A partial regularity result for harmonic maps into a Finsler manifold

In this paper, we consider the energy of maps from an Euclidean space into a Finsler space and study the partial regularity of energy minimizing maps. We show that the -dimensional Hausdorff measure of the singular set of every energy minimizing map is 0 for some , when m=3,4. Received: 6 June 2001 / Accepted: 10 July 2001 / Published online: 12 October 2001     

Uniqueness and transport density in Monge's mass transportation problem

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures on , find the measure preserving map s(x) between them which minimizes the average distance transported. Here distance can be induced by the Euclidean norm, or any other uniformly convex and smooth norm on . Although the solution is never unique, we give a geometrical monotonicity condition singling out a particular optimal map s(x). Furthermore, a local definition is given for the transport cost density associated to each optimal map. All optimal maps are then shown to lead to the same transport density . Received: 18 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Scaling properties of Hausdorff and packing measures

Let . Let be a continuous increasing function defined on , for which and is a decreasing function of t. Let be a norm on , and let , , denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions such that the corresponding Hausdorff or packing measure scales with exponent by showing it must be of the form , where L is slowly varying. We also show that for continuous increasing functions and defined on , for which , is either trivially true or false: we show that if , then for a constant c, where is the Lebesgue measure on . Received June 17, 2000 / Accepted September 6, 2000 / Published online March 12, 2001     

Partitions with prescribed mean curvatures

We consider a certain variational problem on Caccioppoli partitions with countably many components, which models immiscible fluids as well as variational image segmentation, and generalizes the well-known problem with prescribed mean curvature. We prove existence and regularity results, and finally show some explicit examples of minimizers. Received: 7 June 2001 / Revisied version: 8 October 2001     

Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional

The paper is concerned with the higher regularity properties of the minimizers of the Mumford–Shah functional. It is shown that, near to singular points where the scaled Dirichlet integral tends to 0, the discontinuity set is close to an Almgren area minimizing set. As a byproduct, the set of singular points of this type has Hausdorff dimension at most N-2, N being the dimension of the ambient space. Assuming higher integrability of the gradient this leads to an optimal estimate of the Hausdorff dimension of the full singular set. Received: 5 July 2001 / Accepted: 29 November 2001 / Published online: 23 May 2002     

On lower semicontinuity in BH and 2-quasiconvexification

It is proved that if , with p > 1, if is bounded in , , and if in then provided is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BH^p are provided. Received: 10 October 2001 / Accepted: 8 May 2002 / Published online: 17 December 2002     

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.     

The $\mu$-measure as a tool for classifying computational complexity

Two simply typed term systems and are considered, both for representing algorithms computing primitive recursive functions. is based on primitive recursion, on recursion on notation. A purely syntactical method of determining the computational complexity of algorithms in , called $\mu$ -measure, is employed to uniformly integrate traditional results in subrecursion theory with resource-free characterisations of sub-elementary complexity classes. Extending the Schwichtenberg and Müller characterisation of the Grzegorczyk classes for , it is shown $\mathcal{E}_{n+1} = \mathcal{R}^n_1n\ge 1\mathcal{R}^n_i$ denotes the \emph{th modified Heinermann class} based on . The proof does not refer to any machine-based computation model, unlike the Schwichtenberg and Müller proofs. This is due to the notion of modified recursion lying on top of each other provided by . By Ritchie's result, characterises the linear-space computable functions. Using the same method, a short and straightforward proof is presented, showing that characterises the polynomial time computable functions. Furthermore, the classes and coincide at and above level 2. Received: 22 September 1997 / Revised version: 12 May 1999     

A characterization of graphs which can be approximated in area by smooth graphs

For vector valued maps, convergence in W ^1,1 and of all minors of the Jacobian matrix in L ¹ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension n≥ 3 can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain. Received April 29, 1999 / final version received July 21, 2000?Published online September 25, 2000     

Uniform partial regularity of quasi minimizers for the perimeter

Quasi minimizers for the perimeter are measurable subsets G of such that for all variations of G with and for a given increasing function such that . We prove here that, given , G a reduced quasi minimizer, and , there are , with , and , homeomorphic to a closed ball with radius t in , such that for some absolute constant . The constant above depends only on n, and . If moreover for some , we prove that we can find such a ball such that is a dimensional graph of class . This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini can be applied to get the required regularity properties. Received: July 12, 1999 / Accepted: October 1, 1999     

Regularity of components in optimal design problems with perimeter penalization

We consider minimal energy configurations of mixtures of two materials in , where the energy includes a penalty on the length of the interface between the materials. We show that, for one of the materials, the boundary of each component is smooth, and we prove the existence of an upper bound on the relative distances between components. Received: 24 March 2000 / Accepted: 25 October 2001 / Published online: 29 April 2002     

The calibration method for the Mumford-Shah functional and free-discontinuity problems

We present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results. Received: 29 November 2001 / Published online: 29 April 2002     

Egoroff’s theorem and maximal run length

We construct a sequence of measurable functions converging at each point of the unit interval, but the set of points with any given rate of convergence has Hausdorff dimension one. This is used to show that a version of Egoroff’s theorem due to Taylor is best possible. The construction relies on an analysis of the maximal run length of ones in the dyadic expansion of real numbers. It is also proved that the exceptional set for a limit theorem of Renyi has Hausdorff dimension one.     

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     

A proof of the basic measure extension theorem of Carathéodory

In this note a direct elementary proof of Carathéodory's measure extension theorem is presented. It is based on an approximation argument for outer measures where elements of the -algebra are approached by elements of the underlying algebra of sets with respect to the symmetric difference. Received: 3 April 2000 / Accepted: 20 September 2000     

Cardinality of the set of extreme extensions¶of a quasi-measure

Cardinals that arise as the number of extreme quasi-measure extensions of a quasi-measure [resp., measure] μ defined on an algebra [resp., σ-algebra] of sets to a larger algebra [resp., σ-algebra] of sets are characterized in the general case as well as under some natural assumptions on μ. Received: 19 September 2000     

Empirical multifractal moment measures of self-similar measure for <Emphasis Type="Italic">q</Emphasis> < 0<Superscript>*</Superscript>

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L ^q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L ^q - spectrum for q < 0 is generally considered significantly more difficult since the L ^q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L ^q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0. Authors’ addresses: Jiaqing Xiao, School of Science, Wuhan University of Technology, Wuhan 430070, China; Wu Min, School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China     

A free boundary problem for $\infty$–Laplace equation

We consider a free boundary problem for the p-Laplacian describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function. Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001     

A note on the Hodge theory for functionals with linear growth

We study the Hodge decomposition of L ¹-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS]. Received: 19 November 1997 / Revised version: 18 May 1998     

Regularity of quasi-minimizers on metric spaces

Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasi-minimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally H?lder continuous, if the space is doubling and supports a Poincaré inequality. Received: 12 May 2000 / Revised version: 20 April 2001