On unique continuation for Schrödinger operators of fractional and higher orders

In this note we study the property of unique continuation for solutions of , where V is in a function class of potentials including for . In particular, when , our result gives a unique continuation theorem for the fractional Schrödinger operator in the full range of α values.     

Bilinear decompositions for the product space

In this paper, we improve a recent result by Li and Peng on products of functions in and , where is a Schrödinger operator with V satisfying an appropriate reverse Hölder inequality. More precisely, we prove that such products may be written as the sum of two continuous bilinear operators, one from into , the other one from into , where the space is the set of distributions f whose grand maximal function satisfies      

Determinants of classical SG‐pseudodifferential operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG‐pseudodifferential operators on and suitable manifolds, using a finite‐part integral regularization technique. This allows us to define a zeta‐regularized determinant for parameter‐elliptic operators , , . For , the asymptotics of as and of as are derived. For suitable pairs we show that coincides with the so‐called relative determinant .     

Riesz minimal energy problems on ‐manifolds

In , , we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where , for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless ‐dimensional ‐manifolds , , where , each being charged with Borel measures with the sign prescribed. We show that the Gauss variational problem over a convex set of Borel measures can alternatively be formulated as a minimum problem over the corresponding set of surface distributions belonging to the Sobolev–Slobodetski space , where and . An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Γ. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. Numerical results are presented to illustrate the approach.     

On the ‐action on G‐semistable locally free sheaves on

We show that the divisor of jumping lines of any , the moduli space of Gieseker‐semistable locally free sheaves of rank 2 on with , is reduced for . By a lemma of Artamkin this implies, that there are exactly ‐orbits in , the subset of those , which are trivial at a certain line .     

Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces , the splitting subspaces , the quasi‐splitting subspaces and the orthogonally‐closed subspaces . It is well‐known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion i.e. then and . On the other extreme, when i.e. then and . Motivated by this and in contrast to it, we show that in general the codimension of S in bears very little relation to the properties of these families. In particular, we show that the equalities and can hold for inner product spaces with arbitrary codimension in . At the end we also contribute to the study of the algebraic structure of by testing it for the Riesz interpolation property. We show that may fail to enjoy the Riesz interpolation property in both extreme situations when S is “very small” (i.e. and when S is ‘very big’ (i.e. .     

Uniqueness results for semilinear elliptic systems on

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form As an application we consider the nonlinear Schrödinger system for and exponents q which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi‐component nonlinear Schrödinger system is given.     

Oscillation criteria for higher order nonlinear dynamic equations

In this paper, we will consider the higher‐order functional dynamic equations of the form on an above‐unbounded time scale , where and , . The function is a rd‐continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.     

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.     

Fractional integration operators of variable order: continuity and compactness properties

Let be a Lebesgue‐almost everywhere positive function. We consider the Riemann‐Liouville operator of variable order defined by as an operator from to . Our first aim is to study its continuity properties. For example, we show that is always bounded (continuous) in provided that . Surprisingly, this becomes false for . In order to be bounded in L₁[0, 1], the function has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of . We characterize functions for which is a compact operator and for certain classes of functions we provide order‐optimal bounds for the dyadic entropy numbers .     

Cohomological characterisation of monads

Let X be an n‐dimensional smooth projective variety with an n‐block collection , with , of coherent sheaves on X that generate the bounded derived category . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the form where . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V₅. We construct a family of monads on a Segre variety and apply our main result to this family.     

Operators of Hadamard type on spaces of smooth functions

For an open set we study the algebra of continuous linear operators on admitting the monomials as eigenvectors. We give a concrete representation of these operators and evaluate it explicitly for the unit ball and the whole of . We also study the topology of and the algebra of eigenvalue sequences.     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .     

On the order of the automorphism group of foliations

Let be a holomorphic foliation with ample canonical bundle on a smooth projective surface X. We obtain an upper bound on the order of its automorphism group which depends only on and provided this group is finite. Here, and are the canonical bundles of and X, respectively.     

Nonlinear differential identities for cnoidal waves

This article presents a family of nonlinear differential identities for the spatially periodic function , which is essentially the Jacobian elliptic function with one non‐trivial parameter . More precisely, we show that this function fulfills equations of the form for all . We give explicit expressions for the coefficients and for given s. Moreover, we show that for any s the set of functions constitutes a basis for . By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well‐known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.     

Regularity theory for a class of quasilinear equations

This paper addresses the analysis of the weak solution of in a bounded domain Ω subject to the boundary condition on , when the data f belongs to and . We prove existence and uniqueness of solution for this problem in the Nikolskii space . Moreover, we obtain energy estimates regarding the Nikolskii norm of ω in terms of the norm of f.     

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for d‐variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d‐dimensional torus or the d‐dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci‐lattice for bivariate periodic functions from in the case , and . A non‐periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst‐case error.     

Kolmogorov equations for randomly perturbed generalized Newtonian fluids

We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.e. a polynomial of degree , , i.e. the shear thinning case. We prove that the associated Kolmogorov operator K admits at least one infinitesimally invariant measure μ satisfying certain exponential moment estimates. Moreover, K is L²‐unique w.r.t. μ provided , where is the second root of , approximately .     

On the influence of second order uniformly elliptic operators in nonlinear problems

The purpose of the present paper is to discuss the role of second order elliptic operators of the type on the existence of a positive solution for the problem involving critical exponent where Ω is a smooth bounded domain in , , and λ is a real parameter. In particular, we show that if the function has an interior global minimum point x₀ such that is comparable to , where and is the identity matrix of order n, then the range of values of λ for which the problem above has a positive solution can change drastically from to .     

Agmon‐type estimates for a class of jump processes

In the limit we analyse the generators of families of reversible jump processes in associated with a class of symmetric non‐local Dirichlet‐forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being or just Lipschitz. Our estimates are analogous to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice . Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques.