Book: On the partial differential equations of mathematical physics

Context and purpose
Henri Poincaré completed a rigorous study of the partial differential equations that arise in physical problems at the end of the 19th century. The mathematical physics of the period demanded clearer, more systematic treatments of equations governing heat, waves, gravitation and electrostatics, together with precise formulations of boundary conditions. The thesis addressed these needs by laying foundations for existence and uniqueness theory and by organizing methods for handling boundary-value problems that were central to potential theory.
The goal was both conceptual and technical: to understand which equations admit meaningful solutions under physically natural conditions, to characterize the behavior of those solutions near boundaries and singularities, and to develop constructive representations that could be used in applications.

Core contributions
A principal achievement was the clarification of existence and uniqueness phenomena for a broad class of linear partial differential equations relevant to physics. Poincaré identified conditions under which solutions are uniquely determined by boundary or initial data and described the obstructions that may prevent existence. Attention was given to elliptic equations such as the Laplace and Poisson equations, where boundary-value problems play the central role in potential theory, and to equations of hyperbolic and parabolic type that model propagation and diffusion.
The thesis also advanced the systematic study of boundary conditions. Poincaré established precise formulations of Dirichlet and Neumann type problems, discussed compatibility conditions at corners and edges, and examined the influence of domain geometry on solvability. By doing so he bridged formal physical reasoning and rigorous mathematical analysis, providing criteria that distinguish well-posed problems from those that are ill-posed.

Methods and techniques
Poincaré developed and used integral representations and potential-theoretic constructions to represent solutions. Layer potentials and integral kernels figure prominently as tools to transform differential relations into boundary integral equations, making the role of boundary data explicit. Such representations also allowed a detailed study of singular behavior and regularity up to the boundary.
Characteristic methods and analytic continuation are employed where appropriate, and series expansions and orthogonal function techniques are used to construct explicit solutions in canonical geometries. A recurrent theme is the reduction of PDE problems to problems about harmonic and subharmonic functions, exploiting the deep properties of potentials and the maximum principle to obtain uniqueness and comparison results.

Examples and applications
Concrete physical models motivate many of the examples and computations: electrostatic potentials and gravitational potentials governed by Laplace's equation, steady-state heat conduction, and wave propagation in simple media. These examples illustrate the translation of physical boundary conditions into mathematically precise requirements and demonstrate how the integral and variational techniques produce physically meaningful solutions.
The emphasis on methods that yield explicit representations made the results accessible to applied scientists of the era and provided templates for treating more complicated systems that arise in elasticity and hydrodynamics.

Legacy and influence
The thesis established a standard for rigorous treatment of boundary-value problems in mathematical physics and became a touchstone for later developments in potential theory and the theory of linear partial differential equations. Techniques introduced or systematized there, integral representations, careful boundary analysis, and the interplay of global and local properties, reappear throughout 20th-century analysis, influencing later work on fundamental solutions, Green's functions, and the functional-analytic approach to PDEs.
Beyond technical results, the work exemplified a methodological clarity that guided subsequent generations of mathematicians and mathematical physicists toward a more rigorous and structurally informed understanding of how physical laws are encoded in partial differential equations.