Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let be an open and closed subset of the boundary of M. We say that has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance from a point to is bounded uniformly in x (and hence, in particular, intersects all connected components of M). For manifolds with finite width, we prove a Poincaré inequality for functions vanishing on , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces , .     

Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let be Hilbert spaces such that V is continuously and densely embedded in H and let be the operator associated with a bounded H‐elliptic form for all . Suppose is closed and convex and the orthogonal projection onto . Given and , we investigate when the solution of the non‐autonomous evolutionary problem remains in and show that this is the case if for a.e. . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator

In this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary conditions on a Banach space X and its associated (abstract) Dirichlet‐to‐Neumann operator N acting on a “boundary” space . Our approach is based on similarity transformations and perturbation arguments and allows to split A into an operator A₀₀ with Dirichlet‐type boundary conditions on a space X₀ of states having “zero trace” and the operator N. If A₀₀ generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that A generates an analytic semigroup on X if and only if N does so on . Here we assume that the (abstract) “trace” operator is bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Exterior Navier–Stokes flows for bounded data

We prove unique existence of mild solutions on for the Navier–Stokes equations in an exterior domain in , subject to the non‐slip boundary condition.     

Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi‐valued hyperbolic differential inclusions with memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: and prove the general decay estimates of equivalent energy.     

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.     

Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Fréchet derivative of at . We prove that is one‐to‐one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W^{2, 2}‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.     

Indirect stability of the wave equation with a dynamic boundary control

In this paper, we consider a damped wave equation with a dynamic boundary control. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case for the unit disk. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach. In a first step, by giving some sufficient conditions on the boundary of our domain and by using the exponential decay of the wave equation with a standard damping, we prove a polynomial decay in of the energy. In a second step, under appropriated conditions on the boundary, called the multiplier control conditions, we establish a polynomial decay in of the energy. Later, we show in a particular case that such a polynomial decay is available even if the previous conditions are not satisfied. For this aim, we consider our system on the unit square of the plane. Using a method based on a Fourier analysis and a specific analysis of the obtained 1‐d problems combining Ingham's inequality and an interpolation method, we establish a polynomial decay in of the energy for sufficiently smooth initial data. Finally, in the case of the unit disk, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained decay is optimal in the domain of the operator.     

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Since the pioneer work of Leray [23] and Hopf [17], Stokes and Navier–Stokes problems have been often studied with Dirichlet boundary condition. Nevertheless, in the opinion of engineers and physicists such a condition is not always realistic in industrial and applied problems of origin. Thus arises naturally the need to carry out a mathematical analysis of these systems with different boundary conditions, which best represent the underlying fluid dynamic phenomenology. Based on the study of the complex and fractional powers of the Stokes operator with pressure boundary condition, we carry out a systematic treatment of the Stokes problem with the corresponding boundary conditions in ‐spaces.     

Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term

In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function , nonlinear function , the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions and .     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Decomposition of integral self‐affine multi‐tiles

In contrast to the situation with self‐affine tiles, the representation of self‐affine multi‐tiles may not be unique (for a fixed dilation matrix). Let be an integral self‐affine multi‐tile associated with an integral, expansive matrix B and let K tile by translates of . In this work, we propose a stepwise method to decompose K into measure disjoint pieces  satisfying in such a way that the collection of sets forms an integral self‐affine collection associated with the matrix B and this with a minimum number of pieces . When used on a given measurable subset K which tiles by translates of , this decomposition terminates after finitely many steps if and only if the set K is an integral self‐affine multi‐tile. Furthermore, we show that the minimal decomposition we provide is unique.     

On functions with one‐dimensional holomorphic extension property in circular domains

In this paper we consider continuous functions given on the boundary of a circular bounded domain D in , , and having the one‐dimensional holomorphic extension property along family of complex lines, passing through a finite number of points of D. We study the problem of existence of holomorphic extension of such functions into D.     

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.     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

The self‐affine measure corresponding to an expanding matrix and the digit set in the space is supported on the spatial Sierpinski gasket, where are the standard basis of unit column vectors in and . In the case and , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.     

Average distances between points in graph‐directed self‐similar fractals

We study several distinct notions of average distances between points belonging to graph‐directed self‐similar subsets of . In particular, we compute the average distance with respect to graph‐directed self‐similar measures, and with respect to the normalised Hausdorff measure. As an application of our main results, we compute the average distance between two points belonging to the Drobot–Turner set with respect to the normalised Hausdorff measure, i.e. we compute where s denotes the Hausdorff dimension of and is the s‐dimensional Hausdorff measure; here the Drobot–Turner set (introduced by Drobot & Turner in 1989) is defined as follows, namely, for positive integers N and m and a positive real number c, the Drobot–Turner set is the set of those real numbers for which any m consecutive base N digits in the N‐ary expansion of x sum up to at least c. For example, if , and , then our results show that where is the unique positive real number such that .     

Operators from the Hardy space to the α‐Bloch space

Let be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space into the α‐Bloch space on . Also, we show the multiplication operator from into is bounded. Finally, we show that there exist no isometric composition operators.