Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Uniqueness of a Bounded Continuous Viscosity Solution on a Hilbert Triple
corollarycor:uniqueness-bounded-continuous-solution-hilbert-triple-2026aAnalysisPDEUnder the hypotheses of the comparison principle, two bounded viscosity solutions that are continuous on the whole space coincide.Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple
theoremthm:bounded-uniformly-continuous-solution-hilbert-triple-2026aAnalysisPDEIf the operator is degenerate elliptic, locally strictly proper and satisfies the first-order structure and shift-continuity conditions, and if two constants are respectively a classical subsolution and a classical supersolution, then the equation has a viscosity solution on the…The Viscosity Property Depends Only on the Values on the Trace of
lemmalem:viscosity-trace-values-hilbert-triple-2026aAnalysisPDETwo locally bounded functions on an open subset of a Hilbert triple that agree on the trace of V have the same delta-envelopes, so one is a viscosity subsolution, supersolution or solution exactly when the other is.Extension of a Uniformly Continuous Real Function from a Dense Subset
lemmalem:uniformly-continuous-dense-extension-2026aAnalysisTopologyA uniformly continuous real function on a dense subset of a metric space extends to a uniformly continuous function on the whole space; the extension is unique among continuous functions, and it inherits any modulus and any bound of the original function.The Dissipative Hamilton-Jacobi Operator on a Hilbert Triple Satisfies the Comparison Hypotheses
propositionprop:dissipative-hamilton-jacobi-hilbert-triple-2026aAnalysisPDEThe operator sending (x,r,p,X) to lambda r + |p|^2/2 + <Ax,p> - g(x), with g bounded and having a modulus of continuity, is a first-order, locally strictly proper second-order equation operator satisfying the first-order structure condition and the shift-continuity condition; com…A Comparison Principle on a Hilbert Triple under the First-Order Structure Condition
theoremthm:comparison-first-order-hilbert-triple-2026aAnalysisPDEFor a locally strictly proper operator satisfying the first-order structure condition and the shift-continuity condition, a bounded viscosity subsolution and a bounded viscosity supersolution on the whole space satisfy a uniform comparison estimate on pairs of nearby points of V,…The Test-Datum Estimate at a Maximum Point of the Doubled Function
lemmalem:doubled-test-estimate-hilbert-triple-2026aAnalysisPDEAt a maximum point of the doubled function of the delta-envelopes of a subsolution and a supersolution, perturbed by a small linear term, the difference of the envelopes is bounded by the properness constant times the sum of the structural moduli evaluated at the test data.Closed Superlevel Sets of a Sum on a Product Space and of a Doubled Function
lemmalem:closed-superlevel-sum-product-hilbert-2026aAnalysisThe sum of two functions bounded above with closed superlevel sets has closed superlevel sets on the product space, and subtracting a multiple of the squared distance between the coordinates preserves this and yields a coercive upper bound.Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence
lemmalem:penalised-supremum-limit-2026aAnalysisFor the suprema of a function penalised by a nonnegative function that vanishes somewhere, the suprema are nonincreasing in the penalty parameter, a near-maximiser has small penalty compared with the drop of the suprema under doubling, and those drops tend to zero along a doublin…The Shift-Continuity Condition on Admissible Test Data
definitiondef:shift-continuity-condition-hilbert-triple-2026aAnalysisPDEIshii's condition (F3): on the admissible sets S^-{delta,R} and S^+{delta,R} the delta-shifts move by at most a modulus of the perturbation when the gradient and form arguments are perturbed.The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple
definitiondef:first-order-structure-condition-hilbert-triple-2026aAnalysisPDEIshii's condition (F2) in its first-order form: at the common doubling gradient the difference of the two delta-shifts is bounded below by three moduli, in the distance of the points, in the penalised distance, and in the shift parameter.A First-Order Equation Operator is Degenerate Elliptic and Its -Shifts Ignore the Form Argument
lemmalem:first-order-operator-hilbert-triple-2026aAnalysisPDEA 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.First-Order Equation Operator on a Hilbert Triple
definitiondef:first-order-operator-hilbert-triple-2026aAnalysisPDEA second-order equation operator is first order when its value does not depend on the form argument.Locally Strictly Proper Second-Order Equation Operator on a Hilbert Triple
definitiondef:locally-strictly-proper-hilbert-triple-2026aAnalysisPDEIshii's condition (F1): on each bounded range of the value argument the operator increases at least linearly in that argument, with a constant depending on the range.Test Data for a Second-Order Equation Operator on a Hilbert Triple and the Admissible Sets
definitiondef:test-data-hilbert-triple-2026aAnalysisPDEFixes the data space on which the delta-shifts of an operator act, the notion of an R-bounded datum, and Ishii's admissible sets and of data on which the structural hypotheses for comparison are impose…Perron's Method on a Hilbert Triple: Existence of a Viscosity Solution Between a Subsolution and a Supersolution
theoremthm:perron-existence-hilbert-triple-2026aAnalysisPDEGiven a viscosity subsolution below a viscosity supersolution of a degenerate elliptic second-order equation on an open subset of a Hilbert triple, the pointwise supremum of all viscosity subsolutions between them is a viscosity solution.The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised Function
lemmalem:perron-bump-hilbert-triple-2026aAnalysisPDEIf a function penalised by satisfies the subsolution inequality for the -shift of a degenerate elliptic operator wherever it exceeds a viscosity subsolution , then the pointwise maximum of the two is again a viscosity subsolution.The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on a Hilbert Triple is a Viscosity Subsolution
propositionprop:sup-of-subsolutions-hilbert-triple-2026aAnalysisPDEIf a nonempty family of viscosity subsolutions of a second-order equation on an open subset of a Hilbert triple is locally uniformly bounded above, then its pointwise supremum is again a viscosity subsolution. No hypothesis is imposed on the equation operator.Restriction of the Equation and the Locality of the Viscosity Sub- and Supersolution Properties on a Hilbert Triple
lemmalem:viscosity-locality-hilbert-triple-2026aAnalysisPDEA second-order equation operator on an open subset of a Hilbert triple restricts to any smaller open set, viscosity sub- and supersolutions restrict with it, and a function that is a viscosity subsolution near each point of the set is one on the whole set.The -Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset
lemmalem:delta-envelopes-local-hilbert-triple-2026aAnalysisPDEThe -envelopes of a function on an open subset of a Hilbert triple are unchanged by restriction to a smaller open set; a continuous function penalised by is its own -envelope; and the -envelope minus a continuous function has closed superlevel…