Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Well-Posedness of the Singular-Cost Dyson Hamilton-Jacobi Equation above the Hardy Threshold
theoremthm:dyson-singular-cost-well-posed-weyl-chamber-2026aAnalysisPDEFor a pair cost c times the sum of inverse squared gaps with c > -kappa^2/(8 theta), a harmonic confinement cost omega|x|^2 and bounded continuous g, the singular-cost Dyson equation on the Weyl chamber satisfies comparison in the class of u with u - P of P-subordinate growth, wh…- The Weyl chamber W_N is the set of points of R^N with strictly decreasing coordinates.
The Dyson Hamilton-Jacobi Equation in Singular-Cost Form on the Weyl Chamber
equationeq:dyson-singular-cost-hamilton-jacobi-weyl-chamber-2026aAnalysisPDEThe Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 - (kappa/2) tr D^2u = c S + omega |x|^2 + g on the Weyl chamber, with S the sum of the inverse squared gaps: a pair cost with reciprocal squares, a harmonic confinement cost and a running cost g, with no drift.Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold
theoremthm:dyson-well-posed-weyl-chamber-2026aAnalysisPDEFor N particles with logarithmic repulsion of strength beta > kappa/2 and a convex confinement of at least linear and regular growth, and bounded continuous running cost g, the Dyson Hamilton-Jacobi equation on the Weyl chamber satisfies comparison in the class of P-subordinate g…The Dyson Hamilton-Jacobi Equation for N Controlled Particles in the Weyl Chamber
equationeq:dyson-hamilton-jacobi-weyl-chamber-2026aAnalysisPDEThe drift-form Hamilton-Jacobi equation on the Weyl chamber for N controlled particles with logarithmic repulsion of strength beta and confinement V_1: the penalty-drift equation with potential P = H_beta + sum V_1(x_k).The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality
lemmalem:confined-log-energy-weyl-chamber-2026aAnalysisFor P = H + sum V_1(x_k) on the Weyl chamber: P is C^2 with explicit gradient and Laplacian; it is a penalty when V_1 grows at least linearly; its gradient is monotone when V_1 is convex; and for kappa < 2 beta and convex V_1 of regular growth it satisfies the dissipation inequal…The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity
lemmalem:log-energy-weyl-chamber-basic-2026aAnalysisThe Weyl chamber is open; the logarithmic energy H is C^2 there with explicit gradient and Laplacian beta S, S the sum of inverse squared gaps; its gradient is monotone, admits a symmetrised form against any vector, satisfies the Calogero identity |DH|^2 = beta^2 S = beta tr D^2H…The Logarithmic Energy of N Ordered Particles on the Weyl Chamber
definitiondef:log-energy-weyl-chamber-2026aAnalysisThe logarithmic energy of strength beta on the Weyl chamber is H(x) = -beta times the sum over pairs i<j of log(x_i - x_j).Well-Posedness of the Langevin Hamilton-Jacobi Equation on Euclidean Space under a Dissipation Inequality
theoremthm:langevin-well-posed-euclidean-2026aAnalysisPDEFor a C^2 potential V on R^n with compact sublevel sets satisfying (kappa/2) tr D^2V <= (1-eps)|DV|^2 + lambda V + C, and bounded continuous g, the Langevin Hamilton-Jacobi equation satisfies comparison in the class of V-subordinate growth and has exactly one viscosity solution t…The Hamilton-Jacobi Equation for Controlled Langevin Dynamics in a Potential on Euclidean Space
equationeq:langevin-hamilton-jacobi-euclidean-2026aAnalysisPDEThe discounted Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 + DV.Du - (kappa/2) tr D^2u = g on R^n of a controlled Langevin particle in a C^2 potential V.The Gradient of a Function on Euclidean Space is One-Sided Lipschitz on Bounded Sets
lemmalem:c2-gradient-one-sided-lipschitz-euclidean-2026aAnalysisFor a C^2 function V on R^n and a bounded set B there is c >= 0 with (DV(x)-DV(y)).(x-y) >= -c|x-y|^2 for all x, y in B.Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality
theoremthm:penalty-drift-well-posed-euclidean-2026aAnalysisPDEIf P is a penalty whose gradient is one-sided Lipschitz on sublevel sets and which satisfies the dissipation inequality (kappa/2) tr D^2P <= (1-eps)|DP|^2 + lambda P + C, then for bounded continuous g the penalty-drift Hamilton-Jacobi equation satisfies comparison in the class of…The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space
equationeq:penalty-drift-hamilton-jacobi-euclidean-2026bAnalysisPDEThe discounted viscous Hamilton-Jacobi equation lambda u + (theta/2)|Du|^2 + DP.Du - (kappa/2) tr D^2u = g on an open set D, whose drift is the gradient of a C^2 potential P.The Viscous Hamilton-Jacobi Operator with a Drift that is One-Sided Lipschitz on Sublevel Sets Satisfies the Operator Hypotheses of the Weighted-Penalty Comparison Principle
lemmalem:quadratic-operator-weighted-penalty-euclidean-2026aAnalysisPDEThe operator lambda r + (theta/2)|p|^2 + b(x).p - (kappa/2) tr X - G(x) on the domain of a penalty, with continuous b and G and a drift b that is one-sided Lipschitz on each sublevel set of the penalty (so it may blow up at the boundary), is continuous, strictly proper, convex an…Gauge Covariance under Subtraction of a Multiple of a Function: Viscosity and Classical Sub- and Supersolutions and the Structural Properties of the Operator
lemmalem:gauge-weighted-penalty-euclidean-2026aAnalysisPDESubtracting cP from the unknown turns sub- and supersolutions of F, viscosity or classical, into those of the operator F_c obtained by shifting the value, gradient and Hessian arguments by cP, cDP and cD^2P; continuity, degenerate ellipticity, strict properness and convexity pass…Perron's Method with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables: Existence and Uniqueness of a Viscosity Solution of Given Weight
theoremthm:perron-weighted-penalty-convex-euclidean-2026aAnalysisPDEUnder the hypotheses of the weighted-penalty comparison principle (in particular F convex in (r,p,X), with no convexity in x), if a viscosity subsolution and a viscosity supersolution of weight w exist, then exactly one function of weight w is both; it is continuous and lies betw…Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables
theoremthm:comparison-weighted-penalty-convex-euclidean-2026aAnalysisPDELet F be continuous, strictly proper, convex in (r,p,X) at each x (no convexity in x), and satisfy the structure condition on the sublevel sets of a penalty P, and let sP-K be a classical subsolution for some s<w. Then a viscosity subsolution u with u-wP of P-subordinate growth f…A Convex Combination of a Viscosity Subsolution and a Classical Subsolution is a Viscosity Subsolution, for an Operator Convex in the Value, Gradient and Matrix Variables
lemmalem:convex-combination-subsolution-euclidean-2026aAnalysisPDEFor an operator convex in (r,p,X), the combination (1-t)u + t phi of a viscosity subsolution u and a classical subsolution phi, with 0 <= t < 1, is again a viscosity subsolution.Restriction of Viscosity Sub- and Supersolutions to an Open Subset
lemmalem:viscosity-locality-euclidean-2026aAnalysisPDEThe restriction of a viscosity subsolution (supersolution) of an operator on an open set to an open subset is a viscosity subsolution (supersolution) of the restricted operator.Second-Order Equation Operator Convex in the Value, Gradient and Matrix Variables
definitiondef:operator-convex-value-gradient-hessian-2026aAnalysisPDEAn operator F(x,r,p,X) is convex in (r,p,X) if, at each fixed point x, it is jointly convex in the value r, the gradient p and the matrix X; nothing is assumed about its dependence on x.