In this paper, the existence, uniqueness, and stability of (weighted) Stepanov-like pseudo almost automorphic solutions are investigated for the non-autonomous Oseen–Navier–Stokes equations (ONSE) in an exterior domain. Our method is based on the one hand on a combination of differential inequalities and interpolations functors for the case of linearized equations, and on the other hand on fixed-point arguments for semilinear equations.
Let 1 < p <= 2. Using the layer potential method, we establish the unique solvability in L-p-spaces of the Schr & ouml;dinger equation-triangle u+Vu = 0 on a Lipschitz domain subject to a Robin boundary condition. The potential V belongs to the reverse H & ouml;lder class. The case of C-1-domains is also discussed.
The Hardy-Hilbert integral inequality has been the subject of extensive study in recent decades. In this article, we present a two-parameter exponential generalization of this inequality. The associated constant factor involves the upper incomplete gamma function. Interestingly, it can also be expressed in terms of the classical error function. Through precise analysis, we prove the optimality of this constant. We then derive several integral inequalities of various types, some involving primitive functions, while others incorporating auxiliary functions.
We investigate the complex Monge-Amp & egrave;re operator on a bounded strongly pseudoconvex domain of a closed, connected, singular, and locally irreducible complex-analytic subvariety. We first examine the classes epsilon(p) for p > 0 and establish a characterization of their images under the complex Monge-Amp & egrave;re operator. This result answers a question posed by N. Q. Dieu, T. V. Long. We then turn to the weighted energy classes epsilon(chi) (ohm), consisting of negative plurisubharmonic functions with finite chi-energy, and provide a precise characterization of their images under the complex Monge-Amp & egrave;re operator, where chi is a convex increasing function satisfying chi(0) = 0 and chi(-infinity) = -infinity.
Let f is an element of C{X, Y } be a convergent series and let f = 0 be a germ of an isolated plane curve singularity. To describe the singularity we apply the Eggers-P & lstrok;oski tree which is graphically equivalent to the Eggers tree. Instead of the contact exponent between branches (as in the original construction) we use P & lstrok;oski's logarithmic distance between them. An advantage of this approach is that the tree can be constructed without fixing any coordinate system. We call a germ a dandelion if all branches go through all the black vertices of the tree of f. Then these balls form a single chain. We consider the jacobian Newton polygon NJ(f) introduced by Bernard Teissier. We say that the jacobian Newton polygon determines the equisingularity class of f = 0 among all singularities if for any germ g = 0 the equality NJ(f) = NJ(g) implies the equisingularity of the germs f = 0 and g = 0. Evelia Garcia Barroso and Janusz Gwo & zacute;dziewicz proved that every branch f = 0 satisfies this property. In this way they obtained a new criterion of irreducibility. We present an analogous result for dandelions.
We propose the velocity-vorticity model of the three-dimensional (3D) generalized Navier-Stokes equations with exponential damping terms and then study the asymptotic behavior of solutions in a periodic bounded domain. First, we study the global well-posedness using the Faedo-Galerkin approximation method. Then, we investigate the asymptotic behavior of weak solutions via attractors and their properties using the theory of the evolutionary system which was recently developed by Cheskidov, Foias and Lu. Finally, we investigate determining wavenumbers.
We prove that singular integrals T with standard Calder & oacute;n-Zygmund kernel and S with variable kernel are bounded on appropriate weighted Hardy spaces. Similar results hold for the commutators Tb and Sb when b belongs to a suitable subspace of BMO(Rn).
Catenoids are rotationally symmetric hypersurfaces with zero mean curvature in Minkowski space. This paper considers three generalizations of catenoids. First, we construct a generalization by replacing the rotational orbits of catenoids with minimal submanifolds within these orbits. Second, we present another generalization: O(m) & times; O1(n)invariant hypersurfaces in Lm+n+2 with zero mean curvature, where O1(n) is the group of Lorentz transformations, and we classify all profile curves. Finally, we consider two types of birotationally symmetric functions. These functions are the sum of two functions, each depending on a radial variable, and their graphs have zero mean curvature. If the graph is not a hyperplane, one of the functions is linear, while the other represents a catenoid of the corresponding dimension under rotation.
We deal with some embedding properties between the homogeneous Besov-Triebel-Lizorkin-type spaces of bounded functions and the homogeneous Sobolev spaces built on the Morrey scale.
Oscillatory properties of perturbed half-linear differential equations are investigated. We make use of the modified Riccati technique. A certain linear differential equation associated with the modified Riccati equation plays an important part. For a perturbed half-linear Riemann-Weber differential equation, new oscillation criteria are obtained.
We study the thinness of sets with respect to weighted local Riesz capacities, where the associated weights are of local Muckenhoupt class. The thinness is characterized in terms of Wolff potentials. We also obtain the Kellogg property and Choquet theorem as applications.
Several recent papers investigate the way in which the number of zeros of a complex-valued harmonic function depends on its coefficients by analyzing specific simple families. These families share two features: (1) the critical curve separating the sense-preserving and sense-reversing regions is a circle, and (2) the image of that circle is a well-understood parametric curve. In such cases, the harmonic analogue of the Argument Principle can be applied to count the zeros. In this paper, we illustrate the strengths and limitations of these techniques; we construct a new family of complex-valued harmonic functions with poles having some of these features. We obtain detailed zero-counting theorems for two subfamilies, and we illustrate how to obtain less detailed zero-counting theorems for the general family.
Consider a semi-quasi-homogeneous complex plane curve singularity f with Milnor number mu (f ). In Theorem 1.1 we list integers which can be attained as Milnor numbers of deformations of the given semi-quasi-homogeneous singularity. The result follows mainly from combinatorial considerations and geometric interpretation of Euclid's algorithm via the classic relation between Milnor and Newton numbers.
Let (X, omega) be a compact Hermitian manifold of dimension n. We show that all (omega, m)-subharmonic functions are L-p-integrable on X, for any p < n/n-m.
The boundary of every relatively compact Stein domain in a complex manifold of dimension at least 2 is connected. No assumptions on the boundary regularity are necessary. The same proofs hold also for q-complete domains, and in the context of almost complex manifolds as well.
It is well known that the reduction of an m-primary ideal q in the Noetherian local ring (R, m) with infinite residue field R/m may be given in the form of a sufficiently general linear combination of its generators. We give a condition for the existence of such a reduction in terms of the sum of the degrees of the prime divisors of the ideal fiber cone Fq(R) in the case of any Noetherian local ring.
We establish a connection between Pick bodies and invariant functions. We demonstrate that an invariant function can be associated with any Pick body, which determines the solvability of a given Pick interpolation problem and generalizes the Carathéodory pseudodistance. A complete description of this invariant function is provided for the open unit disc, and it is shown that it leads to another invariant function which can be regarded as a generalized Lempert function. It is also proved that these two invariant functions are equal if certain geodesics can be found. Lastly, we show that, in a very special case, a result analogous to Lempert’s theorem holds for the bidisc and the tridisc.
We study graded analytic expansions associated with suitable Laurent power series using Brion type maps. These methods allow us to analytically compute Poincare series, both in the algebraic setting of polyhedra and in the topological setting of singularities.
