oblique derivative problem, Lame system, Stokes system, Navier–Stokes system, Wiener test, p-Laplace operator, Heat operator, Wave operator, Surface waves, Hybrid iterative methods. Contents 1. Introduction 4 2. Basic Notations 4 3. Sobolev Spaces of Integer and Fractional Orders 5 3.1. Isoperimetric Problem for Fractional Perimeter 5 3.2. Strong Capacitary Inequality for Functions with Arbitrary Boundary Data 6 3.3. Boundary Traces of Functions in a Sobolev Space 6 3.4. Embedding of a Sobolev Space into the Space of Distributions 7 3.5. An Approximation Problem in the Theory of Sobolev Spaces 7 3.6. Extensions of Sobolev Functions Outside of an Angle 7 3.7. Best Constants in Some Classical Inequalities 8 3.8. Hardy Inequality with Capacitary Distance to ∂Ω 8 The author would like to thank O.O.
  • Sharp Pointwise Estimates for Solutions of Weakly Coupled Second Order Parabolic System in a Layer

    Sharp Pointwise Estimates for Solutions of Weakly Coupled Second Order Parabolic System in a Layer

    Sharp pointwise estimates for solutions of weakly coupled second order parabolic system in a layer Gershon Kresina∗ and Vladimir Maz’yab † aDepartment of Mathematics, Ariel University, Ariel 40700, Israel bDepartment of Mathematical Sciences, University of Liverpool, M&O Building, Liverpool, L69 3BX, UK; Department of Mathematics, Link¨oping University,SE-58183 Link¨oping, Sweden; RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Russia In memory of great mathematician S.L. Sobolev Abstract. We deal with m-component vector-valued solutions to the Cauchy problem for linear both homogeneous and nonhomogeneous weakly coupled second order parabolic Rn+1 Rn system in the layer T = (0, T ). We assume that coefficients of the system are real and depending only on t, n 1× and T < . The homogeneous system is considered with initial data in [Lp(Rn)]m, 1 ≥p . For the∞ nonhomogeneous system we suppose that the ≤ ≤∞ p Rn+1 m α Rn+1 m initial function is equal to zero and the right-hand side belongs to [L ( T )] [C T ] , α (0, 1). Explicit formulas for the sharp coefficients in pointwise estimates ∩for solutions of these∈ problems and their directional derivative are obtained. Keywords: Cauchy problem, weakly coupled parabolic system, sharp pointwise estimates, directional derivative of a vector field AMS Subject Classification: Primary 35K45, 35A23; Secondary 47A30 arXiv:2004.07942v1 [math.AP] 14 Apr 2020 1 Introduction Parabolic equations and systems are classical subjects of mathematical physics (e.g. [1], [2], [14], [15]). The present paper is a continuation of our recent work [10] on sharp pointwise estimates for the gradient of solutions to the Cauchy problem for the single parabolic equation of the second order with constant coefficients.