PDEs and Green Functions

A partial differential equation becomes a determinate problem only after its operator, domain, function spaces, data, and boundary or support prescription are fixed. This record is a compact checklist for choosing and testing that problem. For diagnostic examples and a bounded preparation route, see PDEs and Green Functions for Gravitation.

The complete PDE problem

For a second-order scalar differential expression

$$P=A^{\mu\nu}(x)\partial_\mu\partial_\nu +B^\mu(x)\partial_\mu+C(x),$$

the principal symbol is

$$\sigma_P(x,\xi)=A^{\mu\nu}(x)\xi_\mu\xi_\nu.$$

It identifies elliptic, hyperbolic, parabolic, or mixed principal behavior. Lower-order terms do not change this classification, although degeneracy of $A^{\mu\nu}$ can. For gauge field equations, classify a declared reduction and separately check its constraints; the unreduced equations need not form a hyperbolic evolution system.

Before selecting a method, record all of the following:

Part of the problem	What must be stated
Differential operator	Coefficients, principal symbol, bundle or component fields, source normalization, and gauge reduction
Space and domain	Regularity or test space, measure and pairing, geometric region, and operator domain
Data	Cauchy data, boundary data, decay, compatibility conditions, and freely specifiable homogeneous data
Selection rule	Elliptic boundary condition, retarded or advanced support, radiation condition, self-adjoint extension, or another named prescription
Constraints	Constraint equations, propagation system, and boundary conditions that preserve them

The same differential expression on two domains can define different operators, kernels, and solutions.

Fundamental solutions and Green operators

A distributional fundamental solution satisfies

$$P_x E(x,x')=\delta_g(x,x'),$$

where the invariant delta is normalized against the declared volume measure. This local identity does not by itself choose causal support, boundary behavior, or a unique inverse. Fourier constructions and hyperbolic examples make the distinction between a fundamental solution and a selected Green kernel explicit in Duistermaat and Kolk 2010, pp. 115–152 and 271–285.

For a normally hyperbolic operator $P$ on a smooth globally hyperbolic spacetime without timelike boundary, acting on sections of a specified finite-rank bundle, the unique retarded and advanced Green operators obey, for compactly supported smooth $f$,

$$\begin{aligned} P G_{\mathrm{ret}}f&=f, &G_{\mathrm{ret}}P f&=f, &\operatorname{supp}(G_{\mathrm{ret}}f)&\subseteq J^+(\operatorname{supp}f),\\ P G_{\mathrm{adv}}f&=f, &G_{\mathrm{adv}}P f&=f, &\operatorname{supp}(G_{\mathrm{adv}}f)&\subseteq J^-(\operatorname{supp}f). \end{aligned}$$

These hypotheses and both inverse identities are essential; existence and uniqueness with the stated support follow in Bär, Ginoux, and Pfäffle 2007, Ch. 3, especially Cor. 3.4.3 and Thm. 3.4.7, pp. 67–101. Their difference is a homogeneous causal propagator. A timelike boundary, a non-globally-hyperbolic region, or a changed operator class requires additional conditions and can admit several inequivalent inverses.

For an elliptic problem, causal language is normally inappropriate. Boundary conditions and the operator kernel control inversion; zero modes can require an orthogonality or compatibility condition. For a hyperbolic problem, Cauchy data and causal support control evolution. A mixed or changing-type problem must be divided into regimes rather than assigned one global label.

Energy, boundaries, and constraints

An energy estimate is useful only with its norm, source work, and boundary flux visible. Maximally dissipative or hierarchical boundary conditions can control incoming characteristic fields, but a theorem for one reduced wave system does not automatically establish constraint preservation or well-posedness for another. The relevant energy estimates, compatibility conditions, and boundary hierarchy are given in Kreiss et al. 2009, §§ II–III and Thms. 1–3, arXiv PDF pp. 2–13.

Before trusting a solution, apply $P$ and recover the source distributionally; verify the retarded or advanced support; count data against characteristics and constraints; retain boundary flux; test for kernels and adjoint compatibility; and compare a limiting case or conserved quantity. A small residual alone does not show that the intended domain or boundary-value problem was solved.

For the flat-space causal construction, continue to Retarded and Advanced Green Functions. For the curved, globally hyperbolic theorem, use Advanced, Retarded, and Causal Propagators. Boundary and constraint questions belong with Characteristics, Constraint Propagation, and Well-Posedness.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society, 2007. DOI 10.4171/037.
• Duistermaat, J. J., and J. A. C. Kolk. Distributions: Theory and Applications. Cornerstones. Boston: Birkhäuser, 2010. DOI 10.1007/978-0-8176-4675-2.
• Kreiss, Heinz-Otto, Oscar Reula, Olivier Sarbach, and Jeffrey Winicour. “Boundary Conditions for Coupled Quasilinear Wave Equations with Applications to Isolated Systems.” Communications in Mathematical Physics 289 (2009): 1099–1129. Open preprint.