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Preliminary Materials

  • Yoshihiro Tonegawa
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Throughout, 1 ≤ k < n are integers, \(\mathbb {N}\) is the set of natural numbers, \(\mathbb {R}^n\) is the n-dimensional Euclidean space and
$$\displaystyle \mathbb {R}^+:=\{x\in \mathbb {R} : x\geq 0\}. $$
Let \(U\subset \mathbb {R}^n\) be an open set. The symbol
$$\displaystyle C^l_c(U) $$
denotes the set of l times continuously differentiable functions with compact support in U and
$$\displaystyle C_c(U):=C_c^0(U). $$
A function \(f:U\rightarrow \mathbb {R}\) is said to be Lipschitz if
$$\displaystyle \mathrm {Lip}(f):=\sup _{x,y\in U}\frac {|f(x)-f(y)|}{|x-y|}<\infty . $$
The set of vector fields with each component in \(C^l_c(U)\) is denoted by
$$\displaystyle C^l_c(U;\mathbb {R}^n) $$
and
$$\displaystyle C_c(U;\mathbb {R}^n):=C_c^0(U;\mathbb {R}^n). $$
For \(g\in C^1_c(U;\mathbb {R}^n)\), we define ∇g as the n × n matrix-valued function whose first row is composed of partial derivatives of the first component of g and so forth. By ∘, we indicate the usual matrix multiplication.

References

  1. 1.
    Allard, W.: On the first variation of a varifold. Ann. Math. (2) 95, 417–491 (1972)MathSciNetCrossRefGoogle Scholar
  2. 4.
    Almgren, F.J., Jr.: Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints. Memoirs of the American Mathematical Society, vol. 4, no.165. American Mathematical Society, Providence (1976)Google Scholar
  3. 7.
    Brakke, K.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)Google Scholar
  4. 12.
    Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, Revised edn. CRC Press, Boca Raton (2015)Google Scholar
  5. 33.
    Simon, L.: Lectures on Geometric Measure Theory. In: Proceedings of Centre for Mathematical Analysis, Australian National University, vol. 3 (1983)Google Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Yoshihiro Tonegawa
    • 1
  1. 1.Tokyo Institute of TechnologyTokyoJapan

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