Uniqueness of least area surfaces in the 3-torus

Given any -periodic metric g on and a plane through the origin, Bangert [4] shows that there exists a properly embedded surface homeomorphic to which is homotopically area-minimizing w.r.t. g, lies in a strip of bounded width around P, and does not have self-intersections when projected to the 3-torus . For the set of such surfaces, we show the following uniqueness theorems: If P is irrational, i.e., is not spanned by vectors in , the action of on by translations has a unique minimal set. If P is totally irrational, i.e., , then the surfaces in are pairwise disjoint. Received: 8 July 1999 / In final form: 14 February 2000 / Published online: 25 June 2001     

Continuity of eigenvalues for Schrödinger operators, $L^p$-properties of Kato type integral operators

Given an arbitrary relatively compact (finely) open subset of -eigenvalues of are studied where is the Dirichlet Laplacian on D and are measures on such that is continuous and is bounded for every ball X in being Green's function for X). Moreover, it is shown that these eigenvalues depend continuously on D and . The results are based on very general compactness and convergence properties of integral operators of Kato type which are developed before. Received: 9 November 2000 / Published online: 24 September 2001     

Elliptic equations with vertical asymptotes in the nonlinear term

We study the existence of solutions of the nonlinear problem {fx349-1} where μ is a bounded measure andg is a continuous nondecreasing function such thatg(0)=0. In this paper, we assume that the nonlinearityg satisfies {fx349-2} Problem (0.1) need not have a solution for every measure μ. We prove that, given μ, there exists a “closest” measure μ^* for which (0.1) can be solved. We also explain how assumption (0.2) makes problem (0.1) different from the case whereg(t) is defined for everyt ∈ ℝ.     

The asymptotic Plateau problem in Gromov hyperbolic manifolds

We solve the asymptotic Plateau problem in every Gromov hyperbolic Hadamard manifold (X,g) with bounded geometry. That is, we prove existence of complete (possibly singular) k-dimensional area minimizing surfaces in X with prescribed boundary data at infinity, for a large class of admissible limit sets and for all . The result also holds with respect to any riemannian metric on X which is lipschitz equivalent to g. Received: 23 January 2001 / Accepted: 25 October 2001 Published online: 28 February 2002     

Manifold constrained variational problems

The integral representation for the relaxation of a class of energy functionals where the admissible fields are constrained to remain on a -dimensional manifold is obtained. If is a continuous function satisfying for and for all then where is open, bounded, and is the tangential quasiconvexification of f at belong to the tangent space to at Received October 10, 1998 / Accepted December 1, 1998     

Partial regularity in non-linear elasticity

We prove partial regularity of minimizers of some polyconvex functionals. In particular our results include models such as ∫_Ω a(x,u)(|Du|²+| det Du|²), where a is a bounded H?lder continuous function, such that a(x,u)≥c for some positive constant c. Received: 2 January 2001 / Revised version: 30 August 2001     

Certain Subsets on Which Every Bounded Convex Function Is Continuous

To guarantee every real-valued convex function bounded above on a set is continuous, how ＂thick＂ should the set be？ For a symmetric set A in a Banach space E,the answer of this paper is： Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold： i） spanA has finite co-dimentions and ii） coA has nonempty relative interior. This paper also shows that a subset A C E satisfying every real-valued convex function bounded above on A is continuous on E if （and only if） every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E.     

Mixed Optimal Stopping and Stochastic Control Problems with Semicontinuous Final Reward for Diffusion Processes

We consider mixed control problems for diffusion processes, i.e. problems which involve both optimal control and stopping. The running reward is assumed to be smooth, but the stopping reward need only be semicontinuous. We show that, under suitable conditions, the value function w has the same regularity as the final reward g, i.e. w is lower or upper semicontinuous if g is. Furthermore, when g is l.s.c., we prove that the value function is a viscosity solution of the associated variational inequality.     

A note on the implicit function theorem for quasi-linear eigenvalue problems

We consider the quasi-linear eigenvalue problem −Δ_pu=λg(u) subject to Dirichlet boundary conditions on a bounded open set Ω, where g is a locally Lipschitz continuous function. Imposing no further conditions on Ω or g, we show that for λ near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by λ depends continuously on the parameter.     

Stability Constants in Linear Spaces

We investigate the stability constants of convex sets in linear spaces. We prove that the stability constants of affinity and of the Jensen equation are of the same order of magnitude for every convex set in arbitrary linear spaces, even for functions mapping into an arbitrary Banach space. We also show that the second Whitney constant corresponding to the bounded functions equals half of the stability constant of the Jensen equation whenever the latter is finite. We show that if a convex set contains arbitrarily long segments in every direction, then its Jensen and Whitney constants are uniformly bounded. We prove a result that reduces the investigation of the stability constants to the case when the underlying set is the unit ball of a Banach space. As an application we prove that if D is convex and every δ-Jensen function on D differs from a Jensen function by a bounded function, then the stability constants of D are finite.     

A Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998     

Vector-valued holomorphic functions revisited

Abstract. Let be open,X a Banach space and . We show that every is holomorphic if and only if every set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that is holomorphic for all , where W is a separating subspace of to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of , uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers). Received January 29, 1998; in final form March 8, 1999 / Published online May 8, 2000     

A Liouville property for spherical averages in the plane

It is shown that any continuous bounded function f on such that , is constant provided r is a strictly positive real function on satisfying The proof is based on a minimum principle exploiting that and on a study of -stable sets, i.e., sets A such that the circle of radius r(x) centered at x is contained in A whenever . The latter reveals that there is no disjoint pair of non-empty closed -stable subsets in unless (taking spheres this holds for any , ). A counterexample is given where . Received November 24, 1999 / Published online December 8, 2000     

Continuity at the extreme points

We prove that a bounded convex lower semicontinuous function defined on a convex compact set K is continuous at a dense subset of extreme points. If there is a bounded strictly convex lower semicontinuous function on K, then the set of extreme points contains a dense completely metrizable subset.     

Regularity of the Level Set Flow

We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed C¹ manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc.     

A singular limit arising in combustion theory: Fine properties of the free boundary

The equation where converges to the Dirac measure concentrated at with mass has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in and have a uniformly bounded support, we study non-negative solutions of the Cauchy problem in as We show that each limit of is a solution of the free boundary problem in on (in the sense of domain variations and in a more precise sense). For a.e. time t the graph of u(t) has a unique tangent cone at -a.e. The free boundary is up to a set of vanishing measure the sum of a countably n-1-rectifiable set and of the set on which vanishes in the mean. The non-degenerate singular set is for a.e. time a countably n-1-rectifiable set. As key tools we introduce a monotonicity formula and, on the singular set, an estimate for the parabolic mean frequency. Received: 8 August 2001 / Accepted: 8 May 2002 / Published online: 5 September 2002 RID="a" ID="a" Partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan.     

BV solutions of nonconvex sweeping process differential inclusion with perturbation

This paper is devoted to the study of differential inclusions, particularly discontinuous perturbed sweeping processes in the infinite-dimensional setting. On the one hand, the sets involved are assumed to be prox-regular and to have a variation given by a function which is of bounded variation and right continuous. On the other hand, the perturbation satisfies a linear growth condition with respect to a fixed compact subset. Finally, the case where the sets move in an absolutely continuous way is recovered as a consequence.     

Relative isoperimetric inequalities for minimal submanifolds outside a convex set

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ?C along ?Σ∩?C and ?Σ ～ ?C is radially connected from a point p ∈ ?Σ∩?C. We introduce a modified volume M_p(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.     

On the equivalence of two variational problems

We consider the following problems where is a convex function, is an open bounded subset of is a closed convex subset of such that and and are suitable obstacles. We give conditions on the function {\it g} under which the two problems are equivalent. Received March 24, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Comparing two methods of resolving homogeneous differential inclusions

Given a compact set we consider the differential inclusion We show how to use the main idea of the method of convex integration [ N], [G], [K] (to control convergence of the gradients of a sequence of approximate solutions by appropriate selection of the sequence) to obtain an optimal existence result. We compare this result with the ones available by the Baire category approach applied to the set of admissible functions with topology. A byproduct of our result is attainment in the minimization problems with integrands L having quasiaffine quasiconvexification that was, in fact, the reason of our interest to differential inclusions. This result can be considered as a first step towards characterization of those minimization problems which are solvable for all boundary data. This problem was solved in [S1] in the scalar case m=1. Received November 5, 1998 / Accepted July 17, 2000 / Published online December 8, 2000