


Document Library: Series 1 Abstracts

[1.29] 
Eydeland, A., Spruck, J. & Turkington, B.
Multiconstrained variational problems of nonlinear eigenvalue type: new formulations and algorithms
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abstract
[From the introduction]
The present work was motivated by a reexamination of some fundamental
constrained variational problems that arise in equilibrium theory in
fluid dynamics and Magnetohydrodynamics. These problems lead
formally to nonlinear eigenvalue problems as variational
equations. Indeed, it is traditional that many fundamental problems of
mathematical physics can b e formulated as nonlinear eigenvalue
problems of the form
\[\Delta u = \Lambda(u), u\in H_0^1(\Omega)\]
The profile function $\Lambda(u) is usually expressed in the form
\$\Lambda(u)=\lambda f(u)$ where $f(u)$ is a known function and
$\lambda$ is a Lagrange multiplier.
 [1.30] 
Baldes, A. & Wohlrab, O.
Computer graphics of solutions of the generalized MongeAmpere equation
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Using the method of Minkowski or Alexandrov one finds simple discretizations
of elliptic MongeAmp\022ere equations, including the equation of graphs
with prescribed positive Gaussian curvature. It is shown how these
discrete problems can b e solved numerically, and computer graphics of the
piecewise linear, convex solutions are presented.
 [1.31] 
Hoffman, D.
New examples of singlyperiodic minimal surfaces and their qualitative behavior
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[Excerpt]
I would like to describe some recent research, concerning properly embedded
minimal surfaces with periodicity that I have been doing with Michael
Callahan and Bill Meeks III. The work includes the construction of new
examples and the characterization of the qualitative behavior of all
examples in an important class. It is based, in turn, on an analysis of the
geometric behavior of such surfaces at infinity. This talk is divided into
two parts; the first discusses the examples and qualitative results; and the
second gives a feeling for the analytical background.
 [1.36] 
Meeks III, W. H.
The theory of triply periodic minimal surfaces
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Working primarily within the conformal category, we develop complementary
existence and rigidity theorems for periodic minimal
surfaces in $R^n$ . Applying this theory, we prove: 1. Every flat
threetorus contains an infinite number of genus 3 embedded minimal
surfaces; 2. Necessary and sufficient conformal conditions for a closed
Riemann surface of genus g to conformally minimally immerse in a
flat $3$ or $(2g1)$torus; 3. The existence of distinct isometric
minimal surfaces in flat tori; 4. Special results on the geometry of
minimal surfaces of genus 3 and of classical examples of minimal
surfaces in flat threetori; 5. The determination of the group of
symmetries of certain minimal surfaces in R^{3}.
 [1.37] 
Evans, L. C. & Spruck, J.
Motion of level sets by mean curvature I
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abstract
[From the introduction]
We set forth in this paper rigorous justification of a new approach
for defining and then investigating the evolution of a hypersurface
in R n moving according to its mean curvature. This problem has been
long studied using parametric methods of differential geometry: see,
for instance, Gage, GageHamilton, Grayson, Huisken, etc., etc. In
this classical setup, we are given at time 0 a smooth hypersurface
$\Gamma_0$ which is, say, the connected boundary of a bounded open subset
of $R^n$. As time progresses we allow the surface
to evolve, by moving each point in the opposite direction to the
mean curvature vector, at a velocity equal to $(n1)$ times the
absolute value of the mean curvature at that point. Assuming this
evolution is smooth, we define thereby for each $t > 0$ a new
hypersurface $\Gamma_t$. The primary problem is then to study geometric
properties of $\{\Gamma_t\}_{t>0} in terms of $\Gamma_0$.
 [1.38] 
Evans, L. C. & Spruck, J.
Motion of level sets by mean curvature II
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abstract
[From the introduction]
We present in this paper a new, elementary, and fairly concise proof
of short time existence for the classical motion of a smooth
hypersurface evolving according to its mean curvature. In this problem we
are given initially a smooth connected hypersurface $\Gamma_0$ which is the
boundary of a bounded open set $U\subset R^n$. We then allow $\Gamma_0$ to
evolve in time into a family of surfaces $\{\Gamma_t\}_{t>0}$ by moving
each point on $\Gamma_t (t\ge 0)$ in the opposite direction to its mean
curvature vector, at a velocity equal to $(n1)$ times the absolute
value of the mean curvature. Our intent is to verify that for small
times at least, the classical motion as envisioned in fact exists and is
unique. This assertion was first proved by R. Hamilton [4], and we
discuss below the relation of our work to his.
 [1.39] 
Caffarelli, L. A. & Spruck, J.
Variational problems with critical Sobolev growth and positive Dirichlet data
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dvi
abstract
[From the introduction]
In this paper, we consider the Dirichlet problem for the conformally
invariant model problem of critical Sobolev growth: \[\] Problem (1.1)
is formally the EulerLagrange equations for the variational problem
\[\] for $u$ in the admissible class \[\] where $h$ is the harmonic
extension of $\phi\ge 0$.


© GANG 2001
