Preliminary Materials

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


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
$$\displaystyle C^l_c(U;\mathbb {R}^n) $$
$$\displaystyle C_c(U;\mathbb {R}^n):=C_c^0(U;\mathbb {R}^n). $$
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.


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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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