Clifford analysis and boundary value problems of partial differential equations (Q2735602)

These (survey) lectures were delivered at the Department of Mathematics, University of Coimbra, Portugal, during the Second European Intensive Course on ``Complex Analysis and its generalizations (with applications to partial differential equations)'' (March 1996). The authors being well-known researchers and specialists on Quaternionic Analysis and Clifford Analysis and their applications to partial differential equations aim at giving a first insight into Clifford analysis and its applications and at raising interest in this fast increasing theory.NEWLINENEWLINENEWLINEThe booklet starts with Hamilton's quaternions, complex quaternions and Pauli matrices, and the definition and examples of Clifford algebras. Among the universal Clifford algebras \(Cl_{p,q}\) the algebra \(Cl_{o,n}\) is preferred, and the definition of Banach spaces over \(Cl_{o,n}\) is given. In view of the Cauchy-Fueter operator (which, in the case \(n=1\), yields the Cauchy-Riemann equations), the practicability of introducing a type of ``analyticity'' in higher dimensions is investigated; here, left Clifford regular (left monogenic), left Clifford analytic, and \(H\)-regular functions are considered \((H\) stands for the algebra of real quaternions). The well-known \(T\)-operator and the Cauchy(-Green) formula of complex function theory are utilized to get the Teodorescu transform and the Borel-Pompeiu formula (respectively), and related inequalities and theorems. Furthermore, Cauchy-type operators and a result by A. W. Bitsadze lead to the introduction of the so-called Cauchy-Bitsadze operator, including the Plemelj-Sokhotzki formula. After a chapter on the decomposition of the Clifford valued Hilbert space \(L_2(G,Cl_{o,n})\), an application is treated, namely, to the Dirichlet boundary value problem for the Laplacian. The last third of these lectures deals with the question of numerical applications of Clifford analytic methods. Here, methods of complex discrete function theory suggest ideas. A ``schedule'' for the desired numerical approach is obtained from a theorem of V. S. Ryabenskij (quoted there); this requires the definition of discrete operators and corresponding function spaces in a suitable manner. The lectures finish with considering, once more, the Dirichlet problem for the Laplacian and, besides, a discrete Stokes system. The bibliography contains 47 entries.NEWLINENEWLINENEWLINEFor brevity, only some (important) proofs are given in full length, but, in the other cases, a sketch of proofs or at least a hint to the literature are mentioned. Of great value are (i) many comparisons with similar situations in (usual) complex function theory, (ii) many examples, historical remarks, and motivations, (iii) a thorough discussion (demonstrating advantages, disadvantages, difficulties, and limitations) why definitions, methods etc. used here, are preferable in contrast to other possible ways. Altogether, this is a very interesting and informative booklet.