Bernstein Theorem Research Articles

Min-max theory for capillary surfaces

Abstract We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 𝑐, and with smooth boundary contacting at any given constant angle 𝜃. Moreover, if 𝑐 is nonzero and 𝜃 is not π 2 \frac{\pi}{2} , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

Global Nonlinear Stability of Traveling Wave Solution to Time-Like Extremal Hypersurface in Minkowski Space

Abstract There are a few results about the global nonlinear stability of nontrivial large solution to quasilinear wave equations. Time-like extremal surface in Minkowski space is an important model of quasilinear wave equation. There are two folds in this paper. Firstly, we get the existence of traveling wave solution to the time-like extremal hypersurface in $\mathbb {R}^{1+(n+1)}$, which can be considered as the generalized Bernstein theorem. For $n=2$, we are also concerned with global stability of traveling wave solutions with speed of light to time-like extremal hypersurface equation in $1+(2+1)$ dimensional Minkowski space, which is corresponding with quasilinear wave equation in two space dimensions.

Bernstein theorems for nonlinear geometric PDEs

Bernstein theorems for nonlinear geometric PDEs

Banach’s theorem in higher-order reverse mathematics

In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schröder–Bernstein theorem. Some additional results address statements in higher-order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher-order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice.

On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of for almost every α where is ranging over a long interval and f is a 1-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for N in a set of positive upper density, the order of is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy–Koksma inequality for such f, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel–Bernstein Theorem that allows very general modularity conditions.

Tracking control based on adaptive Bernstein polynomial approximation for a class of unknown nonlinear dynamic systems

In this article, an adaptive tracking control approach using Bernstein polynomial approximation is firstly proposed for an unknown nonlinear dynamic system. Bernstein polynomial approximation aims to compensate the unknown nonlinear dynamic function. However, if Bernstein theorem is directly used, the Bernstein polynomial's coefficients need to be derived from the system dynamic function. Nevertheless, the dynamic function is presumed to be unknown, hence the polynomial approximation still cannot be used for designing this control. In order to obtain the available function approximation, adaptive strategy is considered to estimate these coefficients. Finally, by learning from the classical adaptive algorithm, the undetermined coefficient problem is addressed, so that the valid tracking control is found for the unknown nonlinear dynamic system. According to Lyapunov stability analysis and simulation experiment, it is concluded that the new adaptive scheme can realize the control objective.

On Lipschitz Geometry at infinity of complex analytic sets

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein’s problem. Moser’s Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser’s Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $$\mathbb {C}^n$$ which is Lipschitz regular at infinity must be an affine linear subspace of $$\mathbb {C}^n$$ . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here.

Stable anisotropic minimal hypersurfaces in

AbstractWe show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in$\mathbf {R}^4$has intrinsic cubic volume growth, provided the parametric elliptic integral is$C^2$-close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in$\mathbf {R}^4$. The new proof is more closely related to techniques from the study of strictly positive scalar curvature.

A Cantor–Bernstein theorem for infinite matroids

A Cantor–Bernstein theorem for infinite matroids

Spacelike minimal surfaces which are graphs in [formula omitted

In this work we study a system of two second order nonlinear partial differential equations corresponding to the equations of minimal spacelike graphs in the four-dimensional Minkownski space R14. We obtain a set of entire solutions of these equations, adapting the method of Karl Kommerell used for the analogous problem in R4. In our technique we replace ‘‘graphs of holomorphic functions” by ‘‘graphs of generalized harmonic curves” and then we obtain a class of minimal spacelike surfaces in R14 which are also entire graphs with non-zero Gauss curvature. We then provide two versions of the Bernstein Theorem corresponding for the two types of graphs which can occur in R14. Several explicit examples are given.

On the entire self-shrinking solutions to Lagrangian mean curvature flow II

We prove the rigidity of entire graphic Lagrangian self-shrinkers in \(({\mathbb {R}}^{2n}, g_\tau )\), where \(g_\tau =\sin \tau \,\delta _0+\cos \tau \,g_0\) is a linear combination of the Euclidean metric \(\delta _0\) and the pseudo metric \(g_0=2\sum _i dx_idy_i\) with \(\tau \in (0,\frac{\pi }{2})\), complementing the previous results for \(\tau =0\) and \(\tau =\frac{\pi }{2}\); actually we obtain Bernstein theorems for three corresponding nonlinear elliptic equations between the Monge–Ampère equation (\(\tau =0\)) and the special Lagrangian equation (\(\tau =\frac{\pi }{2}\)). Moreover, we find Bernstein theorem fails when \(\tau \in (-\frac{\pi }{4},0)\) and entire graphic spacelike self-shrinker in Minkowski spaces share this non-rigidity property.

Interior estimates for the Monge–Ampère type fourth order equations

In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial Legendre transform. As an application, we obtain a new proof of the Bernstein theorem without using Caffarelli-Guti\'errez's estimate, including the Chern conjecture on affine maximal surfaces. For the inhomogeneous equation, we also obtain a new proof in dimension two by an integral method relying on the Monge-Amp\`ere Sobolev inequality. This proof works even when the right hand side is singular. In higher dimensions, we obtain the interior regularity in terms of integral bounds on the second derivatives and the inverse of the determinant.

A quasiconformal Hopf soap bubble theorem

We show that any compact surface of genus zero in {mathbb {R}}^3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon’s quasiconformal Bernstein theorem. The result generalizes, among others, Hopf’s theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.

Necessary and sufficient conditions to Bernstein theorem of a Hessian equation

The Hessian quotient equations S k , l ( D 2 u ) ≡ S k ( D 2 u ) S l ( D 2 u ) = 1 , ∀ x ∈ R n \begin{equation} S_{k,l}(D^2u)\equiv \frac {S_k(D^2u)}{S_l(D^2u)}=1, \ \ \forall x\in {\mathbb {R}}^n \end{equation} were studied for k − k- th symmetric elementary function S k ( D 2 u ) S_k(D^2u) of eigenvalues λ ( D 2 u ) \lambda (D^2u) of the Hessian matrix D 2 u D^2u , where 0 ≤ l > k ≤ n 0\leq l>k\leq n . For l = 0 l=0 , (0.1) is reduced to a k − k- Hessian equation S k ( D 2 u ) = 1 , ∀ x ∈ R n . \begin{equation} S_k(D^2u)=1, \ \ \forall x\in {\mathbb {R}}^n. \end{equation} Two quadratic growth conditions were found by Bao-Cheng-Guan-Ji [American J. Math. 125 (2013), pp. 301–316] ensuring the Bernstein properties of (0.1) and (0.2) respectively. In this paper, we will drop the point wise quadratic growth condition of Bao-Cheng-Guan-Ji and prove three necessary and sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse isoperimetric type inequality, volume growth or L p L^p -integrability respectively. Our new volume growth or L p − L^p- integrable conditions improve largely various previously known point wise conditions provided Bao et al.; Chen and Xiang [J. Differential Equations 267 (2019), pp. 52027–5219]; Cheng and Yau [Comm. Pure Appl. Math. 39 (1986), pp. 8397–866]; Li, Ren, and Wang [J. Funct. Anal. 270 (2016), pp. 26917–2714]; Yuan [Invent. Math. 150 (2002), pp. 1177–125], etc.

Liouville-type results in two dimensions for stationary points of functionals with linear growth

We consider elliptic systems generated by variational integrals of linear growth satisfying the condition of \(\mu\)-ellipticity for some exponent \(\mu >1\) and prove that stationary points \(u\colon\mathbb{R}^2\to\mathbb{R}^N\) with the property
 \(\limsup_{|x|\to \infty} \frac{|u(x)|}{|x|} < \infty\)
 must be affine functions. The latter condition can be dropped in the scalar case together with appropriate assumptions on the energy density providing an extension of Bernstein's theorem.

METRIC THEORY OF PARTIAL QUOTIENTS OF N-CONTINUED FRACTIONS

On analogy of the regular continued fractions, for any fixed positive integer [Formula: see text], every [Formula: see text] can be expanded into an [Formula: see text]-continued fraction, denoted by [Formula: see text], where [Formula: see text] are called the partial quotients. In this paper, we concern the metric theory of the partial quotients. More precisely, let [Formula: see text], the Borel–Bernstein theorem and Hausdorff dimension of the set [Formula: see text] for infinitely many [Formula: see text] are determined. This generalizes the results of regular continued fractions.

A complex Borel–Bernstein theorem

A complex Borel–Bernstein theorem

NON-CLASSICAL FOUNDATIONS OF SET THEORY

AbstractIn this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.