Distributions and Fundamental Solutions - Applications

Distribution theory and fundamental solutions enable rigorous treatment of singular data, generalized solutions, and provide computational tools for explicit solution formulas.

TheoremWeak Solutions via Distributions

A function $u \in L^1_{\text{loc}}(\Omega)$ is a weak solution to $Lu = f$ if: $\langle u, L^*\phi \rangle = \langle f, \phi \rangle$ for all test functions $\phi \in \mathcal{D}(\Omega)$ , where $L^*$ is the formal adjoint of $L$ .

Equivalently, $u$ satisfies $Lu = f$ in the sense of distributions.

This formulation allows solutions with lower regularity than required for classical derivatives, essential for treating shocks, discontinuities, and measure-valued solutions.

ExampleApplications in Physics

Point charges: Electrostatic potential from charge $q$ at origin: $\nabla^2\phi = -q\delta_0$ , solution $\phi = qE$ where $E$ is the fundamental solution
Impulse response: Systems described by PDEs have impulse response equal to the fundamental solution
Singular sources: Heat equation with instantaneous point source: $u_t = k\Delta u + Q\delta(x - x_0)\delta(t - t_0)$

TheoremFundamental Solution and Green's Identity

For solutions to $Lu = f$ via fundamental solution $E$ : $u(x) = \int_\Omega E(x-y)f(y)\,dy + \text{boundary terms}$

This extends Green's representation formula to distributional sources and provides explicit solution formulas.

Remark

Parametrix: For variable-coefficient operators where exact fundamental solutions are unknown, a parametrix $E$ satisfying $LE = \delta + R$ with $R$ smooth provides approximate solutions and is key to regularity theory and existence proofs via iteration.

ExampleNumerical Methods

Boundary element methods (BEM) discretize integral equations derived from fundamental solutions, reducing boundary value problems to surface integrals. This is especially efficient for exterior problems and infinite domains.

TheoremDistribution Theory in Quantum Field Theory

In quantum field theory, fields are operator-valued distributions. Correlation functions like: $\langle 0|T\{\phi(x_1)\phi(x_2)\}|0\rangle = D_F(x_1 - x_2)$ are Green's functions (Feynman propagators) for the field equation. Distribution theory provides the rigorous mathematical framework.

These applications demonstrate that distribution theory is not merely technical machinery but essential for modeling physical singularities and solving PDEs with generalized data.