Perturbation Expansions on Perturbed Domains

Expansions in powers of a perturbation parameter are considered for partial differential equations to be solved on a domain that is also perturbed. Two kinds of expansions—Lagrange-like and Euler-like—are developed and shown to be equivalent to one another under a certain transformation of dependent variables. This equivalence is used to justify the hierarchy of partial differential equations produced by the Euler-like expansion. In addition to partial differential equations, integrals extended over a perturbed domain are also considered. Such integrals suffer perturbations due to the domain perturbation. A formula for these perturbations is given that involves the domain-mapping function only at the boundary of the unperturbed domain. This contrasts with the usual change-of-domain formula, which involves the Jacobian of the domain-mapping function throughout the unperturbed domain. The formula in question is compatible with Euler-like expansions (and the usual change-of-domain formula is compatible with Lagrange-like expansions), and is therefore appropriate in problems involving both partial differential equations and integrals on perturbed domains, for which the Euler-like viewpoint is preferable. Application is made to finding the gravitational potential of a perturbed mass, and related applications in astrophysical problems are indicated.