DOCSLIB.ORG

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

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. We say that the estimate is sharp if the coefficient in front of the norm in the majorant part of the inequality cannot be diminished. ∗Corresponding author. E-mail: [email protected] †E-mail: [email protected] 1 Sharp pointwise estimates for solutions to the Laplace, modified Helmholtz, Lam´e, Stokes, and heat equations , as well as for the analytic functions were obtained earlier in [5] - [9]. In this paper we study solutions of the Cauchy problem for parabolic weakly coupled system with real coefficients of the form ∂u n ∂2u n ∂u = a (t) + b (t) + C(t)u + f(x, t) (1.1) ∂t jk ∂x ∂x j ∂x j k j=1 j j,kX=1 X Rn+1 Rn in the layer T = (0, T ) with initial condition u t=0 = ϕ, considering separately two cases, f = 0 and ϕ = 0.× Here and henceforth T < , n | 1, u(x, t)=(u (x, t),...,u (x, t)) ∞ ≥ 1 m and f(x, t)=(f1(x, t),...,fm(x, t)). Throughout the article, we assume that A(t) = ((ajk(t))) is a symmetric positive defi- nite (n n)-matrix-valued function on [0, T ], which elements satisfy the H¨older condition × with exponent α/2 (0 <α< 1), b1(t),...,bn(t) are continuous functions on [0, T ], C(t) is continuous on [0, T ] matrix-valued function of order m. We obtain sharp pointwise estimates for u , ∂u/∂ℓ and max|ℓ|=1 ∂u/∂ℓ , where u solves | | | Rn+1| | | the Cauchy problem for system (1.1) in the layer T , denotes the Euclidean length of a vector and ℓ is a unit n-dimensional vector. By ∂u|·|/∂ℓ we mean the derivative of a vector-valued function u(x, t) in the direction ℓ: ∂u u(x + λℓ, t) u(x, t) = lim − ∂ℓ λ→0+ λ m ∂u = (ℓ, )u = j e , (1.2) ∇x ∂ℓ j j=1 X where x =(∂/∂x1,...,∂/∂xn) and ej means the unit vector of the j-th coordinate axis. The∇ present paper consists of five sections, including Introduction. Section 2 is auxiliary. Rn+1 Section 3 is devoted to explicit formulas for solutions of the Cauchy problem in T for system (1.1). In Section 4, we consider a solution of the Cauchy problem u n 2u n u ∂ ∂ ∂ Rn+1 = ajk(t) + bj(t) + C(t)u in T , ∂t ∂xj ∂xk ∂xj j,k=1 j=1 (1.3) X X u ϕ t=0 = , where ϕ [Lp(R n)]m, p [1, ]. The norm in the space [Lp(Rn)]m is defined by ∈ ∈ ∞ k·kp 1/p p ϕ p = ϕ(x) dx k k Rn | | Z for 1 p< , and ≤ ∞ ϕ = ess sup ϕ(x) : x Rn . k k∞ {| | ∈ } 2 In this section we obtain two groups of sharp estimates. First of them concerns modulus to solution u of problem (1.3). Namely, we derive the inequality u(x, t) (t) ϕ (1.4) | | ≤ Hp k kp with the sharp coefficient |||eIC∗ (t)||| p(t)= 1/p , (1.5) H 1/2 ′ (2√π)n/p det (t) (p′)n/(2p ) IA Rn+1 t where (x, t) is an arbitrary point in the layer T . Here and henceforth F (t)= 0 F (t)dt, where F can be a vector-valued or matrix-valued function, p−1 + p′−1 =I 1, the symbol ∗ R denotes passage to the transposed matrix and |||B||| = max z Bz means the spectral norm | |=1 | | Rl 1/2 of the l l real-valued matrix B, z . It is known (e.g. [11], sect. 6.3) that |||B||| = λB , × ∈ ∗ where λB is the spectral radius of the matrix B B. As a special case of (1.5) one has (t)= |||eIC∗ (t)||| . (1.6) H∞ For the single parabolic equation ∂u n ∂2u n ∂u = a (t) + b (t) + c(t)u + f(x, t) (1.7) ∂t jk ∂x ∂x j ∂x j k j=1 j j,kX=1 X with f = 0, formula (1.6) becomes t (t) = exp c(t)dt . H∞ Z0 The second group of sharp estimates concerns modulus of ∂u/∂ℓ, where u is solution of problem (1.3). Namely, the formula for the sharp coefficient ′ ′ 1/p −1/2 p +1 IC∗ (t) Γ A (t)ℓ ||| e ||| 2 ℓ(t)= I (1.8) p, 1/2 1/p ′(n+p′)/2 K 2nπ (n+p−1)/2 det (t) p IA in the inequality ∂u (x, t) ℓ(t) ϕ (1.9) ∂ℓ ≤Kp, k kp is obtained, where (x, t) is an arbitrary point in the layer Rn+1. T As a consequence of (1.8), the sharp coefficient p(t) = max p,ℓ(t) (1.10) K |ℓ|=1 K 3 in the inequality ∂u max (x, t) p(t) ϕ p (1.11) |ℓ|=1 ∂ℓ ≤K k k is found. In particular, 1 − 1/2 (t)= ||| (t)|||||| eIC∗ (t)||| . (1.12) K∞ √π IA For the single parabolic equation (1.7) with f = 0, the previous formula takes the form 1 t (t)= ||| −1/2(t)||| exp c(t)dt . (1.13) K∞ √π IA Z0 As a special case of (1.13) one has t 1 exp 0 c(t)dt ∞(t)= 1/2 K √π tnR o 0 a(t)dt n o for A(t)= a(t)I, where I is the unit matrix of orderR n. In Section 5 we consider a solution of the Cauchy problem u n 2u n u ∂ ∂ ∂ Rn+1 = ajk(t) + bj(t) + C(t)u + f(x, t) in T , ∂t ∂xj ∂xk ∂xj j,k=1 j=1 (1.14) X X u 0 t=0 = , where f [ Lp(Rn+1)]m [Cα Rn+1 ]m, α (0, 1). By [Cα Rn+1 ]m we denote the space of ∈ T ∩ T ∈ T f Rn+1 m-component vector-valued functions (x, t) which are continuous and bounded in T and locally H¨older continuous with exponent α in x Rn, uniformly with respect to t [0, T ]. p Rn+1 m ∈ ∈ The space [L T ] is endowed with the norm T 1/p p f p,T = (f(x, τ) dxdτ k k Rn Z0 Z for 1 p< , and ≤ ∞ f = ess sup f(x, τ) : x Rn, τ (0, T ) . k k∞,T {| | ∈ ∈ } In this section we also obtain two groups of sharp estimates. As before, first of them concerns modulus to solution u of problem (1.14). Namely, we prove the inequality u(x, t) (t) f (1.15) | |≤Np k kp,t with the sharp coefficient ′ p′ 1/p 1 t eIC∗ (t,τ)z (t)= max dτ , (1.16) p n/p ′ n/(2p′) 1/2 p′−1 N (2√π) p |z|=1 ( 0 det (t, τ ) ) Z IA 4 Rn+1 t where (x, t) is an arbitrary point in the layer T . Here and henceforth F (t, τ)= τ F (t)dt, where F can be a vector-valued or matrix-valued function. Formula (1.16)I is obtained under assumption that the integral in (1.16) converges. For instance, integral in (1.16) is convergentR for p> (n + 2)/2 in the case of single parabolic equation with constant coefficients. As a special case of (1.16) one has t IC∗ (t,τ) ∞(t) = max e z dτ. (1.17) N |z|=1 Z0 For the single parabolic equation (1.7) the previous formula takes the form t t (t)= exp c(t)dt dτ . N∞ Z0 Zτ The second group of sharp estimates concerns ∂u/∂ℓ. The explicit formula for the sharp coefficient ′ ′ ′ 1/p p ′ 1/p p′+1 −1/2 p t ℓ IC∗ (t,τ)z 1 Γ 2 A (t, τ) e (t)= max I dτ (1.18) p,ℓ 1/p ′ (n+p′)/2 p′−1 C n (n+p−1)/2 p |z|=1 0 1/2 2 π Z det (t, τ) IA in the inequality ∂u (x, t) ℓ(t) f (1.19) ∂ℓ ≤ Cp, k kp,t for solutions of the Cauchy problem (1.14) is found, where (x, t) Rn+1.
Load more

Recommended publications
  • 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.
    [Show full text]
  • $ 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • Advances in Nonlinear Pdes ! Book$Of$Abstracts$
    ! ! 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.
    [Show full text]
  • 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.
    [Show full text]
  • Abstract the Dirichlet Boundary Value Problem for the Stokes Operator
    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 (2 ¡ "; 1] and also its solvability in C(D) for the data in C(@D). 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 May¯eld 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, LinkÄopingUniversity, SE-581 83 LinkÄoping,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 ¯xed domain D ½ Rn, for n ¸ 3. In fact, we establish our result for a both Lam¶esystem (º < 1=2) and the Stokes system (º = 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 ¯elds f 2 Lp(@D) there is a pair of (u; p) (here u : D ! Rn, p : D ! R) such that ¡¢u + rp = 0; div u + (1 ¡ 2º)p = 0 in D; (1.1) ¯ ¯ u @D = f almost everywhere; u¤ 2 Lp(@D); ¤Mathematics Subject Classi¯cations: 35J57, 35J47 2 and moreover for some C > 0 independent of f the estimate ¤ ku kLp(@D) · CkfkLp(@D) holds.
    [Show full text]
  • Arxiv:1711.06996V1 [Math.AP]
    The Lp-dissipativity of certain differential and integral operators Alberto Cialdea and Vladimir Maz’ya In Memory of Selim G. Krein Abstract. The first part of the paper is a survey of some of the results pre- viously obtained by the authors concerning the Lp-dissipativity of scalar and matrix partial differential operators. In the second part we give new necessary and, separately, sufficient conditions for the Lp-dissipativity of the “complex oblique derivative” operator. In the case of real coefficients we provide a nec- essary and sufficient condition. We prove also the Lp-positivity for a certain class of integral operators. 1. Introduction In a series of papers [2, 3, 4, 6] we have studied the problem of characterizing the Lp-dissipativity of scalar and matrix partial differential operators. The main result we have obtained is that the algebraic condition (1.1) p 2 I m A ξ, ξ 2 p 1 Re A ξ, ξ | − | |h i| ≤ − h i for any ξ Rn, is necessary and sufficient forp the Lp-dissipativity of the Dirichlet problem for∈ the scalar differential operator t(A ), where A is a matrix whose entries are complex measures, not necessarily∇ absolutely∇ continuous, and whose imaginary parts is symmetric. Specifically we have proved that condition (1.1) is necessary and sufficient for the Lp-dissipativity of the related sesquilinear form arXiv:1711.06996v1 [math.AP] 19 Nov 2017 L (u, v)= A u, v . ZΩh ∇ ∇ i Such condition characterizes the Lp-dissipativity individually, for each p, while in the literature previous results dealt with the Lp-dissipativity for any p [1, + ).
    [Show full text]
  • (ISAAC) – June 2015 10Th ISAAC Congress in Macau in Summer 2015
    Newsletter of the International Society of Analysis, Applications and 1 Computation (ISAAC) – June 2015 10th ISAAC congress in Macau in summer 2015 The congress takes place in Macau 2015 August 2 to August 8. Please spread this information within your community. Dear ISAAC members, Dear All, I am happy to invite you to attend the next ISAAC International Congress in Macau, 2-8 August 2015. Participants are welcome to contribute to the Special Sessions, devoted to recent results in specific fields of interest. As in the preceding ISAAC meetings, the Special Sessions will give the opportunity to meet colleagues, to continue or start international collaborations. Your participation and your own scientific contribution will be essential for the success of the Congress! I look forward to meet you, and wish you a pleasant stay in Macau. Yours, Luigi Rodino, ISAAC president Organizing Committee: Jun Long Chen, Honorary Co-Chair of Local Organizing Committee (University of Macau) Yang Chen (University of Macau) Pei Dang (Secretary) (Macao University of Science and Technology) Jin-Yuan Du (Wuhan University) Kit Ian Kou (Secretary) (University of Macau) Tao Qian, Chair of Local Organizing Committee (University of Macau) Plenary Speakers: Prof. Darren Crowdy, Imperial College London, [email protected] Prof. Alan Mclntosh, Australian National University, Canberra [email protected] Prof. Ngaiming Mok, The University of Hong Kong, [email protected] Prof. Yoshinori Morimoto, Kyoto University, Japan, [email protected] Prof. Zhong-Shan Ou-Yang, ITP, CAS, China, [email protected] Edit by the board of ISAAC – society Content: M.
    [Show full text]
  • Erwin Schrödinger International Institute for Mathematical Physics Contents
    2013 Erwin Schrödinger International Institute for Mathematical Physics CONTENTS Erwin Schrödinger International Institute Preface—Turning Over the Page .................................................................................................................................................................... 2 for Mathematical Physics Boltzmanngasse 9 Erwin Schrödinger International Institute 1090 Vienna / Austria for Mathematical Physics—Structure and Activities .......................................................................................... 5 T +43-1-4277-28301 F +43-1-4277-9283 [email protected] At the Interface of Mathematics and Physics www.esi.ac.at Some Thoughts on the Fruitful and Ongoing Interaction Between Mathematics and Physics by Wendelin Werner ................................................................................................................................................................................................................ 10 Algebras, Groups and Strings by Peter Goddard ........................................................................................................................................................................................................................... 12 A Deeper Union? Poincaré on Mathematics and Mathematical Physics by Jeremy Gray ...................................................................................................................................................................................................................................
    [Show full text]
  • Pointwise Inequalities for Elliptic Boundary Value Problems
    Pointwise Inequalities for Elliptic Boundary Value Problems Guo Luoa,∗, Vladimir Maz’yab aDepartment of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong bDepartment of Mathematics, Link¨opingUniversity, Link¨oping, Sweden Abstract We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar second-order elliptic equation, the 3D Lame´ system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic bound- ary value problems provided that the corresponding weighted integral inequalities are satisfied. Keywords: Pointwise inequality, Weighted positivity 1. Introduction An important open problem in the mathematical theory of linear elasticity is whether solutions of the elasticity system, when supplemented with homogeneous Dirichlet boundary conditions and sufficiently smooth right-hand side data, are uniformly bounded in arbitrary domains. A similar question stands in the theory of hydrostatics, where the uniform boundedness of solutions of the Stokes system in general domains remains unknown. For bounded domains Ω with smooth boundaries @Ω, inequalities of the form k a b j j 1 ≤ j j j j u L (Ω) CΩ D u Lq(Ω) Lu Wl;p(Ω) (1) can often be obtained, by combining appropriate a priori estimates with Sobolev inequalities. The problem of such inequalities is that the constant CΩ generally depends on the smoothness of the domain Ω, and can blow up if the boundary of Ω contains geometric singularities. Efforts have been devoted to the study of inequalities of the type (1) with constants independent of the domain Ω.
    [Show full text]