All Questions
Tagged with geometric-measure-theory calculus-of-variations
44 questions
6
votes
1
answer
172
views
Mass minimizing current in real homology class
It is a well-known results by Federer and Fleming that there exists at least one mass-minimizing normal current in every real homology class of a closed $n$-dimensional Riemannian manifold $M$. Their ...
- 61
0
votes
0
answers
38
views
A general rule for approximating the perimeter of a set with finite perimeter in terms of the volume
I want to know if it is possible to have a general rule for approximating the perimeter of a set $E\subset \mathbb{R}^n$ with finite perimeter in terms of the volume (Lebesgue measure) of a sequence ...
- 747
1
vote
0
answers
96
views
Periodic orbits in planar smooth billiard table with large periods
Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period.
Formulation of my question: We are considering ...
- 441
4
votes
1
answer
181
views
Is every area-minimizing cone a level set of a least-gradient function?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
- 5,038
5
votes
0
answers
114
views
Varifold convergence of images of Sobolev maps
Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that:
$f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$,
The images $\Sigma_k:...
- 221
1
vote
0
answers
97
views
Reference request: theory for local minimizers in the calculus of variations
Let $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be the Lagrangian. We say that $f \in X$ is a local minimizer of the variational integral if for all compact sets $C \subset \...
- 161
2
votes
0
answers
65
views
Defining minimality 'through deformations'
Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
- 5,038
6
votes
1
answer
634
views
What is the current status on bad tangent cones at isolated singularities?
Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.
Question.
...
- 5,038
5
votes
1
answer
504
views
Tangent cones at zero and infinity to minimal surfaces
Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth:
$\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \...
- 5,038
4
votes
0
answers
162
views
Density of smooth functions in the calculus of variations
In the non-convex calculus of variations, in the context of non-linear elasticity, the following classes of mappings $u:\Omega\to\mathbb{R}^n$, $\Omega\subset\mathbb{R}^n$, were introduced by John ...
- 28k
5
votes
1
answer
500
views
Is every set with finite $\mathcal{H}^{n-1}$ measure a set of locally finite perimeter?
Given a measurable set $E \subset \mathbb{R}^d$, with $\mathcal{H}^{d-1} (\partial E) < +\infty$, is it true in general that $E$ is a set of locally finite perimeter? that is, is it true that $\...
- 747
6
votes
1
answer
425
views
Lipschitz property of the symmetric rearrangement
I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...
- 457
0
votes
0
answers
43
views
Minimal condition on set for an optimisation problem
We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{E}(\...
- 181
3
votes
0
answers
102
views
When is the least-area surface unique?
Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...
- 5,038
8
votes
3
answers
804
views
How to interpret this quote of Lin?
I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42].
It is a well-known fact that a weakly converging sequence of stationary integral currents may have a ...
- 5,038
4
votes
2
answers
286
views
Area-minimising hypersurface with unbounded area growth
Let $T$ be an $n$-dimensional area-minimising hypersurface in $\mathbf{R}^{n+1}$. If $T$ has bounded area growth in the sense that there is a constant $C > 0$ so that $\mathcal{H}^n(T \cap B_R) \...
- 5,038
1
vote
1
answer
213
views
Stability of isoperimetric inequality
Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1.
Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$,
where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...
- 2,576
10
votes
1
answer
696
views
How to shrink a square with minimal distortion?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\euc}{\mathfrak{e}}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\al}{\alpha}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\Lip}{\text{Lip}_{\text{inj}}...
- 6,741
1
vote
0
answers
157
views
Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
- 6,741
1
vote
1
answer
201
views
Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
1
vote
1
answer
146
views
Is a locally invertible weak limit of injective maps injective almost everywhere?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...
- 6,741
2
votes
0
answers
77
views
Area of minimising surface
I am interested in calculating the least area of a surface spanning the boundary of an octant on the unit sphere; and short of precise values I am looking for upper bounds for this area.
In $\mathbf{S}...
- 5,038
3
votes
1
answer
138
views
Boundary regularity of rectifiable multiplicity 1 hypercurrents
Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
- 31
6
votes
0
answers
249
views
Do asymptotically conformal maps converge to a weakly conformal map?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, ...
- 6,741
6
votes
0
answers
171
views
The distributional gradient of the closest isometry to the differential of a smooth map
The setting-a "linear algebra" fact:
Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
- 6,741
7
votes
1
answer
504
views
Anisotropic perimeter and regularity of anisotropic minimal surfaces
1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...
- 980
2
votes
0
answers
82
views
Volume of critical points decreases under symmetric decreasing rearrangement
In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...
- 21
5
votes
1
answer
412
views
Continuous deformation of soap films
Let $S$ be a soap film bounded by an unknotted wireframe cycle (in $R^3$). Why is it the case that as we deform the wireframe in $R^3$, $S$ deforms continuously?
- 51
2
votes
0
answers
115
views
Currents with mean curvature
so let $T=\tau(M,\theta,\xi)$ a rectifiable current in $U\subset\mathbb{R}^{n+k}$, then $T$ has mean curvature vector $H$ if for every $X\in C^1_c(U\setminus\partial T,\mathbb{R}^{n+k})$ the variation ...
- 43
1
vote
1
answer
269
views
Existence of stationary tangent cones
My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42
So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)&...
- 43
1
vote
0
answers
64
views
decomposition of codim 1 currents
Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...
- 43
8
votes
3
answers
1k
views
Almgren's mimeographed lectures notes on varifolds
I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
- 1,616
4
votes
1
answer
241
views
Is a function $u\in \mathrm{SBV}(\Omega)$ with these additional properties essentially bounded?
Some related earlier discussion can be found here.
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary, $\mathcal H^{N-1}(\partial\Omega)<\infty$ and $u\in SBV(\Omega)$. Then
$$
...
- 679
2
votes
0
answers
285
views
Method of proving the regularity of the minimizer of geometric variational problems
Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer.
We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants $...
- 1,350
1
vote
1
answer
83
views
Convergent algorithm for dividing a body into two regions of equal volume
Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ ...
6
votes
2
answers
722
views
Stability of minimal surfaces
Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
5
votes
1
answer
234
views
Minimizing the perimeter around an obstacle
Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all Caccioppoli subsets ...
- 2,270
2
votes
0
answers
175
views
An isoperimetric type maximization problem with a barrier.
I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details:
Let $(r(\theta), \theta)$ be a ...
- 2,641
3
votes
0
answers
278
views
What are the most general types of curves in $\mathbb{R}^2$ for which Gauss-Bonnet holds?
I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has
$...
- 2,641
4
votes
1
answer
352
views
Is there a characterization of generalized constant mean curvature surfaces?
It is a well known result of Alexandrov that the only compact, connected, constant mean curvature surface is the ball.
There is a generalized notion of curvature known as generalized mean curvature ...
- 2,641
6
votes
2
answers
656
views
Minimal surface which divides a convex body into two regions of equal volume
Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume?
Background/motivation.
A 2D version of the ...
- 22.3k
3
votes
0
answers
401
views
Generalization of First Variation of Area
The area of an $m$-rectifiable varifold in $n$-dimensional space can be expressed in terms of the surface divergence. More precisely, if $M$ is $m$-rectifiable, $\Omega$ is open, $\eta$ is a $C^1_c$ ...
- 320
0
votes
1
answer
154
views
optimize with respect to domain shape
Let $\Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $\Omega$ be the set of all subsets of $\mathbb{R}^2$ that are enclosed by some curve in $\Gamma$. Now ...
16
votes
6
answers
3k
views
Smallest area shape that covers all unit length curve
On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this ...
- 613