SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.

ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS

We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary?Ωis split into two disjoint parts and possesses different transmission conditions.Using the variational method,we obtain the well posedness of the interior transmission problem,which plays an important role in the proof of the discreteness of eigenvalues.Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n≡1,where a fourth order differential operator is applied.In the case of n■1,we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

UNIQUENESS OF INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION BY PHASELESS FAR FIELD PATTERN

In this paper,we establish the unique determination result for inverse acoustic scattering of a penetrable obstacle with a general conductive boundary condition by using phaseless far field data at a fixed frequency.It is well-known that the modulus of the far field pattern is invariant under translations of the scattering obstacle if only one plane wave is used as the incident field,so it is impossible to reconstruct the location of the underlying scatterers.Based on some new research results on the impenetrable obstacle and inhomogeneous isotropic medium,we consider different types of superpositions of incident waves to break the translation invariance property.

Volume Difference Inequalities for the Polars of Mixed Complex Projection Bodies

In this paper, based on the notion of mixed complex projection and generalized the recent works of other authors, we obtain some volume difference inequalities containing Brunn-Minkowski type inequality, Minkowski type inequality and Aleksandrov-Fenchel inequality for the polars of mixed complex projection bodies.

The Shephard Type Problems for General L_p-Centroid Bodies

In this paper, combining with the L_p-dual geominimal surface area and the general L_p-centroid bodies, we research the Shephard type problems for general L_p-centroid bodies.

Numerical simulation of two droplets impacting upon a dynamic liquid film

The impact of droplets on the liquid film is widely involved in industrial and agricultural fields.In recent years,plenty of works are limited to dry walls or stationary liquid films,and the research of multi-droplet impact dynamic films is not sufficient.Based on this,this paper employs a coupled level set and volume of fluid(CLSVOF) method to numerically simulate two-droplet impingement on a dynamic liquid film.In our work,the dynamic film thickness,horizontal central distance between the droplets,droplets’ initial impact speed,and simultaneously the flow velocity of the moving film are analyzed.The evolution phenomenon and mechanism caused by the collision are analyzed in detail.We find that within a certain period of time,the droplet spacing does not affect the peripheral crown height;when the droplet spacing decreases or the initial impact velocity increases,the height of the peripheral crown increases at the beginning,and then,because the crown splashed under Rayleigh-Plateau instability,this results in the reduction of the crown height.At the same time,it is found that when the initial impact velocity increases,the angle between the upstream peripheral jet and the dynamic film becomes larger.The more obvious the horizontal movement characteristics,the more restrained the crown height;the spread length increases with the increase of the dynamic film speed,droplet spacing and the initial impact velocity.When the liquid film is thicker,more fluid enters the crown,due to the crown being unstable,the surface tension is not enough to overcome the weight of the rim at the end of the crown,resulting in droplets falling off.

A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION

In this paper,we consider a class of fractional problem with subcritical perturbation on a bounded domain as follows:(P_(k)){(−△)^(s)u = g(x)[(u − k)+]^(q−1) +u^(2^(∗)_(s)−1), x ∈Ω ,u > 0, x∈Ω ,u = 0, x ∈Ω R^(N)\Ω. We prove the existence of nontrivial solutions u_(k) of(P_(k))for each k∈(0,∞).We also investigate the concentration behavior of the solutions u_(k) as k→∞.

The Generators and Relations for Little q-Schur Algebra u_(k)(3.5)

In this paper,we investigate the presentation for little q-Schur algebra u_(k)(3.5)at odd roots.The Poincaré-Birkhoff-Witt(PBW)basis of u_(k)(3.5)is obtained through the basis of q-Schur algebra U(3.5).Then we present a new set for generators and relations for u_(k)(3.5)by PBW basis.

Busemann-Petty Type Problem for the General L_(p)-Centroid Bodies

Lutwak showed the Busemann-Petty type problem(also called the Shephard type problem)for the centroid bodies.Grinberg and Zhang gave an affirmation and a negative form of the Busemann-Petty type problem for the L_(p)-centroid bodies.In this paper,we obtain an affirmation form and two negative forms of the BusemannPetty type problem for the general L_(p)-centroid bodies.

Regular and p-Regular Orbits of Solvable Linear Groups,Ⅱ

Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.

A Type of Shephard Problems for Lp Blaschke-Minkowski Homomorphisms

The Blaschke-Minkowski homomorphisms was defined by Schuster.Recently,Wang extended its concept to Lp version.In this paper,we obtain affirmative and negative forms of the Shephard type problems for Lp geominimal surface areas with respect to the Lp Blaschke-Minkowski homomorphisms.

A Type of Busemann-Petty Problems for Blaschke-Minkowski Homomorphisms

Schuster introduced the notion of Blaschke-Minkowski homomorphisms and first considered Busemann-Petty type problems. In this paper, we study the Busemann-Petty type problems for the geominimal surface area with respect to Blaschke-Minkowski homomorphisms.

Closure Extended Pseudocomplemented MS-Algebras

The variety CPMS of closure extended pseudocomplemented MS-algebras consists of the algebras(L;∧,∨,°,*,+,0,1)of type(2,2,1,1,1,0,0),where(L;∧,∨,°,*,+,0,1)is a pseudocomplemented MS-algebra,+is a lattice endomorphism on L with x≤x^(+)=x^(++)and the operations x→x°,x→x^(*)and x→x^(+)are linked by the identities x^(+*)=x^(*+)and x^(+)°=x°^(+).In this paper,we characterize congruences on a CPMS-algebra,and show that there are precisely eight non-isomorphic subdirectly irreducible nontrivial algebras in the class of these algebras and give a complete description of them.

Star Dual of Radial Blaschke-Minkowski Homomorphism

The notion of radial Blaschke-Minkowski homomorphism was introduced by Schuster. In this paper, we study the radial Blaschke and radial Minkowski linear combinations, respectively, and give several inequalities under the notion of radial Blaschke Minkowski homomorphism in forms of star dual.

Extremum Values of Asymmetric Lp-Difference Bodies for Quermassintegrals and Dual Quermassintegrals

Based on Lutwak＇s the notion of Lp-difference bodies, Wang and Ma introduced asymmetric Lp-difference bodies and gave their extremum values for volumes. In this paper, we establish the extremum value inequalities for the quermassintegrals and dual quermassintegrals of asymmetric Lp-difference bodies and their polars, respectively.

Some Brunn-Minkowski Type Inequalities for the(p,q)-Mixed Volumes

Lutwak,Yang and Zhang defined the(p,q)-mixed volumes very recently.In this paper,some Brunn-Minkowski type inequalities on the(p,q)-mixed volumes are established based on the general Lp-projection bodies and the general Lp-centroid bodies.

Ground States of K-component Coupled Nonlinear Schrödinger Equations with Inverse-square Potential

In this paper,the authors study ground states for a class of K-component coupled nonlinear Schrödinger equations with a sign-changing potential which is periodic or asymptotically periodic.The resulting problem engages three major difficulties:One is that the associated functional is strongly indefinite,the second is that,due to the asymptotically periodic assumption,the associated functional loses the Z^(N)-translation invariance,many effective methods for periodic problems cannot be applied to asymptotically periodic ones.The third difficulty is singular potentialμ/(|x|)^(2),which does not belong to the Kato’s class.These enable them to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential.

Hierarchical Absorbing Interface Conditions for Wave Propagation on Non-Uniform Meshes

In this paper,we propose hierarchical absorbing interface conditions to solve the problem of wave propagation in domains with a non-uniform space discretization or grid size inhomogeneity using Pad´e Via Lanczos(PVL)method.The proposed interface conditions add an auxiliary variable in the wave system to eliminate the spurious reflection at the interface between regions with different mesh sizes.The auxiliary variable with proper boundary condition can suppress the spurious reflection by cancelling the boundary source term produced by the space inhomogeneity in variational perspective.The new hierarchical interface conditions with the help of PVL implementation can effectively reduce the degree of freedom in solving the wave propagation problem.

Large Time Behavior of Solutions to the Generalized Hall-Magnetohydrodynamics System

In this paper,we investigate the large time behavior of solutions to the three dimensional generalized Hall-magnetohydrodynamics(Hall-MHD)system in the spacesχ^(s)(R^(3)).We obtain that the temporal decay rate is‖(u,b)(t)‖χ^(1−2α)+‖(u,b)(t)‖χ^(1−2β)+‖(u,b)(t)‖χ^(2−2α)+‖(u,b)(t)‖χ^(2−2β)≤(1 t)^(-(5-4max{α,β}/4max{α,β})with 1/2≤α,β≤1 for the small global solution by using Fourier splitting method.The parametersαandβare the fractional dissipations corresponding to the velocity and magnetic field,respectively.