Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in ${\mathbb{R}}^n$, with weights that are positive powers of the distance to the boundary.     

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$-framework

We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the $L^r$-setup. The mathematical model used here is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated using three types of boundary conditions: the conditions of periodicity on the “lower” and “upper” parts of the boundary, the Dirichlet boundary conditions on the “inflow” and on the profile and an artificial “do nothing”-type boundary condition on the “outflow.” Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in ${\mathbf{W}}^{1,r}(\mathrm{\Omega })$ and its continuous dependence on the data. We explain the sense in which the “do nothing” boundary condition on the “outflow” is satisfied.     

Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of Oden, Sung, and Wang [Trans. Amer. Math. Soc. 351 (1999), no. 9, 3533–3548] to $L^p$-Ricci curvature assumptions, $p>n/2$. To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest.     

Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces

We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces. To do this, we prove the weak–weak type modular inequality of the Hardy–Littlewood maximal operator with respect to the Young function. Orlicz–Morrey spaces contain $L^p$ spaces ($1\le p\le \infty$), Orlicz spaces, and generalized Morrey spaces as special cases. Hence, we get necessary and sufficient conditions on these function spaces as corollaries.     

Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces

We show that if $S={\{f_t:Y\to Y\}}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space $(Y,d)$ that satisfies some geometric properties, then there exists $\xi \in \partial Y$ such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics.     

Foliations by curves on threefolds

We study the conormal sheaves and singular schemes of one-dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ-stable whenever the tangent bundle $TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on ${\mathbb{P}}^3$ and on a smooth quadric hypersurface $Q_3\subset {\mathbb{P}}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q₃.     

Unbounded operators having self-adjoint,subnormal, or hyponormal powers

We show that if a densely defined closable operator A is such that the resolvent set of A² is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\ge 2$, then T is normal. Hence a closed subnormal operator T such that $T^n$ is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that $A^p$ and $A^q$ are self-adjoint for some coprime numbers p and q, then A must be self-adjoint.     

On the convergence properties of sampling Durrmeyer-type operators in Orlicz spaces

Here, we provide a unifying treatment of the convergence of a general form of sampling-type operators, given by the so-called sampling Durrmeyer-type series. The main result consists of the study of a modular convergence theorem in the general setting of Orlicz spaces $L^{\phi }(\mathbb{R})$. From the latter theorem, the convergence in $L^p(\mathbb{R})$, in $L^{\alpha }{\log }^{\beta }L$, and in the exponential spaces can be obtained as particular cases. For the completeness of the theory, we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, in case of bounded continuous and bounded uniformly continuous functions; in this context, we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.     

Besov estimates for weak solutions of a class of quasilinear parabolic equations with general growth

In this paper, we obtain the local higher fractional differentiability regularity estimates in Besov spaces of weak solutions for the quasilinear parabolic equations with general growth under some proper conditions on $a,A$, and $\mathbf{F}$. Moreover, we would like to point out that the results in the present work improve the known results for such parabolic equations.     

On the volume functional of compact manifolds with harmonic Weyl tensor

The main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary $\partial M$ and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere ${\mathbb{S}}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form ${\mathbb{R}}^n$, ${\mathbb{H}}^n$, and ${\mathbb{S}}^n$. To this end, we first conclude the classification of such critical metrics under the Bach-flat assumption and then we prove that both geometric conditions are equivalent in this situation.     

A characterization of well-posedness for the second order abstract Cauchy problems with finite delay

In this work, we introduce a strongly continuous one-parameter family of bounded linear operators that completely describes the well-posedness of a second order abstract differential delay equation in the initial history space $L^p([-r,0];X)$, $r>0$. This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation.     

Curvature loci of 3-manifolds

We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular 3-manifolds in ${\mathbb{R}}^6$ and singular corank one 3-manifolds in ${\mathbb{R}}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by four ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3-manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.     

Transience of symmetric nonlocal Dirichlet forms

We establish transience criteria for symmetric nonlocal Dirichlet forms on $L^2({\mathbb{R}}^d;\mathrm{d}x)$ in terms of the coefficient growth rates at infinity. Applying these criteria, we find a necessary and sufficient condition for recurrence of Dirichlet forms of symmetric stable-like with unbounded/degenerate coefficients. This condition indicates that both of the coefficient growth rates of small and big jump parts affect the sample path properties of the associated symmetric jump processes.     

Dimension-free square function estimates for Dunkl operators

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in ${\mathbb{R}}^d$ is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the $L^p$ boundedness is studied for all $p\in (1,\infty )$. For $p\in (1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in [2,\infty )$, we restrict to a particular case when the corresponding Weyl group is isomorphic to ${\mathbb{Z}}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.     

Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities

In this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term: $$-\mathrm{\Delta }u-\frac{\mu }{{|x|}^2}u=\lambda u+{|u|}^{2^{\ast }-2}u+\nu {|u|}^{p-2}u\text{in}{\mathbb{R}}^N,N\ge 3,$$ with prescribed mass $${\int }_{{\mathbb{R}}^N}{u}^2=a^2,$$ where 2* is the Sobolev critical exponent. For a L²-subcritical, L²-critical, or L²-supercritical perturbation ${\nu |u|}^{p-2}u$, we prove several existence results of normalized ground state when $\nu \ge 0$ and non-existence results when $\nu \le 0$. Furthermore, we also consider the asymptotic behavior of the normalized solutions u as $\mu \to 0$ or $\nu \to 0$.     

On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let $X^{•}=(X,D_X)$ be a pointed stable curve over an algebraically closed field of characteristic $p>0$ and ${\mathrm{\Pi }}_{X^{•}}$ the admissible fundamental group of $X^{•}$. We prove that there exists a group-theoretical algorithm, whose input datum is the admissible fundamental group ${\mathrm{\Pi }}_{X^{•}}$, and whose output data are the topological and the combinatorial structures associated with $X^{•}$. This result can be regarded as a mono-anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.     

Optimal control of generalized multiobjective games with application to traffic networks modeling

The purpose of this paper is to study some new results on the existence and convergence of the solutions to controlled systems of generalized multiobjective games, controlled systems of traffic networks, and optimal control problems (OCPs). First, we introduce the controlled systems of generalized multiobjective games and establish the existence of the solutions for these systems using Browder-type fixed point theorem in the noncompact case and the $C_i$-quasi-concavity. Results on the convergence of controlled systems of the solutions for such problems using the auxiliary solution sets and the extended $C_i$-convexity of the objective functions are studied. Second, we investigate OCPs governed by generalized multiobjective games. The existence and convergence of the solutions to these problems are also obtained. Finally, as a real-world application, we consider the special case of controlled systems of traffic networks. Many examples are given for the illustration of our results.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.     

Taylor's inequalities in Orlicz–Sobolev type spaces

In this paper, we obtain inequalities involving the Taylor polynomial and weak derivatives of a function in an Orlicz–Sobolev type space. Moreover, we show that any such function can be expanded in a finite Taylor series almost everywhere. As a consequence, we prove that the coefficients of any extended best polynomial $L^{\mathrm{\Phi }}$-approximation of a function on a ball almost everywhere converge to the weak derivatives of such a function when the radius tends to 0. Lastly, we get a mean convergence result of such coefficients.