Appendix J Notation
| Symbol | Description | Location |
|---|---|---|
| \(\delta_{ij}\) | Kronecker delta | Notation 2.1 |
| \(e_i\) | canonical basis vector | Notation 2.1 |
| \(\langle x,y\rangle\) | dot product (inner product) | Example 2.3 |
| \(x \cdot y\) | dot product (inner product) | Example 2.3 |
| \(\abs x\) | norm of a vector | Definition 2.5 |
| \(\abs A\) | norm of a matrix | Definition 2.5 |
| \(\abs\alpha\) | order of a multiindex | Definition 2.7 |
| \(\Omega\) | non-empty open subset of \(\R^N\) | Notation 2.8 |
| \(\partial_j f\) | \(\partial f/\partial x_j\) | Notation 2.9 |
| \(\nabla f\) | gradient of a function | Notation 2.9 |
| \(DF\) | Jacobian matrix of a (vector-valued) function | Notation 2.9 |
| \(\nabla\cdot F\) | divergence of a vector field | Notation 2.9 |
| \(\partial_{ij} f\) | \(\partial_i \partial_j f\) | Notation 2.10 |
| \(D^2 f\) | Hessian of a function | Notation 2.10 |
| \(\Delta f\) | Laplacian of a function | Notation 2.10 |
| \(\partial^\alpha\) | \(\partial_1^{\alpha_1} \cdots \partial_N^{\alpha_N}\) | Notation 2.11 |
| \(C^0(A)\) | continuous functions \(A \to \R\) | Notation 2.19 |
| \(C^k(\Omega)\) | \(C^k\) functions \(\Omega \to \R\) | Notation 2.19 |
| \(C^k(\overline \Omega)\) | partials extend to elements of \(C^0(\overline\Omega)\) | Notation 2.19 |
| \(\n f_{C^k(\Omega)}\) | \(C^k\) norm of a function | Notation 2.19 |
| \(C^\infty(\Omega)\) | \(\bigcap_{k \in \N} C^k(\Omega)\) | Notation 2.19 |
| \(C^\infty(\overline \Omega)\) | \(\bigcap_{k \in \N} C^k(\overline \Omega)\) | Notation 2.19 |
| \(B_r(x_0)\) | open ball with radius \(r\) and centre \(x_0\) | Definition 2.23 |
| \(B\) | some open ball | Definition 2.23 |
| \(\partial\Omega\) | boundary of a set | Definition 2.24 |
| \(\max_A f\) | maximum of a function over a set | Definition 2.32 |
| \(\min_A f\) | minimum of a function over a set | Definition 2.32 |
| \(\inf_A f\) | infimum of a function over a set | Definition 2.33 |
| \(\sup_A f\) | supremum of a function over a set | Definition 2.33 |
| \(n\) | unit normal vector | Definition 2.47 |
| \(\frac{\partial u}{\partial n}\) | normal derivative \(n \cdot \nabla u\) | Definition 2.47 |
| \(L\) | an elliptic operator \(a_{ij} \partial_{ij} + b_i \partial_i + c\) | Section 3.1 |
| \(a_{ij},b_i,c\) | coefficients of an elliptic operator \(L\) | Section 3.1 |
| \(\lambda_0\) | ellipticity constant | Definition 3.1 |
| \(\tdashint\) | average | Notation 4.9 |
| \(\abs B\) | volume of a ball | Notation 4.9 |
| \(\abs{\partial B}\) | surface area of a sphere | Notation 4.9 |
| \(T_\lambda(e)\) | hyperplane \(\{ \xi : \xi\cdot e = \lambda \}\) | Definition 5.2 |
| \(x^{\lambda,e}\) | reflection of \(x\) through \(T_\lambda(e)\) | Definition 5.2 |
| \(T_\lambda\) | shorthand for \(T_\lambda(e_1)\) | Notation 5.5 |
| \(\Sigma(\lambda)\) | \(B \cap \{x_1 \gt \lambda\}\) | Notation 5.5 |
| \(x^\lambda\) | shorthand for \(x^{\lambda,e_1}\) | Notation 5.5 |
| \(u^\lambda\) | reflected function \(u^\lambda(x)=u(x^\lambda)\) | Notation 5.5 |
| \(D\) | parabolic interior | Paragraph |
| \(Q\) | usual interior | Paragraph |
| \(\Sigma\) | parabolic boundary | Paragraph |
| \(L\) | a parabolic operator \(-\partial_t + a_{ij} \partial_{ij} + b_i \partial_i + c\) | Paragraph |
| \(\mathcal S(u)\) | a semilinear parabolic operator | Paragraph |
