Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem
theoremthm:perron-existence-2026aAnalysisPDEIf comparison holds for the Dirichlet problem and there are a bounded subsolution and a bounded supersolution whose envelopes attain the boundary data, then the supremum of all subsolutions between them is a viscosity solution of the Dirichlet problem.The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality
lemmalem:perron-bump-2026aAnalysisPDEIf the lower semicontinuous envelope of a viscosity subsolution fails the supersolution inequality at an interior point, then the subsolution can be raised strictly somewhere near that point, by a modification supported in an arbitrarily small ball, without losing the subsolution…The Maximum of Two Viscosity Subsolutions is a Viscosity Subsolution
corollarycor:max-two-subsolutions-2026aAnalysisPDEThe pointwise maximum of two viscosity subsolutions of a continuous second-order equation operator is again a viscosity subsolution.The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution
lemmalem:sup-of-subsolutions-2026aAnalysisPDEIf a nonempty family of viscosity subsolutions of a continuous second-order equation operator is locally uniformly bounded above, then the upper semicontinuous envelope of its pointwise supremum is again a viscosity subsolution.A Viscosity Solution of the Dirichlet Problem is Continuous and Attains the Boundary Data
propositionprop:dirichlet-solution-basic-2026aAnalysisPDEA viscosity solution of the Dirichlet problem equals the boundary data on the boundary, is continuous on the closure, and restricts to a viscosity solution of the equation on the open set.Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem
definitiondef:dirichlet-problem-viscosity-2026aAnalysisPDEA viscosity subsolution of the Dirichlet problem is a viscosity subsolution up to the boundary that lies below the boundary data there; dually for supersolutions, and a solution is both.The Viscosity Sub- and Supersolution Properties are Local
lemmalem:viscosity-subsolution-local-2026aAnalysisPDEA viscosity subsolution restricts to a viscosity subsolution on any open subset, and conversely a function that is a viscosity subsolution on an open neighbourhood of each point of the domain is a viscosity subsolution on the whole domain. The same holds for supersolutions.Sequential Form of the Continuity of a Second-Order Equation Operator
lemmalem:operator-sequential-continuity-2026aAnalysisPDEIf a second-order equation operator is continuous at a quadruple and each of the four arguments is approached by a convergent sequence, then the values of the operator converge to its value at that quadruple.Restriction of a Second-Order Equation Operator to an Open Subset
lemmalem:operator-restriction-2026aAnalysisPDERestricting the spatial variable of a second-order equation operator to an open subset again gives a second-order equation operator, and degenerate ellipticity and continuity are inherited.The Structure Condition Implies Degenerate Ellipticity
propositionprop:structure-condition-implies-elliptic-2026aAnalysisPDEA continuous second-order equation operator that satisfies the structure condition of the comparison principle for the Dirichlet problem is automatically degenerate elliptic, so the ellipticity hypothesis of that comparison principle is redundant. Only the openness of the domain…Uniqueness for the Dirichlet Problem for a Viscous Hamilton-Jacobi Equation
corollarycor:uniqueness-viscous-hamilton-jacobi-2026aAnalysisPDETwo continuous viscosity solutions on the closure of a bounded domain of the equation that agree on the boundary agree everywhere.The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition
propositionprop:viscous-hamilton-jacobi-operator-2026aAnalysisPDEFor positive , nonnegative and continuous , the operator of the viscous Hamilton-Jacobi equation is continuous, strictly proper with constant , and satisfies th…Structure Condition: the Trace Operator of a Lipschitz Diffusion Coefficient
exampleex:structure-condition-trace-diffusion-2026aAnalysisPDEThe linear second-order operator satisfies the structure condition of the comparison principle with the linear modulus , whenever the coefficient is Lipschitz with constant in the row-sum-…Structure Condition: a Linear First-Order Operator with an Almost Monotone Drift
exampleex:structure-condition-monotone-drift-2026aAnalysisPDEThe first-order operator satisfies the structure condition of the comparison principle, with the linear modulus , whenever , that is, whenever becomes monotone after adding times the id…Structure Condition: a Degenerate Elliptic Operator with a Continuous Inhomogeneity
exampleex:structure-condition-continuous-inhomogeneity-2026aAnalysisPDEIf is degenerate elliptic in the matrix variable and is continuous on the closure of the domain, then satisfies the structure condition of the comparison principle, with a modulus of continuity for as modulus.Uniqueness for the Dirichlet Problem for Second-Order Equations
corollarycor:uniqueness-dirichlet-second-order-2026aAnalysisPDETwo continuous viscosity solutions on a bounded domain that agree on the boundary agree throughout the closure, for a continuous, strictly proper operator satisfying the structure condition.Comparison Principle for the Dirichlet Problem for Second-Order Equations
theoremthm:comparison-dirichlet-second-order-2026aAnalysisPDEFor a continuous, strictly proper operator satisfying the structure condition, a viscosity subsolution up to the boundary that lies below a viscosity supersolution on the boundary of a bounded domain lies below it throughout the closure.Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference
lemmalem:ishii-doubling-2026aAnalysisPDEAt a local maximum of there are symmetric matrices and giving test data from above for and from below for , both with first-order coefficient , and satisfying the two-sided bound…- A quadruple is approximable by test data from above for a function exactly when the sign-reversed quadruple is approximable from below for the negated function, and symmetrically.
Continuity of a Second-Order Equation Operator
definitiondef:continuous-second-order-operator-2026aAnalysisPDEDefines continuity of a second-order equation operator at a quadruple , and continuity on its whole domain, in terms of the Euclidean distance, the absolute value, the Euclidean norm and the distance between symmetric matrices.