Equivalence of two definitions of a generalized <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> solution to the initial-boundary value problem for the wave equation

In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫ ₀ ^T (T ? t)|µ(t)| ^p dt exists. Here we prove that this solution is a unique solution to the problem in L _p that satisfies the corresponding integral identity.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Weakly nonlinear Schrödinger equation with random initial data

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ _t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈? ^d and t∈?. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ _t (x) has a limit as λ→0 for t=λ ^?2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.     

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Assume that (N, ?) and (M, S) are two Riemann surfaces with conformal metrics ? and S. We prove that if there is a harmonic homeomorphism between an annulus A ? N with a conformal modulus Mod(A) and a geodesic annulus A _S (p, ρ₁, ρ₂)?M, then we have ρ₂/ρ₁ ≥ Ψ _S Mod(A)²+ 1, where Ψ _S is a certain positive constant depending on the upper bound of Gaussian curvature of the metric S. An application for the minimal surfaces is given.     

Finding the coefficients in the new representation of the solution of the Riemann–Hilbert problem using the Lauricella function

The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function F _D ^(N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.     

Construction of hyperbolic Riemann surfaces with large systoles

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface S_max a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Unexpected spectral asymptotics for wave equations on certain compact spacetimes

We study the spectral asymptotics of wave equations on certain compact spacetimes, where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S¹×S². For the Laplacian on S¹×S², theWeyl asymptotic law gives a growth rate O(s^3/2) for the eigenvalue counting function N(s) = #{λ_j: 0 ≤ λ_j ≤ s}. For the wave operator, there are two corresponding eigenvalue counting functions: N^±(s) = #{λ_j: 0 < ±λ_j ≤ s}, and they both have a growth rate of O(s²). More precisely, there is a leading term π²s²/4 and a correction term of as^3/2, where the constant a is different for N^±. These results are not robust in that if we include a speed of propagation constant to the wave operator, the result depends on number theoretic properties of the constant, and generalizations to S¹ × S^q are valid for q even but not q odd. We also examine some related examples.     

Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator

We obtain an upper estimate N?_χ(M) for the sum Q _N of singular zero multiplicities of the Nth eigenfunction of the Laplace-Beltrami operator on the two-dimensional, compact, connected Riemann manifold M, where _χ M is the Euler characteristic ofM. Stronger estimates, but equivalent asymptotically (N å ∞), are given for the cases of the sphere S ² and the projective plane ?². Asymptotically sharper estimates are shown for the case of a domain on the plane.     

Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R¹⁺¹. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H^s1 and wave data in H^s2 ×H^s2?1 for 3/4?α < s₁ ≤ 0 and ?1/2 < s₂ < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α ≤ 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s₁ = 0 and ?1/2 < s₂ < 1/2 by using the conservation law for the L² norm of u.     

The Riemann constant for a non-symmetric Weierstrass semigroup

The zero divisor of the theta function of a compact Riemann surface X of genus g is the canonical theta divisor of Pic\({^{(g-1)}}\) up to translation by the Riemann constant \({\Delta}\) for a base point P of X. The complement of the Weierstrass gaps at the base point P gives a numerical semigroup, called the Weierstrass semigroup. It is classically known that the Riemann constant \({\Delta}\) is a half period, namely an element of \({\frac{1}{2}\Gamma_\tau}\) , for the Jacobi variety \({\mathcal{J}(X)=\mathbb{C}^{g}/\Gamma_\tau}\) of X if and only if the Weierstrass semigroup at P is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor D₀, we express the relation between the Riemann constant \({\Delta}\) and a half period in the non-symmetric case. We point out an application to an algebraic expression for the Jacobi inversion problem. We also identify the semi-canonical divisor D₀ for trigonal pointed curves, namely with total ramification at P.     

Ergodicity of the 2D Navier-Stokes equations with degenerate multiplicative noise

Consider the two-dimensional, incompressible Navier-Stokes equations on torus T² = [?π, π]² driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS): dw _t = νΔw _t dt + B(Kw _t ,w _t )dt + Q(w _t )dW _t . We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {P _t }_t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140: 41–82 (2005) with a different method, we get an exponential ergodicity under a stronger norm.     

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.     

Parabolic equation with a singular potential

We study the existence of a nonnegative generalized solution of an initial-boundary value problem for the heat equation with a singular potential in an arbitrary bounded domain Ω ? R ⁿ , n ≥ 3, containing the unit ball. We show that if the condition ∝ _Ω V ^n/2+s |x| ^s dx ≤ c _n is satisfied for some s ≥ 0 and c _n = c _n (n, s, Ω) > 0, then the problem in question has a nonnegative solution.     

On the problem of determining the structure of a layered medium and the shape of an impulse source

For the equation of wave propagation in the half-space ? ₊ ² + = {(x, y) ∈ ?² | y > 0} we consider the problem of determining the speed of wave propagation that depends only on the variable y and the shape of a point impulse source on the boundary of the half-space. We show that, under some assumptions on the shape of the source and the structure of the medium, both unknown functions of one variable are uniquely determined by the displacements of boundary points of the medium. We estimate stability of a solution to the problem.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

Grauert’s line bundle convexity,reduction and Riemann domains

We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H⁰(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pⁿ, where n is the complex dimension of X and L is the pull-back of O(1).     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S ₃ onto a genus 2 Riemann surface S ₂ by Hol(S ₃, S ₂). Call two mappings f and g in Hol(S ₃, S ₂) equivalent whenever there exist conformal automorphisms α and β of S ₃ and S ₂ respectively with f ? α = β ? g. It is known that Hol(S ₃, S ₂) always consists of at most two equivalence classes.We obtain the following results: If Hol(S ₃, S ₂) consists of two equivalence classes then both S ₃ and S ₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S ₃, S ₂) there exist anticonformal automorphisms α? and β? with f ? α? = β? ? g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S ₃, S ₂) such that Hol(S ₃, S ₂) consists of two equivalence classes.     

Locally One-Dimensional Difference Scheme for the Third Boundary Value Problem for a Parabolic Equation of the General Form with a Nonlocal Source

We consider a locally one-dimensional scheme for an equation of parabolic type of the general form in a p-dimensional parallelepiped, obtain an a priori estimate for its solution, and prove that the solutions of this scheme converge to a solution of the equation at the rate O(|h|² + τ), where |h|² = h ₁ ² + · · · + h _p ² and p_α, α = 1,..., p, and τ are the steps in the space and time variables. We do not assume that the operator in the leading part of the equation is sign definite.