Kato's equality and spectral decomposition for positive C0-groups

Let A be the generator of a C₀-semigroup {T(t); t0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies <¦x¦, Ax>= (xD(A), x D(A)) (where q: EE is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.     

A priori estimates of the solution of the dirichlet problem for the Monge-ampére equation in weight spaces

One proves that a priori boundedness of the norm of the solution of the problem det(U_xx)=f(x,u,u_x)>>0,u¦=0. The magnitudes of the exponents,() depends on whether the arguments u p occur or not in f (x,u,p).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 74–90, 1983.     

Chaos in almost periodic systems

Let be an almost periodic differential equation with a hyperbolic almost periodic solutionu(t) and another hyperbolic solutionv(t) satisfying ¦v(t)–u(t)¦0 as ¦t¦. It is shown that the solutions of such an equation exhibit chaotic behavior.

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Zusammenfassung Sei eine fastperiodische Differentialgleichung mit einer fastperiodischen hyperbolischen Lösungu(t) und seiv(t) eine weitere hyperbolische Lösung für welche ¦v(t)–u(t)¦0 für ¦t¦. Es wird gezeigt, daß die Differentialgleichung unter diesen Bedingungen chaotisches Verhalten aufweist.

     

Existence and regularity of solutions for a semilinear first-order equation on the torus

The problem is considered on the N-dimensional torus with a_i and g a continuous function satisfying a growth condition as ¦u¦. We show the existence of bounded solutions that are continuous if g is strictly increasing in u.     

On Wagner Connections and Wagner Manifolds

Let (M, E) be a Finsler manifold. A triplet (¯D, ¯h, ) is said to be a Wagner connection on M if (¯D, ¯h) is a Finsler connection, C (M) and the axioms (W1)–(W4), formulated originally by M. Hashiguchi, are satisfied. Then ¯h is called a Wagner endomorphism on M. We establish an explicit relation between the (canonical) Barthel endomorphism of (M, E) and a Wagner endomorphism ¯h. We show that the second Cartan tensors ¯C, ¯C _b belonging to ¯h are symmetric and totally symmetric, respectively. An explicit relation between the "canonical" tensors C, C _b and the "Wagnerian" ones is also derived. We can conclude that the rules of calculation with respect to a Wagner connection are formally the same as those with respect to the classical Cartan connection. We establish some basic curvature identities concerning a Wagner connection, including Bianchi identities. Finally, we present a new, intrinsic definition as well as several tensorial characterizations of Wagner manifolds.     

Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

Stabilization of solutions of initial-boundary problems for quasilinear parabolic equations

Lower bounds are obtained for solutions of the initial-boundary Dirichlet problem for high order equations. Sharp bounds are also obtained for ess sup¦u(x, t)¦ of the Neumann initial-boundary problem for a second- order equation in D=x(t >0), where (Rⁿ, n 2 is a domain with noncompact convex boundary.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 44, No. 10, pp. 1441–1450, October, 1992.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

The problem of conformal transformations of a circle into nonoverlapping regions

Let a, a0, a, be a fixed point in the z-plane, (a, 0, ), the class of all systemsf _k()_l ³ of functions z=f ^k(), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦¦<1, and the third maps in a similar manner the region ¦¦>1, into pair-wise nonintersecting regions B_k, k=1, 2, 3, containing the points a, 0, and , respectively, so thatf ₁(0)=a,f ₂(0)=0 andf ₃()=. The region of values (a, 0, ) of the system M(¦f ₁'(0)¦, ¦f ₂'(0)¦, 1/¦f ₃'()¦) in the class (a, 0, ) is determined.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

     

Phragmén-Lindelöf theorems for second-order quasilinear elliptic equations

Analogues are formulated of the well-known, in the theory of analytic functions, Phragmen-Lindelöf theorem for the gradients of solutions of a broad class of quasilinear equations of elliptic type. Examples are given illustrating the accuracy of the results obtained for the gradients of solutions of the equations of the form div(|U|^–2u)=f(x, u, u), where f(x, u, u) is a function locally bounded in ²ⁿ⁺¹. f(x, 0, u)=0, uf(x, u, u) c¦u¦1+q(1+ ¦u|), > 1, c > 0, q > 0, is an arbitrary real number, and n >- 2. The basic role in the technique employed in the paper is played by the apparatus of capacitary characteristics.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1376–1381, October, 1992.The author sincerely appreciates E. M. Landis's permanent attention and numerous useful discussions.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

On the behavior of solutions of quasilinear elliptic equations of second order in unbounded domains

Analogues of the well known in the theory of analytic functions Phragmén-Lindelöff theorem are formulated for the solutions of a wide class of quasilinear equations of elliptic type. Examples are given which illustrate the sharpness of the obtained results for solutions of equations of the form div(|u|^–2u)=f(x, u), where the function f(x, u) is locally bounded in IRⁿ⁺¹,f(x, 0)=0,uf(x, u)a¦u¦^1+q,a>0,>1,-1>q0, n>/2.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 279–283, February, 1992.     

Differentiation of fractional order onp-groups,approximation properties

G- p- . [5] - (G) L _r(G) (1r<), . . , - . , , , . . , X. , . (. [1], [2] [4]).     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Lévy homeomorphic parametrization and exponential families

Summary Let P={P : } be an exponential family of probability distributions with the canonical parameter and consider the one to one mapping : P . It is shown that, under mild regularity assumptions, and ^–1 are continuous with respect to the Lévy metric in P and Euclidean metric in .     

Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.