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Results tagged with calculus-of-variations
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user 121665
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
5
votes
Does there exist energy-minimizing immersions?
I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A_*=A(r …
- 26.5k
5
votes
Accepted
Uniqueness of minimizers in a problem in the Calculus of Variations - Part II
Existence of minimizers should not be a severe issue...
The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in:
E. G …
- 26.5k
2
votes
Hausdorff dimension of the non-differentiability set a convex function
This is true and it follows from the following result:
Lemma. Suppset that $f:W\to\mathbb{R}$ is convex and $L$-Lipschitz, where $W\subset\mathbb{R}^n$ is convex. Then,
$$
\tilde{f}(x)=\inf_{z\in W}\ …
- 26.5k
6
votes
Accepted
Is $L^1$ strong convergence of Jacobians valid for maps between manifolds?
You actually do not need to assume that the mappings are Lipschitz as it is true for general $W^{1,n}$ mappings
Theorem. If $\mathcal{M}$ and $\mathcal{N}$ are smooth compact and oriented manifolds, …
- 26.5k
7
votes
Accepted
Bounded deformation vs bounded variation
Example 7.7 in
L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso,
Fine properties of functions with bounded deformation.
Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.
- 26.5k
12
votes
Accepted
An $L^1$ function but (really) no better?
There is a much more general result of Vallée-Poussin from which a negative answer to your question follows.
Let $(X,\mu)$ be a measure space. We say that a family of function $\mathcal{F}\subset L^1( …
- 26.5k