Second-order elliptic semijets and their value-sensitive closures

Let $U\subseteq {\mathbb{R}}^m$ be open, $u:U\to \mathbb{R}$, and $z\in U$. Write ${\mathbb{S}}^m$ for the real symmetric $m$ by $m$ matrices. The second-order elliptic superjet consists of pairs $(p,X)\in {\mathbb{R}}^m\times {\mathbb{S}}^m$ such that $u(z+h)\le u(z)+p\cdot h+\frac{1}{2}{h}^{\mathsf{T}}Xh+o(\mathrm{\mid }h{\mathrm{\mid }}^2)(h\to 0)\mathrm{.}$ Here the inequality means that its positive remainder divided by $\mathrm{\mid }h{\mathrm{\mid }}^2$ has limit zero. Denote this set by $J^{2,+}u(z)$. Define $J^{2,-}u(z)=-J^{2,+}(-u)(z)\mathrm{.}$ The closed superjet ${\bar{J}}^{2,+}u(z)$ consists of all limits $(p,X)$ for which there are $z_n\in U$ and $(p_n,X_n)\in J^{2,+}u(z_n)$ satisfying $(z_n,u(z_n),p_n,X_n)⟶(z,u(z),p,X)\mathrm{.}$ Define ${\bar{J}}^{2,-}u(z)=-{\bar{J}}^{2,+}(-u)(z)$. The value convergence is part of the definition. The same definitions work for functions allowed to equal $-\mathrm{\infty }$ away from the finite base point, with all approximating base values finite. A two-sided quadratic expansion means that both the upper and lower inequalities hold with the same $(p,X)$; it is also called a second-order Peano expansion. Matrix order means $X\le Y$ iff $h^{\mathsf{T}}Xh\le h^{\mathsf{T}}Yh$ for every $h$.

For space-time $z=(t,x)$ this is the full Euclidean jet, whose matrix also has time and mixed entries. No assertion about parabolic jets is included in this definition.