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This monograph covers the existing results regarding Green’s functions for differential equations with involutions (DEI).The first part of the book is devoted to the study of the most useful aspects of involutions from an analytical point of view and the associated algebras of differential operators. The work combines the state of the art regarding the existence and uniqueness results for DEI and new theorems describing how to obtain Green’s functions, proving that the theory can be extended to operators (not necessarily involutions) of a similar nature, such as the Hilbert transform or projections, due to their analogous algebraic properties. Obtaining a Green’s function for these operators leads to new results on the qualitative properties of the solutions, in particular maximum and antimaximum principles.

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations. --- Involutes (Mathematics) --- 517.91 Differential equations --- Differential equations --- Mathematics. --- Physics. --- Ordinary Differential Equations. --- Mathematical Methods in Physics. --- Curves --- Inversions (Geometry) --- Differential Equations. --- Mathematical physics. --- Physical mathematics --- Physics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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Inversions (Geometry) --- Conformal mapping. --- Conformal mapping --- 517.54 --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- Inversion geometry --- Circle --- Geometry, Modern --- Sphere --- Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- 517.54 Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations