Regularity for minimal boundaries in R n ¶with mean curvature in L n

Let be a minimal set with mean curvature in L ⁿ that is a minimum of the functional , where is open and . We prove that if then can be parametrized over the (n−1)-dimensional disk with a C ^α mapping with C ^α inverse. Received: 11 July 1997 / Revised version: 24 February 1998     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

Partial regularity for weak solutions of a nonlinear elliptic equation

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=u ^α in -Δu=u ^α in Ω ⊂ ℝ ⁿ , which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.     

Embedded Rotational Hypersurfaces with Constant Scalar Curvature in Sn: A Correction to a Statement in M. L. Leite, Manuscripta Math. 67 (1990), 285-304.

We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S ⁿ other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC.     

Mapping problems, fundamental groups and defect measures

We study all the possible weak limits of a minimizing sequence, for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy condition. We show that if p is not an integer, then any such weak limit is a strong limit and, in particular, a stationary p-harmonic map which is C ^1,α continuous away from a closed subset of the Hausdorff dimension ≤ n − [p] − 1. If p is an integer, then any such weak limit is a weakly p-harmonic map along with a (n − p)-rectifiable Radon measure μ. Moreover, the limiting map is C ^1,α continuous away from a closed subset Σ=spt μ ∪ S with H ^{n − p} (S)=0. Finally, we discuss the possible varifolds type theory for Sobolev mappings. Partially supported by NSF Grant DMS 9626166     

Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature

Let M be a compact n-dimensional Riemannian orbifold of Ricci curvature ≥n−1. We prove that for 1 ≤k≤n, the k ^th nonzero eigenvalue of the Laplacian on M is equal to the dimension n if and only if M is isometric to the k-times spherical suspension over the quotient S ^{n − k} }Γ of the unit (n−k)-sphere by a finite group Γ⊂O(n−k+1) acting isometrically on S ^{n − k} ⊂ℝ ^{n − k +}. Received: 21 September 1998 / Revised version: 23 February 1999     

Classification ofS n -normal semigroups

IfS _n andC _n denote, respectively, the symmetric group and inverse semigroup onn symbols, thenS _n⊂C_n and a semigroupT⊂C_n isS _n -normal ifα ⁻¹ Tα ⊂Tfor every α∈S _n . TheS _n -normal semigroups are classified.     

Proof of the double bubble curvature conjecture

An area minimizing double bubble in ℝⁿ is given by two (not necessarily connected) regions which have two prescribed n-dimensional volumes whose combined boundary has least (n−1)-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for n = 2, 3,4, but is, for general volumes, unknown for n ≥ 5. Here, for arbitrary n, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in ℝ³, as well as some new component bounds in ℝⁿ.     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

Existence result for minimal hypersurfaces¶with a prescribed finite number of planar ends

Paralleling what has been done for minimal surfaces in ℝ³, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝ ^{n +1} which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature. Received: 1 July 1999 / Revised version: 31 May 2000     

BMO-scale of distribution on ℝ n

Let S′ be the class of tempered distributions. For ƒ ∈ S′ we denote by J ^−α ƒ the Bessel potential of ƒ of order α. We prove that if J ^−α ƒ ∈ BMO, then for any λ ∈ (0, 1), J ^−α (f)_λ ∈ BMO, where (f)_λ = λ⁻ⁿ f(φ(λ⁻¹)), φ ∈ S. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the VMO space.     

Removable singularities and non-uniqueness of solutions of a singular diffusion equation

 We prove that the solution u of the equation u _t =Δlog u, u>0, in (Ω\{x ₀})×(0,T), Ω⊂ℝ², has removable singularities at {x ₀}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ₀, C ₁, C ₂>0, such that C ₁ |x−x ₀|^α≤u(x,t)≤C ₂|x−x ₀|^−α holds for all 0<|x−x ₀|≤ρ₀ and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ²×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ²×(0,T) when 0≤u ₀∉L ¹ (ℝ²) is radially symmetric and u ₀L _loc ¹(ℝ²). Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003 Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65     

The structure of area-minimizing double bubbles

We show that the least area required to enclose two volumes in ℝⁿ orS ⁿ forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝⁿ have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝⁿ. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝⁿ, based on an idea due to Brian White. Supported in part by NSF DMS-9409166.     

On the existence and approximation of zeroes

In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ ⁿ to ℝ ⁿ . In the first part of the paper it is proved thatf has a computable zero iff:C ⁿ →ℝ ⁿ satisfies the nonparallel condition for any two antipodal points on bdC ⁿ, i.e. if for anyx∈bdC ⁿ ,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bdC ⁿ to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extendingf:U→ℝ ⁿ to a function g from a cube containingU to ℝ ⁿ , the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

The size of the shadow boundary projection

Among all embedded closed manifoldsM ^d ⊂ ℝ ^d+1 with positive exterior curvature ≤k the ratio between the (d − 1)-Hausdorff measure of the shadow boundary projection and the volume ofM ^d is maximized by the sphere of radius 1/k.     

Sommets et normales concourantes des courbes convexes de largeur constante et singularités des hérissons

LetC be a convex curve of constant width and of classC ₄ ⁺ . It is known thatC has at least 6 vertices and its interior contains either a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then: (i)C has exactly 6 vertices if, and only if, its evolute is the boundary of a topological disc through each interior point of which pass at least 6 normals; (ii) ifC has more than 6 vertices, then there exists an open set of points through each of which pass at least 10 normals. The proof: (i) expresses the number of normals passing through a point as a function of the index with respect to the evolute; (ii) relates this index to the number of singularities of the evolute (i.e. of vertices). Furthermore, we give formulas for counting singularities of generic hedgehogs in ℝ² and ℝ³.