Potential Theory (Universitext)

Couplage de deux semi-groupes droites. Espaces biharmoniques et couplage de processus de Markov. Axiomatique des fonctions harmoniques. Potential theory for elliptic systems. Potential Theory on Harmonic Spaces. Springer, New York-Heidelberg, Classical Potential Theory and its Probabilistic Counterpart.

Springer, New York, Positivity 6 , — On positivity for the biharmonic operator under Steklov boundary conditions. Polyharmonic boundary value problems.

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Positivity preserving and nonlinear higher order elliptic equations in bounded domains. Lecture Notes in Mathematics, , Springer, Berlin, Positivity properties of elliptic boundary value problems of higher order. For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation.

This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other. Modern potential theory is also intimately connected with probability and the theory of Markov chains.

In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an electrical network on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials.

Even in the finite case, the analogue I-K of the Laplacian in potential theory has its own maximum principle, uniqueness principle, balance principle, and others. A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation.

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Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is linear. This means that the fundamental object of study in potential theory is a linear space of functions. This observation will prove especially important when we consider function space approaches to the subject in a later section.

This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the conformal group or of its subgroups such as the group of rotations or translations.

Proceeding in this fashion, one systematically obtains the solutions of the Laplace equation which arise from separation of variables such as spherical harmonic solutions and Fourier series. By taking linear superpositions of these solutions, one can produce large classes of harmonic functions which can be shown to be dense in the space of all harmonic functions under suitable topologies.

Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as the Kelvin transform and the method of images. Third, one can use conformal transforms to map harmonic functions in one domain to harmonic functions in another domain.

The most common instance of such a construction is to relate harmonic functions on a disk to harmonic functions on a half-plane.

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Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat Riemannian manifolds. More complicated situations can also happen. For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing a multi-valued harmonic function as a single-valued function on a branched cover of R n or one can regard harmonic functions which are invariant under a discrete subgroup of the conformal group as functions on a multiply connected manifold or orbifold. From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions.



• Bilal (Écritures arabes) (French Edition).
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This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function , one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis such as Schwarz's theorem , Morera's theorem , the Weierstrass-Casorati theorem , Laurent series , and the classification of singularities as removable , poles and essential singularities generalize to results on harmonic functions in any dimension.