# Size minimization and approximating problems

## Related Articles

- Existence of integral $$m$$-varifolds minimizing $$\int \!|A|^p$$ and $$\int \!|H|^p,\,p>m,$$ in Riemannian manifolds. Mondino, Andrea // Calculus of Variations & Partial Differential Equations;Jan2014, Vol. 49 Issue 1/2, p431
We prove existence of integral rectifiable $$m$$-dimensional varifolds minimizing functionals of the type $$\int |H|^p$$ and $$\int |A|^p$$ in a given Riemannian $$n$$-dimensional manifold $$(N,g),\,2\le m

m,$$ under suitable assumptions on $$N$$ (in the end of the paper we give many... - Non-unique conical and non-conical tangents to rectifiable stationary varifolds in $$\mathbb R^4$$. Kolář, Jan // Calculus of Variations & Partial Differential Equations;Oct2015, Vol. 54 Issue 2, p1875
We construct a rectifiable stationary 2-varifold in $${\mathbb {R}}^4$$ with non-conical, and hence non-unique, tangent varifold at a point. This answers a question of Simon (Lectures on geometric measure theory, p 243, ) and provides a new example for a related question of Allard (Ann Math (2)...

- Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension. Bangert, Victor; Röttgen, Nena // Calculus of Variations & Partial Differential Equations;Nov2012, Vol. 45 Issue 3/4, p455
For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar...

- Sobolev functions on varifolds. Menne, Ulrich // Proceedings of the London Mathematical Society;Dec2016, Vol. 113 Issue 6, p725
This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts....

- A general regularity theory for weak mean curvature flow. Kasai, Kota; Tonegawa, Yoshihiro // Calculus of Variations & Partial Differential Equations;May2014, Vol. 50 Issue 1/2, p1
We give a new proof of Brakke's partial regularity theorem up to $$C^{1,\varsigma }$$ for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is...

- THE HARDY INEQUALITY WITH ONE NEGATIVE PARAMETER. Kufner, A.; Kuliev, K.; Kulieva, G. // Banach Journal of Mathematical Analysis;2008, Vol. 2 Issue 2, p76
In this paper, necessary and sufficient conditions for the validity of the Hardy inequality for the case q < 0, p > 0 and for the case q > 0, p < 0 are derived.

- THREE-PARAMETER WEIGHTED HARDY TYPE INEQUALITIES. Oinarov, R.; Kalybay, A. // Banach Journal of Mathematical Analysis;2008, Vol. 2 Issue 2, p85
For 0 < r < âˆž and 1 â‰¤ p â‰¤ q < âˆž we find necessary and sufficient conditions for the validity of the following inequality: (Multiple line equation(s) cannot be represented in ASCII text) where u(Â·), v(Â·), and w(Â·) are weight functions.

- Evasion and prediction. Brendle, J�rg; Shelah, Saharon // Archive for Mathematical Logic;May2003, Vol. 42 Issue 4, p349
Say that a function p:n[sup

- The Cendian of a Tree. Win, Z.; Myint, Y. // Southeast Asian Bulletin of Mathematics;2002, Vol. 25 Issue 4, p757
The eccentricity e(x) and the distance sum s(x) of a vertex x of a connected graph G are well-known functions which measure the centrality of the vertex x in G. The set of vertices which minimize e(x) is called the center of G and the set of vertices which minimize s(x) is known as the median....