Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

For a d‐dimensional diffusion of the form dX_t = μ(X_t)dt + σ(X_t)dW_t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where ?? is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y_t = v(t, X_t), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

An equivalence for weighted integrals of an analytic function and ist derivative

In this paper we introduce the class of differentiable weights ω in the unit disc ?? such that where L is a positive constant. The main result in this paper asserts that if ω is one of these weights, then the equivalence holds for all 0 < p < ∞, 0 < q ≤ ∞and f an analytic function in ??. Our results improve others due to Aleman, Siskakis, and Stevi?. We also prove two results on harmonic conjugate functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero distribution of solutions of complex linear differential equations determines growth of coefficients

It is shown that the exponent of convergence λ(f) of any solution f of with entire coefficients A₀(z), …, A_k?2(z), satisfies λ(f) ? λ ∈ [1, ∞) if and only if the coefficients A₀(z), …, A_k?2(z) are polynomials such that for j = 0, …, k ? 2. In the unit disc analogue of this result certain intersections of weighted Bergman spaces take the role of polynomials. The key idea in the proofs is W. J. Kim’s 1969 representation of coefficients in terms of ratios of linearly independent solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Integrability Properties of the Maximal Operator on Partial Sums of Fourier Series in Orlicz Spaces

Let S* (f be the majorant function of the partial sums of the trigonometric Fourier series of f. In this paper we consider the Orlicz space Lπ and give a generalization of Soria's result [S1]. Let π (t) be a concave function with some nice properties and . If there exists a positive constant a₀ < 1 such that then we have .     

Sharp by order estimates of solutions of a simplest singular boundary value problem

We consider a boundary value problem ((0.1)) where f ∈ L_p (?), p ∈ [1, ∞] (L∞ (?) ? C (?)) and 0 ≤ q ∈ L^loc₁ (?). For a given p ∈ [1, ∞], for a correctly solvable problem (0.1) in L_p (?), we obtain minimal requirements to a positive, continuous function Θ(x) for x ∈ ? under which, regardless of f ∈ L_p (?), the solution y ∈ L_p (?) of problem (0.1) satisfies the equality . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Eigenvalue Problem for p-Laplacian in IRN

The nonlinear eigenvalue problem for p-Laplacian is considered. We assume that 1 < p < N and that the function f is of subcritical growth with respect to the variable u. The existence and C^1,α-regularity of the weak solution is proved.     

Global existence for two-velocity models of the Boltzmann equation

If A is a symmetric 2 × 2-matrix, then the initial value problem describes the evolution in time of a fictive gas whose particles can move only with the velocities u₁ and v₂. It is proved that, for continuous initial values vanishing at infinity, (1) has a global solution if an H-Theorem holds for the gas described by (1). The validity of an H-Theorem is expressed by the properties of A.     

Necessary and Sufficient Conditions for the Oscillation of a Second Order Linear Differential Equation

In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory.     

Generalized Hankel operators on the Fock space

In this paper we study generalized Hankel operators ofthe form : ?²(|z |²) → L²(|z |²). Here, (f):= (Id–P_l )( ^kf) and Pl is the projection onto A_l ²(?, |z |²):= cl(span{ ^m zⁿ | m, n ∈ N, m ≤ l }). The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators are not bounded for l < k – 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L² symbols, generalizing arguments from [6], seems possible and is planned for future work (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cylinder Homomorphisms and Chow Groups

Let X be a projective algebraic manifold of dimension n (over C), CH¹(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A¹(X) ? CH¹(X) the subgroup of cycles algebraically equivalent to zero. We say that A¹(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH¹(Γ × X) such that z_*:A¹(Γ)-A¹(X) is surjective. There is the well known Abel-Jacobi map λ₁:A¹(X)-J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A¹(X)→J(X) A¹(X) finite dimensional. Now set with corresponding map A^*(X)→J(X). Also define Level . In a recent book by the author, there was stated the following conjecture: where it was also shown that (?) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A^*(X) finite dimensional ?? Level (H^*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k-planes on X, where ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.     

A Γ‐convergence result for the two‐gradient theory of phase transitions

The generalization to gradient vector fields of the classical double‐well, singularly perturbed functionals, where W(ξ) = 0 if and only if ξ = A or ξ = B, and A ? B is a rank‐1 matrix, is considered. Under suitable constitutive and growth hypotheses on W, it is shown that I_ε Γ‐converge to where K^* is the (constant) interfacial energy per unit area. © 2002 Wiley Periodicals, Inc.     

A topological degree counting for some Liouville systems of mean field type

Let A = (a_ij)_{n × n} be an invertible matrix and A⁻¹ = (a^ij)_{n × n} be the inverse of A. In this paper, we consider the generalized Liouville system (0.1) where 0 < h_j ∈ C¹(M) and \input amssym $\rho_j \in \Bbb R^+$ , and prove that, under the assumptions of (H₁) and (H₂) (see Introduction), the Leray‐Schauder degree of (0.1) is equal to if ρ = (ρ₁, …, ρ_n) satisfies Equation (0.1) is a natural generalization of the classic Liouville equation and is the Euler‐Lagrangian equation of the nonlinear function Φ_ρ: The Liouville system (0.1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems. © 2010 Wiley Periodicals, Inc.     

Nontrivial periodic solutions of a nonlinear beam equation

We consider the existence of a nontrivial solution of the following equation: where g is a nondecreasing function defined on R¹, satisfies g(O) = O, and some other additional conditions. Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]-[8]: .     

Supercritical Branching Brownian Motion and K-P-P Equation In the Critical Speed-Area

If R_t is the position of the rightmost particle at time t in a one dimensional branching brownian motion, whore α is the inverse of the mean life time and m is the mean of the reproduction law. If Z_t denotes the random point measure of particles living at time t, we get in the critical area {c = c₀} The function u(t, x) = P(R_t > x) is studied as a solution of the K-P-P equation for some function f. Conditioned on non-extinction of the spatial tree in the c₀-direction, a limit distribution is obtained and characterized.     

Determinants of Regular Singular Sturm - Liouville Operators

We consider a regular singular Sturm-Liouville operator on the line segment (0,1]. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known (cf. Theorem 1.1 below) that the ζ-function of this operator has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to [RS] the ζ - regularized determinant of L is defined by In this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation L_y = 0 generalizing earlier work of S. Levit and U. Smilansky [LS], T. Dreyfus and H. Dym [DD], and D. Burghelea, L. Friedlander and T. Kappeler [BFK1, BFK2). More precisely we prove the formula Here ? ψ is a certain fundamental system of solutions for the homogeneous equation L_y = 0, W(? ψ), denotes their Wronski determinant, and v₀, v₁ are numbers related to the characteristic roots of the regular singular points 0, 1.     

On non-stationary Navier-Stokes flows with vanishing viscosity associated with a variational inequality

Let g is a positive increasing function with 1?g(0). The existence of a unique solution of the Navier-Stokes flow associated with K_ε,γ and the convergence of the solution to that of the Euler equations as the viscosity goes to zero are established.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Analytic aspects of the Toda system: I. A Moser‐Trudinger inequality

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser‐Trudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (a_ij)_{N × N} the Cartan matrix for SU(N + 1), i.e., We show that has a lower bound in (H¹(Σ))^N if and only if This inequality is optimal. As a direct consequence, if M_j < for 4π for j = 1, 2, …, N, Φ_M has a minimizer u that satisfies © 2001 John Wiley & Sons, Inc.     

Infinitely many solutions for some nonlinear elliptic equations in ℝN

In this paper, we study the multiple solutions for the semilinear elliptic equation where , 1<p<(N + 2)/(N ? 2) for and p>1 for N = 2. We will prove that the problem possesses infinitely many solutions under some assumptions on Q(x). Copyright © 2011 John Wiley & Sons, Ltd.