Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.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…Quadratic Test Functions: a Global Majorant and Minorant Matching a Function to Second Order at a Point
lemmalem:taylor-test-function-hilbert-2026aAnalysisThe second-order Taylor polynomial of a function at a point, perturbed by , is of class on the whole space, majorises the function near the point when and minorises it when , and inherits its local maxima and minima.The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum
lemmalem:envelope-maximum-metric-2026aAnalysisTopologyThe upper semicontinuous envelope of a pointwise maximum of two functions is the pointwise maximum of their upper envelopes, and dually for minima and lower envelopes.Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set
lemmalem:envelopes-open-trace-metric-2026aAnalysisTopologyFor an open set meeting a set in a metric space, restriction to preserves local boundedness, gives the same semicontinuous envelopes at points of , and has the same local maxima and minima there.Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on a Hilbert Triple are Viscosity Sub- and Supersolutions
propositionprop:classical-implies-viscosity-hilbert-triple-2026aAnalysisPDEFor a degenerate elliptic second-order equation operator on an open subset of a Hilbert triple, every classical subsolution (supersolution, solution) of class C^2 is a viscosity subsolution (supersolution, solution).Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of are Supersolutions of
lemmalem:viscosity-sign-reversal-hilbert-triple-2026aAnalysisPDEWith F̃(x,r,p,X) = −F(x,−r,−p,−X): F̃ is degenerate elliptic iff F is, the δ-shifts of F̃ are the negated δ-shifts of F at negated data, and u is a classical or viscosity subsolution of F iff −u is a classical or viscosity supersolution of F̃ (and conversely), so every statement…Basic Properties of the -Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity
lemmalem:delta-envelopes-basic-hilbert-triple-2026bAnalysisPDEThe delta-envelope u^-_delta is upper semicontinuous on V∩U, dual to u^+_delta under negation, has superlevel sets closed in (U, d_H), is dominated by C − delta h when u ≤ C, is monotone in u and antitone in delta with a gain of (delta' − delta) h, and equals u − delta h when u i…First- and Second-Order Conditions at a Local Extremum of a Function Penalised by on the Small Space
lemmalem:penalised-maximum-c2-hilbert-triple-2026aAnalysisPDEIf ψ is C^2 on an open subset U of H and ψ − λh has a local maximum on V∩U at x̂, then x̂ lies in D(A), λAx̂ = Dψ(x̂), and the restriction of D^2ψ(x̂) to V is at most λI_V; dually at a local minimum of ψ + λh.Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple
definitiondef:viscosity-sub-supersolution-hilbert-triple-2026aAnalysisPDEA locally bounded above function u on an open subset U of the large space is a viscosity subsolution of F if, for every δ > 0, every C^2 test function φ and every local maximum x̂ of u^-_δ − φ on V∩U, the shifted operator F^-_δ can be made ≤ ε at data in W×R×H×Sym(H) within ε of…Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple
definitiondef:classical-sub-supersolution-hilbert-triple-2026aAnalysisPDEA function of class C^2 on an open subset U of the large space is a classical subsolution of F if F(x, u(x), Du(x), D^2u(x)|_V) ≤ 0 at every point of W = D(A)∩U, a classical supersolution if the reverse inequality holds, and a classical solution if it is both.The -Envelopes and of a Function on an Open Subset of a Hilbert Triple
definitiondef:delta-envelopes-hilbert-triple-2026aAnalysisPDEFor a function u on an open subset U of the large space of a Hilbert triple, locally bounded above (below), and δ > 0, the δ-envelopes are u^-_δ = (u − δh)^* and u^+δ = (u + δh)*, the semicontinuous envelopes taken on V∩U for the metric of H.