Distributional derivative

Depends on

• Distribution
• Test function operations are continuous

Used by

• A step has no locally integrable weak derivative Counterexample
• Cantor function has singular distributional derivative Counterexample
• Product of two distributions is not canonically defined Counterexample
• Convex entropy--entropy flux pairs Definition
• Dirac delta and its derivatives Definition
• Distributional Riesz measure of a plane subharmonic function Definition
• Distributional weak solutions of the Cauchy problem Definition
• Fundamental solution for the positive operator minus Laplacian Definition
• Fundamental solution of a constant-coefficient operator Definition
• Spatial BV and distributional total variation Definition
• The formal adjoint and the adjoint weak Dirichlet problem Definition
• Weak derivative of a locally integrable function Definition
• Absolute value has a Dirac second derivative Example
• Adding an entire harmonic function preserves a Laplace fundamental solution Example
• Derivative of the heaviside function is dirac delta Example
• Derivatives of piecewise smooth functions include jump deltas Example
• Distributional laplacian of the newtonian kernel Example
• One-dimensional Dirichlet Green kernel on an interval Example
• The method of continuity on a constant-coefficient one-dimensional path Example
• The one-dimensional obstacle reaction is supported on the contact set Example
• A distribution with a vanishing derivative is a polynomial Lemma
• A zero distributional gradient is constant on a connected domain Lemma
• An additive constant in an entropy flux does not change the entropy inequality Lemma
• Distributional derivatives commute with test-function convolution Lemma
• Integrated pressure recovery and MINI pressure-primitive convergence Lemma
• L² forcing and divergence data embed in H⁻¹ with a quantitative bound Lemma
• Polynomial convolutions force activation polynomiality almost everywhere Lemma
• Quantitative CGO construction for C1alpha conductivity potentials Lemma
• Support, integrability, and divergence identity for the Bogovskii kernel Lemma
• The Kruzhkov doubling inequality for two entropy solutions Lemma
• The outgoing three-dimensional Helmholtz kernel is a fundamental solution Lemma
• Sobolev weak derivatives belong to pde Remark
• Weak derivatives are represented distributional derivatives Remark
• A distribution with zero derivatives on a connected open set is constant Theorem
• Associativity of distribution convolution under compact support Theorem
• Convolution with a test function is smooth Theorem
• Differentiation and polynomial multiplication preserve tempered distributions Theorem
• Distributional differentiation is continuous and commutes Theorem
• Leibniz rule for distributions Theorem
• Local structure of distributions as derivatives of continuous functions Theorem

…and 5 more results.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)