# Generalities

- For a brief description, it may be written as $vw$ by omitting ``$\cdot $'' in the inner product $v\cdot w$ of the vector valued functions $v,w$.

- The Einstein summation is used, for example $a_ib_i =\sum _{i} a_ib_i$, and the Frobenius innter product of matrices $A = (a_{ij}),B = (b_{ij})$ is written by $A : B = a_{ij}b_{ij} = tr(A^TB)$ where $tr$ is the trace operator and $A^T$ the transpose of $A$.
- $\partial _k$ is $\partial /\partial x_k$.

# Geometry

- $\Omega $: Bounded domains defining boundary value problems for partial differential equations.
- $D$: General domain. In crack problems, a domain without crack surface $\Sigma $, In crack problem $\Omega =D\setminus D$.
- $\mu , \mu _{\varphi }, \mu _C$: Vector fields, $\mu _{\varphi }=d\varphi _t/dt |_{t=0}$, and the vector field obtained from a crack growth.
- $\gamma $: Singular point. $\Gamma $: Set of singular points, for example, the boundary $\partial \Omega $ is always a set of singularities.

# Function space

- $C^{0,1}(D)$ is the space whose functions are Lipschitz continuous.
- For a Banach space $X$, $C^0([0,T]; X)$: space of continuous functions of $[0,T] →X$, with the norm $\textrm{sup}_{t\in [0,T]}\| f(t)\|_X$.
- $C^k([0,T]; X)$: space of differentiable functions of $[0,T] →X$, such that $d^kf(t)/dt^k \in C^0([0,T];X)$.

## Sobolev space

- $L^{\infty }(D)$ stands for essentially bounded Lebesgue functions in $D$.
- $W^{s,p}(D)$ is the Sobolev spaces defined on a domain (surface) $D$ whose functions are (an extended) generalized derivatives up to the order $s\in [0,\infty ]$ in $L^p(D), p\in [1,\infty ]$ with the norm $\|\cdot \|_{s,p,D}\,(\|\cdot \|_{0,p,D} =\|\cdot \|_{L^p(D)})$.
- $W^{s,p}(D;\mathbb{R}^m)$ is the Sobolev space whose elements are vector-valued functions $v = (v_1, …, v_m),m \gt 1$ satisfying $v_i\in W^{s,p}(D), i = 1,…,m$ with the norm $\|v\|_{s,p,D}$ without distinction from $W^{s,p}(D)$.
- $H^s(D) = W^{s,2}(D)$ becomes the Hilbert space with the norm $\|\cdot \|_{s,D}$, and $(\cdot , \cdot )_D$ stands for the inner product in $L^2(D)$. Also in $W^{s,2}(D;\mathbb{R}^m), \|\cdot \|_{s,D}$ and $(\cdot ,\cdot )_D$ are the same as in $W^{s,2}(D)$.
- For a function $g$ defined on $D$, $\textrm{supp}_Dg = \{x \in D : g(x)\neq 0\}$,

# Variational method

- $\widehat{W}(\xi ,z,\zeta )$: Density function $\widehat{W}(x,u,\nabla u)$ of energy functional $\mathcal{E}(u;f,g,\Omega )$ is $\widehat{W}(\xi ,z,\zeta )|_{\xi =x,z=u,\zeta =\nabla u}$.
- For an elastic body given a body force $f$ and a surface force $g$, the energy is $$ \mathcal{E}(u;f,g,\Omega )=\int _{\Omega }\{\widehat{W}(x,u,\nabla u)-fu\} dx-\int _{\Gamma _N}gu~ds$$