Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.- For a nonnegative real constant , the map on the nonnegative reals is a modulus of continuity, and it is nondecreasing.
A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity
lemmalem:modulus-of-continuity-compact-2026aAnalysisTopologyA real-valued continuous function on a nonempty compact subset of a metric space admits a modulus of continuity that is nondecreasing and dominates the oscillation of the function at every scale.The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound
lemmalem:trace-quadratic-forms-2026aLinear AlgebraFor a real matrix with rows and a symmetric , the trace of is the sum of the quadratic forms . Consequences: the trace of a symmetric matrix is the sum of its diagonal quadratic forms, it is monotone for the semidefi…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…The Doubling Matrix and its Elementary Properties
lemmalem:doubling-matrix-2026aAnalysisLinear AlgebraCollects the properties of the block matrix with diagonal blocks and off-diagonal blocks : its action on concatenated vectors, its quadratic form , the bounds , the identity , and the fact that…- 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.Reduction of the Theorem on Sums to a Global Quadratic Bound
lemmalem:theorem-on-sums-reduction-2026aAnalysisPDENormalises the data of the theorem on sums by translation and an affine correction and localises the summands to a closed ball, producing upper semicontinuous functions bounded above on the whole space that vanish at the origin, satisfy a global quadratic bound with matrix…The Theorem on Sums for a Global Quadratic Bound, in Two Groups of Variables
theoremthm:theorem-on-sums-global-quadratic-2026aAnalysisPDEFor upper semicontinuous functions bounded above, vanishing at the origin and satisfying globally, produces for each symmetric matrices that are admissible second-order test d…Second-Order Test Data at a Global Quadratic Maximum of a Semiconvex Function
lemmalem:semiconvex-quadratic-maximum-hessian-2026aAnalysisPDEFor a semiconvex function on whose difference with a quadratic form attains a maximum at the origin, produces points of twice differentiability approaching the origin whose gradients tend to zero and whose Hessians converge to a symmetric matrix with…Transfer of Approximating Test Data from the Sup-Convolution to the Original Function
lemmalem:sup-convolution-test-data-transfer-2026aAnalysisPDEShows that test data approximable from above for the sup-convolution at a point with first-order datum transfers to the same first-order and second-order data for itself at , together with the accompanying identity for the…Concatenation and the Sup-Convolution of a Sum in Separated Variables
lemmalem:sup-convolution-separate-variables-2026aAnalysisMultivariable CalculusRecords that concatenation carries origins to the origin and is compatible with coordinatewise convergence, and shows that the sup-convolution of a function of the form is the sum of the sup-convolutions of and with the same parameter.