The Green function for the mixed problem for the linear Stokes system in domains in the plane

We construct the Green function for the mixed boundary value problem for the linear Stokes system in a two‐dimensional Lipschitz domain.     

Solvability near the characteristic set for a class of first‐order linear partial differential operators

In this work we deal with solvability of first‐order differential equations in the form , where L is a planar complex vector field, elliptic everywhere except along a simple closed curve Σ on which it is tangent and vanishes of order . In contrast with the local solvability, it is shown that the zero order term p has influence in the solvability in a full neighborhood of Σ.     

On a non‐stationary fluid flow problem in an infinite periodic pipe

We study linearized, non‐stationary Navier–Stokes type equations with the given flux in an infinite pipe periodic of period length L with respect to . The existence and uniqueness of the solution is proved. Moreover, the convergence of the solution in a finite pipe of length to the L‐periodic solution as is investigated.     

Multiple solutions for a noncooperative Kirchhoff‐type system involving the fractional p‐Laplacian and critical exponents

In this paper, we use the Limit Index Theory due to Li 19 and the fractional version of concentration compactness principle to study the multiplicity of solutions for a class of noncooperative fractional p‐Laplacian elliptic system with homogeneous Dirichlet boundary conditions involving the critical exponents.     

Linear Sturm–Liouville problems with general homogeneous linear multi‐point boundary conditions

In this paper, we study second order linear Sturm–Liouville problems involving one or two homogeneous linear multi‐point boundary conditions in the most general form. We obtain conditions for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Furthermore, we reveal the interlacing relations between the eigenvalues of such Sturm–Liouville problems and those of Sturm–Liouville problems with certain two‐point separated boundary conditions.     

A Szegő limit theorem for translation‐invariant operators on polygons

We prove Szeg?‐type trace asymptotics for translation‐invariant operators on polygons. More precisely, consider a Fourier multiplier on with a sufficiently decaying, smooth symbol . Let be the interior of a polygon and, for , define its scaled version . Then we study the spectral asymptotics for the operator , the spatial restriction of A onto : for entire functions h with we provide a complete asymptotic expansion of as . These trace asymptotics consist of three terms that reflect the geometry of the polygon. If P is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.     

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k‐Einstein (in the sense that their 2k‐Ricci tensor is constant) or have constant 2k‐Gauss‐Bonnet curvature. The results hold for a family of manifolds containing all non‐flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

Korovkin‐type results for multivariate functions which are periodic with respect to one variable

Let K be a compact metric space and let denote the real Banach space of all continuous functions which are 2π‐periodic with respect to the second variable. We prove the following Korovkin‐type result: Let be a continuous algebraic separating function such that for all , and let be a sequence of positive linear operators. If uniformly with respect to and uniformly on for all , then uniformly on for every . As a corollary we deduce: If , then uniformly on for every if and only if uniformly on for every , where and .     

Limit‐point/limit‐circle classification for Hain–Lüst type equations

Hain–Lüst equations appear in magnetohydrodynamics. They are Sturm–Liouville equations with coefficients depending rationally on the eigenvalue parameter. In this paper such equations are connected with a 2 × 2 system of differential equations, where the dependence on the eigenvalue parameter is linear. By means of this connection Weyl's fundamental limit‐point/limit‐circle classification is extended to a general setting of Hain–Lüst‐type equations.     

Characterizing the Phragmén‐Lindelöf condition for evolution on algebraic curves

For an algebraic curve V in ${mathbb C}^k times {mathbb C}^n$ it is investigated when it satisfies the Phragmén‐Lindelöf condition PL(ω) of evolution in certain classes of ultradifferentiable functions. Necessary as well as sufficient conditions are obtained which lead to a complete characterization for curves in ${mathbb C}times {mathbb C}^n$. Some examples illustrate how these results can be applied. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Well‐posedness for the Cauchy problem of the modified Hunter–Saxton equation in the Besov spaces

This paper is concerned with the Cauchy problem of the modified Hunter‐Saxton equation. The local well‐posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces H^s) by using Littlewood‐Paley decomposition and transport equation theory. Moreover, the local well‐posedness in critical case (with ) is considered.     

Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

On the Hyers–Ulam stabilities of functional equations on ‐Banach spaces

The purpose of this article is to generalize the theory of stability of functional equations to the case of n‐Banach spaces. In this article, we prove the generalized Hyers–Ulam stabilities of the Cauchy functional equations, Jensen functional equations and quadratic functional equations on n‐Banach spaces.     

Interpolation of compact bilinear operators among quasi‐Banach spaces and applications

We study the interpolation properties of compact bilinear operators by the general real method among quasi‐Banach couples. As an application we show that commutators of Calderón–Zygmund bilinear operators are compact provided that , and .     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.