Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Second-Order Structure Condition for an Equation Operator on a Hilbert Triple
definitiondef:second-order-structure-condition-hilbert-triple-2026aAnalysisPDEThe structure inequality of the first-order condition, but required only for those pairs of symmetric forms that the doubling lemma produces at a quadratic penalty of a given strength.Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference
lemmalem:lions-doubling-hilbert-2026aAnalysisPDEThe infinite-dimensional counterpart of Ishii's lemma: at a sequentially strict maximum of a quadratically penalised difference on a Hilbert space, both functions admit test data whose second-order parts are controlled by a pair of forms supported on a finite orthonormal tuple, u…Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space
definitiondef:approximable-test-data-hilbert-2026aAnalysisPDEThe Hilbert-space analogue of approximability by test data: a point, value, gradient and Hessian that are matched, to within any prescribed accuracy, by a test function touching the given function from above or from below at a nearby point of a subset.The Gradient of the Quartic Energy on the First Sobolev Space of the Torus
lemmalem:phi4-energy-frechet-derivative-torus-2026aAnalysisPDEThe quartic energy is differentiable everywhere on the first Sobolev space of the torus, with gradient expressed through the Riesz map of the Sobolev Hilbert triple; on the domain of the form operator the gradient is twice the Riesz image of the Allen-Cahn drift.Well-Posedness of the Allen-Cahn Hamilton-Jacobi Equation on the Torus
corollarycor:allen-cahn-hamilton-jacobi-well-posed-torus-2026aAnalysisPDEOn the torus of dimension at most three, the Hamilton-Jacobi equation whose drift is the Allen-Cahn field has a unique bounded continuous viscosity solution on the square-integrable space, obtained by instantiating the general Hilbert-triple well-posedness theorem at the Sobolev…The Quartic Energy Functional on the First Sobolev Space of the Torus
definitiondef:phi4-energy-torus-2026aAnalysisPDEOn the torus of dimension at most three, the energy of a Sobolev class is the sum of the squared square-integrable norms of its weak partial derivatives, plus a multiple of the integral of the fourth power of a representative, minus a multiple of its squared square-integrable nor…Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three
lemmalem:quartic-product-torus-2026aAnalysisPDEOn the torus of dimension at most three, the pointwise product of representatives of four members of the first Sobolev space is integrable, with integral bounded by the product of their Sobolev norms and independent of the representatives; in particular fourth powers are integrab…The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus
lemmalem:cube-map-monotone-nonlinearity-torus-2026aAnalysisPDEIn dimensions at most three the map sending a Sobolev class on the torus to a nonnegative multiple of its pointwise cube is a monotone nonlinearity for the Hilbert triple formed by the square-integrable and Sobolev spaces.- Composing a class in the Sobolev space of the torus with a continuously differentiable function of bounded derivative again lies in that space, and its weak partial derivatives are given by the classical chain rule formula.
The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple
lemmalem:sobolev-hilbert-triple-torus-2026aAnalysisPDEThe square-integrable space and the Sobolev space of the torus, with the Sobolev inner product, form a Hilbert triple satisfying the standing separability hypothesis; on twice continuously differentiable periodic functions its form operator is the identity minus the Laplacian.The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space
theoremthm:sobolev-h1-torus-2026aAnalysisPDEThe Sobolev space of once weakly differentiable square-integrable classes on the torus is complete, has a countable dense subset, contains the smooth periodic classes densely, and is itself dense in the square-integrable space. Mollification approximates a Sobolev class in the So…The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus
definitiondef:sobolev-space-h1-torus-2026aAnalysisPDENames the space of square-integrable classes on the torus all of whose first weak partial derivatives are square-integrable, together with its inner product, norm and distance.Elementary Properties of the Weak Partial Derivative on the Torus
lemmalem:weak-derivative-torus-basic-2026aAnalysisPDEThe weak partial derivative extends the classical one on continuously differentiable periodic functions, is linear, may be tested against continuously differentiable periodic functions, commutes with mollification, is preserved by limits in the square-integrable space, and equips…- A function on the torus is an i-th weak partial derivative of another when integration against every smooth periodic test function obeys the integration-by-parts identity; the notion descends to almost-everywhere classes, where the weak derivative is unique.
The Fundamental Lemma of the Calculus of Variations on the Torus
lemmalem:fundamental-lemma-torus-2026aAnalysisPDEA function integrable on the torus whose integral against every smooth periodic test function vanishes is zero almost everywhere.Vanishing Mean of a Derivative and Integration by Parts on the Torus
lemmalem:periodic-integration-by-parts-2026aAnalysisPDEA partial derivative of a continuously differentiable periodic function has integral zero over the torus, so integration by parts on the torus carries no boundary term.- Standing notation for analysis on the flat torus: the half-open unit cube with the restriction of Lebesgue measure, the resulting Lebesgue spaces, the periodic function classes, and the gradient and Laplacian.
The Scope of the First-Order Structure Condition
remarkrem:first-order-structure-condition-scope-2026aAnalysisPDEThe first-order structure condition asks for its inequality at all pairs of form arguments independently, which forces the operator to depend on that argument boundedly. An operator with an unbounded second-order term needs a structure condition stated only for the coupled pairs…Well-Posedness for a Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple
theoremthm:monotone-hamilton-jacobi-well-posed-hilbert-triple-2026aAnalysisPDEOn a Hilbert triple, the Hamilton-Jacobi equation with a monotone nonlinearity, a Lipschitz drift and a bounded cost that is uniformly continuous for the smaller norm has exactly one bounded continuous viscosity solution on the whole space. That solution is uniformly continuous f…A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses
propositionprop:monotone-hamilton-jacobi-hilbert-triple-2026aAnalysisPDEOn a Hilbert triple, the operator obtained by adding a monotone nonlinearity and a Lipschitz perturbation to the drift of the dissipative Hamilton-Jacobi operator is first order, strictly proper, satisfies the first-order structure condition with an explicit structure pair, satis…