Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Monotone, -Monotone and Locally Bounded Nonlinearities on a Hilbert Triple
definitiondef:monotone-nonlinearity-hilbert-triple-2026aAnalysisPDEA map from the smaller space of a Hilbert triple to the larger one is a monotone nonlinearity when it is monotone, pairs nonnegatively with the form operator on its domain, and is bounded on the sets bounded in the smaller norm.The Form Operator and the Riesz Map of a Diagonal Hilbert Triple
lemmalem:diagonal-hilbert-triple-2026aAnalysisPDEIn a diagonal Hilbert triple the domain of the form operator consists of the vectors whose basis coefficients are square-summable against the squared weights, the operator multiplies the coefficients by the weights and has each basis vector as an eigenvector, and the Riesz map di…The Diagonal Hilbert Triple Determined by an Orthonormal Basis and a Sequence of Weights
definitiondef:diagonal-hilbert-triple-2026aAnalysisPDEThe Hilbert triple whose smaller space is the weighted coefficient subspace determined by an orthonormal basis and a sequence of weights at least one.Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus
settingset:hilbert-space-calculus-2026aAnalysisPDEA layer on the real Hilbert space setting carrying the standing notation for series of real numbers and of vectors, products of inner product spaces, orthonormal bases, and differential calculus on open sets, each with its background results in force. It introduces no new concept…Uniqueness of a Bounded Continuous Viscosity Solution on a Hilbert Triple
corollarycor:uniqueness-bounded-continuous-solution-hilbert-triple-2026cAnalysisPDEUnder the hypotheses of the second-order 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-2026cAnalysisPDEIf the operator is degenerate elliptic, locally strictly proper and satisfies the second-order structure, shift-continuity and tail-insensitivity conditions, and if two constants are respectively a classical subsolution and a classical supersolution, then the equation has a visco…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.The Dissipative Hamilton-Jacobi Operator on a Hilbert Triple Satisfies the Comparison Hypotheses
propositionprop:dissipative-hamilton-jacobi-hilbert-triple-2026bAnalysisPDEThe 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 with an explicit structure pair (the nondecreasing majorant of the modulus of g, and 3…A Comparison Principle on a Hilbert Triple under the First-Order Structure Condition
theoremthm:comparison-first-order-hilbert-triple-2026bAnalysisPDEFor 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-2026bAnalysisPDEAt a maximum point of the doubled function of the delta-envelopes of a subsolution and a supersolution, perturbed by a small linear term, there are test points in D(A) close to the maximum point, with envelope values close to those at the maximum point, at which the properness co…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-2026bAnalysisPDEDefines a structure pair for an operator at a level R: two moduli, the first evaluated at alpha times the squared distance plus 1/alpha, the second depending on alpha and evaluated at delta times (h(x)+h(y)+1), bounding from below the difference of the shifted operators at double…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.