A First-Order Equation Operator is Degenerate Elliptic and Its -Shifts Ignore the Form Argument
lemmaAnalysisPDElem:first-order-operator-hilbert-triple-2026aA first-order equation operator on a Hilbert triple is degenerate elliptic, and each of its delta-shifts takes the same value at all form arguments.
In the setting of Hilbert Triples: Standing Notation and Background, let be open in , with as in Hilbert Triples: Standing Notation and Background §open-sets, let and be as in Hilbert Triples: Standing Notation and Background §restriction, and let be a second-order equation operator on relative to that is first order, with -shifts and . Let be positive. Then the following hold.
1. (Degenerate ellipticity)¶ The operator is degenerate elliptic.
2. (The shifts ignore the form argument)¶ For every , every , every and all ,
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