R-3 Repository :: Browsing by Author "Hardt, Robert M."

Browsing by Author "Hardt, Robert M."

Now showing 1 - 20 of 27

Asymptotics under self-intersection for minimizers of self-avoiding energies
(2009) Dunning, Ryan Patrick; Hardt, Robert M.
A knot energy is a real-valued function on a space of curves which in some sense assigns higher energy values to more complicated curves. The key property of any knot energy is self-repulsiveness: for a sequence of curves approaching a self-intersection, the energy blows up to infinity. While the study of optimally embedded curves as minimizers of energy among a given knot class has been well-documented, this thesis investigates the existence of optimally immersed self-intersecting curves. Because any self-intersecting curve will have infinite knot energy, parameter-dependent renormalizations of the energy remove the singular behavior of the curve. This process allows for the careful selection of an optimally immersed curve.
Axially symmetric harmonic maps and relaxed energy
(1991) Poon, Chi-Cheung; Hardt, Robert M.
Here we investigate some new phenomena in harmonic maps that result by imposing a symmetry condition. A map $u:B\sp3\to S\sp2$ is called axially symmetric if, in cylindrical coordinates, $u(r,\theta,z)$ = $(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$ for some real-valued function $\phi(r,z)$, called an angle function for u. The important notion of the L energy of a map from $B\sp3$ to $S\sp2$ was first studied by H. Brezis, F. Bethuel, and J. M. Coron. In (BBC), the weak $H\sp1$ lower semicontinuity of $\rm E + 8\pi\lambda L$ is proven. Thus, the minimizers of $\rm E + 8\pi\lambda L$ exist. For minimizers of $\rm E + 8\pi\lambda L$, 0 $<$ $\lambda$ $<$ 1, Bethuel and Brezis (BB) prove that the singularities are only isolated points. Note that such minimizers are still weak solutions of the harmonic map equation. In this thesis, we treat these problems in the axially symmetric context. By studying a elliptic equation, we show that there is at most one smooth axially symmetric harmonic map corresponding to any given smooth axially symmetric boundary data. We also show that any minimizer in the axially symmetric class of $\rm E + 8\pi\lambda L$, where 0 $<$ $\lambda$ $\leq$ 1, has only isolate singularities in minimizers may occur even for $\lambda$ = 1. These provide the first examples of isolated singularities of degree 0.
Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets
(2007) Samansky, Eric Michael; Hardt, Robert M.
Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.
Coordinate scans, compactness properties, and area minimization
(2006) Peterson, James Ernest; Hardt, Robert M.
In the framework of geometric measure theory, we investigate compactness results and possible solutions of area-minimizing problems on surfaces using area functionals other than the traditional mass norm. These solutions will be of a relatively new class of objects called rectifiable coordinate scans. We begin by reviewing traditional theory and motivating problems for using non-mass area functionals. Next we set up the basic definitions and theorems, mostly in analogy to the classical theory of currents. Our major result is a rectifiable compactness theory which leads to solutions of Plateau-type problems for scans. Finally, we use our compactness results to construct a Holder continuous area-decreasing flow.
Deformations of Hilbert Schemes of Points on K3 Surfaces and Representation Theory
(2014-04-22) Zhang, Letao; Hassett, Brendan E.; Hardt, Robert M.; Riviere, Beatrice M.
We study the cohomology rings of Kaehler deformations X of Hilbert schemes of points on K3 surfaces by representation theory. We compute the graded character formula of the Mumford - Tate group representation on the cohomology ring of X. Furthermore, we also study the Hodge structure of X, and find the generating series for deducting the number of canaonical Hodge classes in the middle cohomology.
Edge length minimizing polyhedra
(2002) Berger, Scott Byron; Hardt, Robert M.
This thesis investigates two different edge length minimization problems on convex polyhedra: first, minimizing edge length for fixed volume (Melzak's Problem), and second, minimizing edge length for fixed surface area (Aberth's Problem). Specific examples are given that demonstrate the necessity of restricting the problems to convex polyhedra. The right regular 3-prism of height 1 relative to base edge 1 is shown to minimize Melzak's Problem over several families of polyhedra, including Platonic solids, regular pyramids, and general prisms. For Aberth's Problem, the right regular 3-prism of height 2-13 relative to base edge 1 is shown to minimize over the same families. The minimizer of Aberth's Problem cannot have all equal-area faces; similarly, for Melzak's Problem, the minimizer cannot be an equal-faced polyhedron with 10 or more faces. For the minimizing object in both Aberth's Problem and Melzak's Problem, the area of the kth face must be of the order 1/k 2. The minimizer for Melzak's Problem must exist in a more general class which includes infinite-faced objects. Examples are presented of infinite-faced objects that have finite edge length. Although the minimizer might have an infinite number of faces, the edge skeleton of such a minimizer is proved to consist almost entirely of line segments, which means that the bad points of the edging contribute nothing to the total length of the edge skeleton. Specifically, an edge point is called bad if the local edge set is not a line segment. For a minimizing convex object, the 1-dimensional Hausdorff measure of the set of bad edge points is zero. As a corollary, the edge set of the minimizer for Melzak's Problem does not contain any smooth non-linear arcs.
Energy minimizers, gradient flow solutions, and computational investigations in the theory of biharmonic maps
(1998) Ledbetter, Ashley Ann; Hardt, Robert M.
We present the definitions, derive the relevant Euler-Lagrange equations, and establish various properties concerning biharmonic maps. We investigate several classes of examples exhibiting singular behavior. Existence of a weak solution to the associated evolution equation is proved using a penalization argument and the Galerkin method. We prove higher integrability of order greater than two for derivatives of Laplacian energy minimizers contingent upon certain energy constraints. We initiate a numerical analysis of biharmonic maps using a discrete Laplacian energy, a finite difference scheme, and involving spherical coordinates in a variety of dimensions in order to understand isolated singularities. Since singularities of Laplacian energy minimizers first occur in dimension five, where both mathematical and numerical analysis is quite complicated, we also consider an analogous problem for a lower non-integer power, p, of the Laplacian. The Euler-Lagrange equations are now not only nonlinear and fourth-order, but also degenerate. Nevertheless, we are able to analyze a specific solution of the p-biharmonic map equation having a degree two singularity. We prove the non-minimality of the solution by a construction that splits this singularity. In this work, we have developed relevant computer code to expedite biharmonicity testing and energy computations. We have also developed computer code applicable to the initial study of the discretized biharmonic problem. We hope this code will be useful in future research aimed at a more comprehensive numerical analysis.
Evolution problems in geometric analysis
(1991) Cheng, Xiaoxi; Hardt, Robert M.
This thesis studies problems derived from nonlinear partial differential equations of parabolic type. Part I. A mass reducing flow for integral currents. A mass reducing flow of integral current is constructed. The current flow has the property that it is Holder continuous under the flat norm and reduces the mass of the initial current while keeping the boundary fixed. Part II. Estimate of singular set of the evolution problems for harmonic maps. Let $u$: ${\cal M}$ $\times$ R$\sb+$ $\to$ ${\cal N}$ be a weak solution to the evolution problem for harmonic maps. We prove that the singular set of $u$ has at most finite $m$ $-$ 2 dimensional Hausdorff measure on each time slice ${\cal M}$ $\times$ $\{t\}$.
Geometric variational problems with cross-sectional constraints
(2005) Meng, Zheng; Hardt, Robert M.
We study here some geometric variational problems motivated by the modeling of plants' growth. In Chapter 2, we conclude the general existence of an area minimizing surface with given boundary on two parallel hyperplanes and the constraint that the intersection of the surface with each hyperplane parallel to these of the boundary encloses the same area. In Chapter 3, we study the area minimizing surface in R 3 whose intersection with each of the hyperplanes R 2 x {h}, h ∈ [0, 1] encloses a prescribed area. We conclude that, up to a translation, the minimizer exists and is invariant under revolution. In Chapter 4, as a specific case of the problem in Chapter 2, the minimizing surface bounded by two parallel circles of the same size is studied carefully. We conclude that such an area minimizing surface is the skewed cylinder determined by the two circles. In Chapter 5, we study an analogous energy minimizing problem in PDE with a boundary constraint and a cross-sectional constraint on the L1 norm over a rectangular region. The even terms and the conditions for the odd terms in the Fourier expansion of the energy minimizer are given.
Harmonic maps, heat flows, currents and singular spaces
(1995) Li, Ming; Hardt, Robert M.
This thesis studies some problems in geometry and analysis with techniques developed from non-linear partial differential equations, variational calculus, geometric measure theory and topology. It consists of three independent parts: Chapter I. We study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at most n-2, where n is the dimension of the domain. We will also give an example of an energy minimizing map from a surface to a surface that has a singular point. Thus the n-2 dimension estimate is optimal, in contrast to the n-3 dimension estimate of Schoen-Uhlenbeck (SU) for compact targets. Chapter II. Here we study a new intersection homology theory for currents on a space X with cone-like singularities. This homology is given by a new mass functional $M\sb{p}$ associated with the perversity index p. For X, it pairs with the intersection homology of Goresky-MacPherson, as well as the $L\sp2$-cohomology of J. Cheeger. We also give a deformation theorem and then prove the existence of $M\sb{p}$-minimizing currents in a given intersection homology class. Chapter III. We construct a weak solution for the heat flow associated with various quasiconvex functionals into homogeneous spaces, in particular, the p-harmonic map heat flow for any $p > 1.$ Our proof generalizes previous works (CHN), (CH2) which treated the case for $p \ge 2$ where the target is a sphere.
Inverse source problems for time-dependent radiative transport
(2014-03-20) Acosta Valenzuela, Sebastian; Borcea, Liliana; Riviere, Beatrice M.; Symes, William W.; Hardt, Robert M.; Alonso, Ricardo J
In the first part of this thesis, I develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small. Then, using duality arguments, I show that the solvability of the inverse problem leads to exact controllability of the transport field. The solution approach to both of these problems may have applications in areas such as optical imaging and optimization of radiation delivery. The second portion of the work is dedicated to the simultaneous recovery of a source of the form "s(t,x,d) f(x)" (with "s" known) and an isotropic initial condition "u0(x)", using the single measurement induced by these data. This result is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. More precisely, based on exact boundary controllability, I derive a system of equations for the unknown terms "f" and "u0". The system is shown to be of Fredholm type if "s" satisfies a certain positivity condition. This condition requires that the radiation visits the region over which "f" is to be recovered. I show that for generic term "s" and weakly absorbing media, the inverse problem is well-posed.
Minimizers of the vector-valued coarea formula
(2012-09-05) Carroll, Colin; Hardt, Robert M.; Wolf, Michael; Cox, Steven J.
The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.
Minimizing and flow problems for multiple-valued functions and maps
(2007) Zhu, Wei; Hardt, Robert M.; Wolf, Michael
We consider variational problems in the setting of multiple-valued functions (with a fixed number of values) and multiple-valued maps into manifolds. In particular, for an energy minimizing map into a sphere, we prove that the interior singular set is at least of codimension three. We also construct an energy reducing flow for multiple-valued functions, which is H older continuous with respect to its L 2 norms. Some questions concerning regularity and vanishing of branch points are also addressed.
Minimizing the mass of the codimension-two skeleton of a convex, volume-one polyhedral region
(2011) Scott, Ryan Christopher; Hardt, Robert M.
In this paper we establish the existence and partial regularity of a (d-2)-dimensional edge-length minimizing polyhedron in [Special characters omitted.] . The minimizer is a generalized convex polytope of volume one which is the limit of a minimizing sequence of polytopes converging in the Hausdorff metric. We show that the (d-2)-dimensional edge-length ζ d -2 is lower-semicontinuous under this sequential convergence. Here the edge set of the limit generalized polytope is a closed subset of the boundary whose complement in the boundary consists of countably many relatively open planar regions.
On geometry along grafting rays in Teichmuller space
(2012-09-05) Laverdiere, Renee; Wolf, Michael; Hardt, Robert M.; Goldman, Ron
In this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm\"{u}ller space. The main technique is to describe the hyperbolic metric $$\sigma_{t}$$ at a point along the grafting ray in terms of a conformal factor $$g_{t}$$ times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays.
On the approximation of the Dirichlet to Neumann map for high contrast two phase composites
(2013-09-16) Wang, Yingpei; Borcea, Liliana; Riviere, Beatrice M.; Gorb, Yuliya; Symes, William W.; Hardt, Robert M.
Many problems in the natural world have high contrast properties, like transport in composites, fluid in porous media and so on. These problems have huge numerical difficulties because of the singularities of their solutions. It may be really expensive to solve these problems directly by traditional numerical methods. It is necessary and important to understand these problems more in mathematical aspect first, and then using the mathematical results to simplify the original problems or develop more efficient numerical methods. In this thesis we are going to approximate the Dirichlet to Neumann map for the high contrast two phase composites. The mathematical formulation of our problem is to approximate the energy for an elliptic equation with arbitrary boundary conditions. The boundary conditions may have highly oscillations, which makes our problems very interesting and difficult. We developed a method to divide the domain into two different subdomains, one is close to and the other one is far from the boundary, and we can approximate the energy in these two subdomains separately. In the subdomain far from the boundary, the energy is not influenced that much by the boundary conditions. Methods for approximation of the energy in this subdomain are studied before. In the subdomain near the boundary, the energy depends on the boundary conditions a lot. We used a new method to approximate the energy there such that it works for any kind of boundary conditions. By this way, we can have the approximation for the total energy of high contrast problems with any boundary conditions. In other words, we can have a matrix up to any dimension to approximate the continuous Dirichlet to Neumann map of the high contrast composites. Then we will use this matrix as a preconditioner in domain decomposition methods, such that our numerical methods are very efficient to solve the problems in high contrast composites.
On the approximation of the Dirichlet to Neumann map for high contrast two phase composites and its applications to domain decomposition methods
(2014-08-01) Wang, Yingpei; Borcea, Liliana; Riviere, Beatrice M.; Symes, William W.; Hardt, Robert M.
An asymptotic approximation of the Dirichlet to Neumann (DtN) map of high contrast composite media with perfectly conducting inclusions that are close to touching is presented. The result is an explicit characterization of the DtN map in the asymptotic limit of the distance between the inclusions tending to zero. The approximation of DtN map is applied directly to nonoverlapping domain decomposition methods as preconditioners in order to obtain more computational efficiency.
Regularity and Nearness Theorems for Families of Local Lie Groups
(2011) McGaffey, Tom; Hardt, Robert M.
In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.3.1. The second result essentially says that a pair of mappings that almost satisfy the properties defining a local Lie group must have a local Lie group nearby, see proposition 7.2.1. Pairing the above two results, we get the principal standard consequence of the above work which can be roughly described as follows. If we have pointwise equicontinuous family of mapping pairs (potential local Euclidean topological group structures), pointwise approximating a (possibly differentiably unbounded) family of differentiable (sufficiently approximate) almost groups, then the original family has, after appropriate coordinate change, a local Lie group as a limit point. (See corollary 7.2.1 for the exact statement.) The third set of results give nonstandard renditions of equicontinuity criteria for families of differentiable functions, see theorem 9.1.1. These results are critical in the proofs of the principal results of this paper as well as the standard interpretations of the main results here. Following this material, we have a long chapter constructing a Hausdorff topology on the ring of real valued map germs on Euclidean space. This topology has good properties with respect to convergence and composition. See the detailed introduction to this chapter for the motivation and description of this topology.
Regularity of minimizing maps and flows various functionals and targets
(1996) Wang, Changyou; Hardt, Robert M.
This thesis discusses regularity problems of minimizing maps and flows for various functionals and targets. It consists of four parts: Part 1. Energy Minimizing Mappings into Polyhedra. We prove both the partial interior and complete boundary regularities for maps which minimize energy among all maps into a polyhedron. Part 2. Bubbling Phenomena of Certain Palais-Smale Sequences from Surfaces into General Targets. We show that there is no unaccounted loss of energy for certain Palais-Smale sequences from a surface into a general manifold during the process of bubbling. We also discuss the harmonicity of weak limits of general Palais-Smale sequences. Part 3. Maps Minimizing Convex Functionals between Riemannian Manifolds. We show that any map, which minimizes a uniformly strictly convex $C\sp2$ functionals $F$ among all maps from one manifold to another manifold, has Holder continuous first gradient away from a closed subset with Lebesgue measure zero. Part 4. Existence and Partial Regularity of Weak Flows of Convex Functionals. Assume we are given a $C\sp2$ convex functional $F$, we prove the existence of a weak flow associated to it. We also prove that such a weak flow has Holder continuous spatial gradient away from a closed subset with Lebesgue measure zero.
Singularities of subanalytic sets and energy minimizing maps
(1993) Wang, Shiah-Sen; Hardt, Robert M.
This thesis studies some problems derived from differential topology and differential geometry by techniques developed from geometric measure theory, variational calculus, and partial differential equations. It consists of two independent parts: Part I: An isoperimetric type inequality for chains on singular spaces. We find an isoperimetric type inequality for integral chains with support in a subset of $\IR\sp{n},$ which satisfies some structural conditions but is not in the Lipschitz category. We also apply this inequality to derive some results in the subanalytic category for homologically mass minimizing currents. Part II: Energy minimizing sections of a fiber bundle. We show that a Dirichlet p-energy minimizing section of a fiber bundle is Holder continuous everywhere except possibly for a closed subset of Hausdorff dimension at most $m - \lbrack p\rbrack\ - 1$, where m is the dimension of the base space of the fiber bundle and (p) is the greatest integer less than or equal to p.

Browsing by Author "Hardt, Robert M."

Results Per Page

Sort Options