The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$ as in Hilbert Triples: Standing Notation and Background (v), let $\mathrm{Sym}(H)$ with its sums, differences and scalar multiples be as in Hilbert Triples: Standing Notation and Background (vi), and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$, whose test data form the set $\mathcal{W}=W\times\mathbb{R}\times H\times\mathrm{Sym}(H)$; that a test datum is $R$-bounded, for a positive $R\in\mathbb{R}$, is as defined there. Convergence of a sequence in $H$ or in $\mathbb{R}$ is as fixed in Hilbert Triples: Standing Notation and Background (i) and Real Hilbert Spaces: Standing Notation and Background (i).

1. (The tail forms of a basis lying in $V$)(i) Let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ such that $e_{k}\in V$ for every $k\in\mathbb{N}$. For $m\in\mathbb{N}$ let $e^{(m)}\in H^{m}$ be the $m$-tuple with components $e_{1},\dots,e_{m}$, which is orthonormal because $(e_{k})_{k\in\mathbb{N}}$ is, and all of whose components lie in $V$; and let $N_{m}\in\mathrm{Sym}(H)$ be the tail form of $e^{(m)}$. The sequence $(N_{m})_{m\in\mathbb{N}}$ is called the sequence of tail forms of $(e_{k})_{k\in\mathbb{N}}$.

2. (Tail-insensitivity along a basis)(ii) Let $(e_{k})_{k\in\mathbb{N}}$ and $(N_{m})_{m\in\mathbb{N}}$ be as in clause 1. The operator $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$ if the following two conditions hold for all $\beta,R,\rho\in\mathbb{R}$ with $0<\beta$, $0<R$ and $0<\rho$, and every $\delta\in\mathbb{R}$ with $0<\delta<1$.

(a) (Subsolution side)(iii) Let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $W$ that converges in $H$, let $(r_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathbb{R}$ that converges, let $(p_{m})_{m\in\mathbb{N}}$ be a sequence in $H$ that converges in $H$, and let $(X_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathrm{Sym}(H)$ such that the test datum $(x_{m},r_{m},p_{m},X_{m}+\beta N_{m})$ is $R$-bounded for every $m\in\mathbb{N}$. Then there is $m_{0}\in\mathbb{N}$ such that every $m\in\mathbb{N}$ with $m_{0}\le m$ satisfies

$$F^{-}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}\bigr)\ \le\ F^{-}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}+\beta N_{m}\bigr)+\rho .$$

(b) (Supersolution side)(iv) Let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $W$ that converges in $H$, let $(r_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathbb{R}$ that converges, let $(p_{m})_{m\in\mathbb{N}}$ be a sequence in $H$ that converges in $H$, and let $(X_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathrm{Sym}(H)$ such that the test datum $(x_{m},r_{m},p_{m},X_{m}-\beta N_{m})$ is $R$-bounded for every $m\in\mathbb{N}$. Then there is $m_{0}\in\mathbb{N}$ such that every $m\in\mathbb{N}$ with $m_{0}\le m$ satisfies

$$F^{+}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}-\beta N_{m}\bigr)-\rho\ \le\ F^{+}_{\delta}\bigl(x_{m},r_{m},p_{m},X_{m}\bigr).$$

3. (The tail-insensitivity condition)(v) The operator $F$ satisfies the tail-insensitivity condition if there is an orthonormal basis of $H$, all of whose members lie in $V$, along which $F$ is tail-insensitive.

The tail form satisfies $0_{\mathrm{Sym}}\preceq N_{m}$ by Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (iii), so for a degenerate elliptic operator the left-hand side of the inequality in (a) is at least its right-hand side with $\rho$ removed, and the condition asserts that the two sides come together; the same holds in (b). A basis as in clause 1 exists whenever $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional, by An Orthonormal Basis of the Ambient Space Contained in the Form Space of a Hilbert Triple (i); when $H$ is finite-dimensional there is no orthonormal basis in the sense of Orthonormal Basis of a Real Hilbert Space (i) at all, and no operator satisfies the condition of clause 3. This is the counterpart of the condition (F4) of Ishii recorded in the citation, with the admissible sets there replaced by the larger sets of $R$-bounded test data; the requirement is thereby imposed on more sequences.