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In the present paper, the full range Strichartz estimates for homogeneous Schrödinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data, which lies in H1 Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables. The main novelty of this paper relies on careful constructions of a tangentially weighted analytic energy functional and a specially designed good unknown for the reformulated system. The result extends that of Paicu-Zhang in [Paicu, M. and Zhang, P., Global existence and the decay of solutions to the Prandtl system with small analytic data, Arch. Ration. Mech. Anal., 241(1), 2021, 403–446]. from the two dimensional case to the three dimensional axially symmetric case, but the method used here is a direct energy estimates rather than Fourier analysis techniques applied there. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider an approximation to the isentropic planar Magneto-hydrodynamics (MHD for short) equations by a kind of relaxed Euler-type system. The approximation is based on the generalization of the Maxwell law for non-Newtonian fluids together with the Maxwell correction for the Ampère law, hence the approximate system becomes a first-order quasilinear symmetrizable hyperbolic systems with partial dissipation. They establish the global-in-time smooth solutions to the approximate Euler-type equations in a small neighbourhood of constant equilibrium states and obtain the global-in-time convergence towards the isentropic planar MHD equations. In addition, they also establish the global-in-time error estimates of the limit based on stream function techniques and energy estimates for error variables. [ABSTRACT FROM AUTHOR]

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of ℒ (H) , where ℒ (H) denotes the set of all adjointable operators on H. The C*-subalgebras of ℒ (H) generated by elements in {P − P ∧ Q, Q − P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π ˜ , X) of i(P, Q, H) such that π ˜ (P − P ∧ Q) = π (P) − π (P) ∧ π (Q) , π ˜ (Q − P ∧ Q) = π (Q) − π (P) ∧ π (Q). When (P, Q) is semi-harmonious, that is, ℛ (P + Q) ¯ and ℛ (2 I − P − Q) ¯ are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I − Q, I − P, H) are unitarily equivalent via a unitary operator in ℒ (H) . A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I − Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C*-modules are also provided. [ABSTRACT FROM AUTHOR]

In this paper, the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation Δ p , f u + c u σ = 0 on noncompact smooth metric measure space, where c is a nonnegative constant, and p, σ (1 < p ≤ 2, σ ≤ p - 1) are real constants. Moreover, by the gradient estimate, they can get the corresponding Liouville theorem and Harnack inequality. [ABSTRACT FROM AUTHOR]

It is well known that the Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether they can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, the authors will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the following singular Kirchhoff-Schrödinger problem M ( ∫ ℝ N ∣ ∇ u ∣ N + V (x) ∣ u ∣ N d x ) (− Δ N u + V (x) ∣ u ∣ N − 2 u) = f (x , u) ∣ x ∣ η in ℝ N , (P η) where 0 < η < N, M is a Kirchhoff-type function and V(x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity. Combining variational techniques with some estimates, they get the existence of ground state solution for (Pη)- Moreover, they also get the same result without the A-R condition. [ABSTRACT FROM AUTHOR]

In this paper, the authors give the local L2 estimate of the maximal operator S ϕ , γ * of the operator family {St,ϕ, γ} defined initially by S t , ϕ , γ f (x) : = e i t ϕ (− Δ) f (γ (x , t)) = (2 π) − 1 ∫ ℝ e i γ (x , t) ⋅ ξ + i t ϕ (| ξ |) f ^ (ξ) d ξ , f ∈ S (ℝ) , which is the solution (when {itn} = 1) of the following dispersive equations (*) along a curve {itγ}: { i ∂ t u + ϕ (− Δ) u = 0 , (x , t) ∈ ℝ n × ℝ , u (x , 0) = f (x) , f ∈ S (ℝ n) , where {itϕ}: ℝ+ → ℝ satisfies some suitable conditions and ϕ (− Δ) is a pseudo-differential operator with symbol {itϕ}(∣{itξ}∣). As a consequence of the above result, the authors give the pointwise convergence of the solution (when {itn} = 1) of the equation (*) along curve {itγ}. Moreover, a global {itL}2 estimate of the maximal operator S ϕ , γ ∗ is also given in this paper. [ABSTRACT FROM AUTHOR]

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal's asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185]. [ABSTRACT FROM AUTHOR]

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition. Motivated by the works of Thomann and Tzvetkov (Nonlinearity, 23 (2010), 2771–2791), Tzvetkov (Probab. Theory Relat. Fields, 146 (2010), 4679–4714), Burq, Thomann and Tzvetkov (Ann. Fac. Sci. Toulouse Math., 27 (2018), 527–597), the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity, and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem, Skorokhod convergence theorem and Gibbs measure. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions. [ABSTRACT FROM AUTHOR]

This paper is concerned with strictly k-convex large solutions to Hessian equations Sk(D2u(x)) = b(x)f(u(x)), x ∈ Ω, where Ω is a strictly (k − 1)-convex and bounded smooth domain in ℝn, b ∈ C ∞ ( Ω ¯) is positive in Ω, but may be vanishing on the boundary. Under a new structure condition on f at infinity, the author studies the refined boundary behavior of such solutions. The results are obtained in a more general setting than those in [Huang, Y., Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math., 244, 2010, 85–98], where f is regularly varying at infinity with index p > k. [ABSTRACT FROM AUTHOR]

In this paper the authors investigate the boundedness and almost periodicity of solutions of semilinear parabolic equations with boundary degeneracy. The equations may be weakly degenerate or strongly degenerate on the lateral boundary. The authors prove the existence, uniqueness and global exponential stability of bounded entire solutions, and also establish the existence theorem of almost periodic solutions if the data are almost periodic. [ABSTRACT FROM AUTHOR]

Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric. [ABSTRACT FROM AUTHOR]

In the present paper, the rigidity of hypersurfaces in Euclidean space is revisited. The Darboux equation is highlighted and two new proofs of the rigidity are given via energy method and maximal principle, respectively. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the following singular Kirchhoff-Schrödinger problem M ( ∫ ℝ N ∣ ∇ u ∣ N + V (x) ∣ u ∣ N d x ) (− Δ N u + V (x) ∣ u ∣ N − 2 u) = f (x , u) ∣ x ∣ η in ℝ N , (P η) where 0 < η < N, M is a Kirchhoff-type function and V(x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity. Combining variational techniques with some estimates, they get the existence of ground state solution for (Pη)- Moreover, they also get the same result without the A-R condition. [ABSTRACT FROM AUTHOR]

In this paper, for any given observation time and suitable moving observation domains, the author establishes a sharp observability inequality for the Kirchhoff-Rayleigh plate like equation with a suitable potential in any space dimension. The approach is based on a delicate energy estimate. Moreover, the observability constant is estimated by means of an explicit function of the norm of the coefficient involved in the equation. [ABSTRACT FROM AUTHOR]

In this paper, the Tricomi problem and the generalized Tricomi problem for a quasilinear mixed type equation are studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to these problems is proved. The method developed in this paper can be used to study more difficult problems for nonlinear mixed type equations arising in gas dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with smooth boundary. A class of additional boundary conditions for the vorticities are identified so that the solution is unique and stable. [ABSTRACT FROM AUTHOR]

The authors investigate the global behavior of the solutions of the difference equation where the initial conditions x− r, x− r+1, x− r+2, ..., x0 are arbitrary positive real numbers, r = max{ l, k, p, q} is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper. [ABSTRACT FROM AUTHOR]

The authors prove two global existence results of strong solutions of the isen-tropic compressible Navier-Stokes-Poisson equations in one-dimensional bounded intervals. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition. In this paper the initial vacuum is allowed, and T is bounded. [ABSTRACT FROM AUTHOR]

In the present paper, the author studies the existence of sign-changing solutions for nonlinear elliptic equations, which have jumping nonlinearities, and may or may not be resonant with respect to Fučik spectrum, via linking methods under Cerami condition. [ABSTRACT FROM AUTHOR]

In this paper, the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form u t − Δ ∅( u) = 0, where ∅ ∈ C 1(ℝ1) is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of ∅'( · ) is permitted to have zero measure. This is an answer to an open problem in [13, p. 288]. [ABSTRACT FROM AUTHOR]

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n ≥ 4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are. [ABSTRACT FROM AUTHOR]

We identify ℝ7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6is an associative submanifold of ℝ7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a refinement of the Atiyah-Singer families index theorem at the level of differential characters. Also a Riemann-Roch-Grothendieck theorem for the direct image of flat vector bundles by proper submersions is proved, with Chern classes with coefficients in C/Q. These results are much related to prior work of Gillet-Soulé, Bismut-Lott and Lott. [ABSTRACT FROM AUTHOR]

This paper is devoted to the homogenization of a semilinear parabolic equation with rapidly oscillating coefficients in a domain periodically perforated by ε-periodic holes of size ε. A Neumann condition is prescribed on the boundary of the holes. The presence of the holes does not allow to prove a compactness of the solutions in L2. To overcome this difficulty, the authors introduce a suitable auxiliary linear problem to which a corrector result is applied. Then, the asymptotic behaviour of the semilinear problem as ε→0 is described, and the limit equation is given. [ABSTRACT FROM AUTHOR]

In this paper the classical theorem of Zareckiĭ about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckiĭ theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the Urysohn-Nachbin lemma is presented which is quite different from the classical one. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general k-th Kloosterman sums, and give an interesting asymptotic formula. [ABSTRACT FROM AUTHOR]

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is considered. The existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is studied. An efficient algorithm for constructing the point itself and an asymptotic representation of arbitrary-order accuracy to the solution is proposed. For sufficiently small parameter values, the existence theorem is proved by the technique of matching asymptotic expansions. An example is given to show the effectiveness of their method. [ABSTRACT FROM AUTHOR]

The Darboux transformation for the two dimensional A2n−1(2) Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair. The expression of exact solutions of the equation is obtained by using Darboux transformation. [ABSTRACT FROM AUTHOR]

The authors study the p( x)-Laplacian equations with nonlinear boundary condition. By using the variational method, under appropriate assumptions on the perturbation terms f1( x, u), f2( x, u) and h1( x), h2(x), such that the associated functional satisfies the “mountain pass lemma” and “fountain theorem” respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces. [ABSTRACT FROM AUTHOR]

The author obtains some weighted Hardy-type inequalities on H-type groups and anisotropic Heisenberg groups. These inequalities generalize some recent results due to N. Garofalo, E. Lanconelli, I. Kombe and P. Niu et al. [ABSTRACT FROM AUTHOR]

This paper undertakes a systematic treatment of the low regularity local well-posedness and ill-posedness theory in Hs and Ḣs for semilinear wave equations with polynomial nonlinearity in u and ∂u. This ill-posed result concerns the focusing type equations with nonlinearity on u and ∂tu. [ABSTRACT FROM AUTHOR]

This paper undertakes a systematic treatment of the low regularity local well-posedness and ill-posedness theory in Hs and Ḣs for semilinear wave equations with polynomial nonlinearity in u and ∂u. This ill-posed result concerns the focusing type equations with nonlinearity on u and ∂tu. [ABSTRACT FROM AUTHOR]

This paper constructs a polyconvex stored energy function, satisfying the null condition, for isotropic compressible elastic materials with given Lamé constants. The difference between this stored energy function and St Venant-Kirchhoff's is a three order term. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the instantaneous shrinking and localization of the support of functions in Y[sub λ](m,p,q,N) and their applications to some nonlinear parabolic equations including the porous medium equation u[sub t]=Δu[sup m]-u[sup q], m>0, q>0 and the p-Laplace equation u[sub t]=div(|∇u|[sup p-2]∇u)-u[sup q], p>1, q>0. In particular, as an application of the results, the necessary and sufficient condition for the solutions of the above p-Laplace equation with nonnegative finite Borel measures as initial conditions to have the instantaneous shrinking property of the support is obtained. This is an answer to an open problem posed by R. Kersner and A. Shishkov. [ABSTRACT FROM AUTHOR]

When the rotatory inertia is taken into account, vibrations of a linear plate can be described by the Kirchhoff plate equation. Consider this equation with locally distributed control forces and some boundary condition which is the simply supported boundary condition for a rectangular plate. In this paper, the authors establish exact controllability of the system in terms of the equivalence to exact internal controllability of the wave equation, by means of a frequency domain characterization of exact controllability introduced recently in [11]. [ABSTRACT FROM AUTHOR]

In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function. [ABSTRACT FROM AUTHOR]

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u0) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u0) < 0. [ABSTRACT FROM AUTHOR]

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. [ABSTRACT FROM AUTHOR]

An initial boundary-value problem for the Hirota equation on the half-line, 0 < x < ∞, t > 0, is analysed by expressing the solution q(x, t) in terms of the solution of a matrix Riemann-Hilbert (RH) problem in the complex k-plane. This RH problem has explicit (x, t) dependence and it involves certain functions of k referred to as the spectral functions. Some of these functions are defined in terms of the initial condition q(x, 0) = q0(x), while the remaining spectral functions are defined in terms of the boundary values q(0, t) = g0(t), qx(0, t) = g1(t) and qxx(0, t) = g2(t). The spectral functions satisfy an algebraic global relation which characterizes, say, g2(t) in terms of {q0(x), g0(t), g1(t)}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. [ABSTRACT FROM AUTHOR]

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality. [ABSTRACT FROM AUTHOR]

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schrödinger equations. The local and global well-posedness are proved with values in the space Σ(ℝ) = { f ∈ H(ℝ), |·| f ∈ L(ℝ)}. When the nonlinearity is focusing and L-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If ϕ = 0, our results coincide with the ones for the deterministic equation. [ABSTRACT FROM AUTHOR]

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, a second order partial differential equation for the Gauss map is obtained, and it is shown that this equation is the complete integrability condition of the representation. [ABSTRACT FROM AUTHOR]

The authors consider the multidimensional aggregation equation ∂ t ρ-div( ρ▿ K* ρ) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better), and review recent results on this problem concerning well-posedness of nonnegative solutions and finite time blowup in multiple space dimensions depending on the behavior of the kernel at the origin. The problem with bounded initial data, data in L p ∩ L1, and measure solutions are also considered. [ABSTRACT FROM AUTHOR]

Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0 − 1 ∈ Hs+1(ℝ2), u0 ∈ H s(ℝ2) ∩ Ḣ−ε(ℝ2) for s > 2 and 0 < ε < 1, the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid. Furthermore, the L2 decay rate of the velocity field is obtained. [ABSTRACT FROM AUTHOR]

The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman’s result for the Bernstein type theorem is improved. [ABSTRACT FROM AUTHOR]