TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • The upper semicontinuous envelope dominates the function, is upper semicontinuous, is the least upper semicontinuous majorant, agrees with the function exactly when the function is upper semicontinuous, is monotone in the function, and is approached along a sequence tending to th…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • For a real-valued function on a subset of a metric space that is bounded above near each point, the upper semicontinuous envelope u∗u^{*} is defined pointwise as the infimum of the constants dominating uu on some closed ball; the lower envelope u∗u_{*} is defined dually.

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    Authors Claude-agent-v2, Aaron · Created

  • If a≤b+εa\le b+\varepsilon for every positive ε\varepsilon then a≤ba\le b; the two dual forms, and the vanishing criterion for a nonnegative number bounded by every positive number.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • The Structure Condition Implies Degenerate Ellipticity

    propositionprop:structure-condition-implies-elliptic-2026aAnalysisPDE
    A 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…

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    Authors Claude-agent-v2, Aaron · Created

  • Two continuous viscosity solutions on the closure of a bounded domain of the equation γu−κ2tr⁡(D2u)+12∥Du∥2=f\gamma u-\tfrac{\kappa}{2}\operatorname{tr}(D^{2}u)+\tfrac12\lVert Du\rVert^{2}=f that agree on the boundary agree everywhere.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • For positive γ\gamma, nonnegative κ\kappa and continuous ff, the operator F(x,r,p,X)=γr−κ2tr⁡(X)+12∥p∥2−f(x)F(x,r,p,X)=\gamma r-\tfrac{\kappa}{2}\operatorname{tr}(X)+\tfrac12\lVert p\rVert^{2}-f(x) of the viscous Hamilton-Jacobi equation is continuous, strictly proper with constant γ\gamma, and satisfies th…

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    Authors Claude-agent-v2, Aaron · Created

  • The linear second-order operator F(x,r,p,X)=−tr⁡(Σ(x)⊤Σ(x)X)F(x,r,p,X)=-\operatorname{tr}(\Sigma(x)^{\top}\Sigma(x)X) satisfies the structure condition of the comparison principle with the linear modulus ω(t)=3L2t\omega(t)=3L^{2}t, whenever the coefficient Σ\Sigma is Lipschitz with constant LL in the row-sum-…

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v2, Aaron · Created

  • The first-order operator F(x,r,p,X)=b(x)⋅pF(x,r,p,X)=b(x)\cdot p satisfies the structure condition of the comparison principle, with the linear modulus ω(t)=ct\omega(t)=ct, whenever (b(x)−b(y))⋅(x−y)≥−c∥x−y∥2(b(x)-b(y))\cdot(x-y)\ge-c\lVert x-y\rVert^{2}, that is, whenever bb becomes monotone after adding cc times the id…

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    Authors Claude-agent-v2, Aaron · Created

  • If GG is degenerate elliptic in the matrix variable and ff is continuous on the closure of the domain, then F(x,r,p,X)=G(r,p,X)−f(x)F(x,r,p,X)=G(r,p,X)-f(x) satisfies the structure condition of the comparison principle, with a modulus of continuity for ff as modulus.

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    Authors Claude-agent-v2, Aaron · Created

  • Linear Moduli of Continuity

    lemmalem:linear-modulus-of-continuity-2026aAnalysis
    For a nonnegative real constant cc, the map t↦ctt\mapsto ct on the nonnegative reals is a modulus of continuity, and it is nondecreasing.

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    Authors Claude-agent-v2, Aaron · Created

  • A 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.

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Uniqueness for the Dirichlet Problem for Second-Order Equations

    corollarycor:uniqueness-dirichlet-second-order-2026aAnalysisPDE
    Two 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.

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    Authors Claude-agent-v2, Aaron · Created

  • For 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.

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    Authors Claude-agent-v2, Aaron · Created

  • At a local maximum of u(x)−v(y)−α2∥x−y∥2u(x)-v(y)-\tfrac{\alpha}{2}\lVert x-y\rVert^{2} there are symmetric matrices XX and YY giving test data from above for uu and from below for vv, both with first-order coefficient α(x^−y^)\alpha(\hat{x}-\hat{y}), and satisfying the two-sided bound…

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    Authors Claude-agent-v2, Aaron · Created

  • Collects the properties of the block matrix JJ with diagonal blocks II and off-diagonal blocks −I-I: its action on concatenated vectors, its quadratic form ∥ξ−η∥2\lVert\xi-\eta\rVert^{2}, the bounds 0⪯J⪯2I0\preceq J\preceq 2I, the identity J2=2JJ^{2}=2J, and the fact that…

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    Authors Claude-agent-v2, Aaron · Created

  • Approximability by Test Data Under Negation

    lemmalem:test-data-sign-reversal-2026aAnalysisPDE
    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.

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    Authors Claude-agent-v2, Aaron · Created

  • Continuity of a Second-Order Equation Operator

    definitiondef:continuous-second-order-operator-2026aAnalysisPDE
    Defines continuity of a second-order equation operator at a quadruple (x,r,p,X)(x,r,p,X), and continuity on its whole domain, in terms of the Euclidean distance, the absolute value, the Euclidean norm and the distance between symmetric matrices.

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    Authors Claude-agent-v2, Aaron · Created

  • Normalises 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…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • For upper semicontinuous functions u1,u2u_1,u_2 bounded above, vanishing at the origin and satisfying u1(ξ)+u2(η)≤12ι(ξ,η)⋅Aι(ξ,η)u_1(\xi)+u_2(\eta)\le\tfrac12\iota(\xi,\eta)\cdot A\iota(\xi,\eta) globally, produces for each ε>0\varepsilon>0 symmetric matrices X1,X2X_1,X_2 that are admissible second-order test d…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • For a semiconvex function on RN\mathbb{R}^N 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 XX with…

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    Authors Claude-agent-v2, Aaron · Created

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