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On Maz’ya’s work in potential theory and the theory of function spaces. (English) Zbl 0939.31001
Rossmann, Jürgen (ed.) et al., The Maz’ya anniversary collection. Vol. 1: On Maz’ya’s work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31-September 4, 1998. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 109, 7-16 (1999).
The author presents parts of V. G. Maz’ya’s work, beginning with 1959. He divides Maz’ya’s mathematical contributions in some principal directions of study, where he obtained the most remarkable results. The first direction is “Embeddings and isoperimetric inequalities”, where V. G. Maz’ya published his first paper (1959), giving embedding theorems of Sobolev type for functions defined in domains with irregular boundaries, a problem studied many times by him and other authors. A second direction of concern is the “Regularity of solutions”, where he studied the continuity of solutions to elliptic partial differential equations starting from a result obtained by E. de Giorgi and J. Nash in 1957. He himself and, independently, other researchers showed that the result obtained by these two authors couldn’t be extended to higher order equations and systems, by giving counterexamples. A third field investigated by Maz’ya is “Boundary regularity”, concerning the Dirichlet solutions regularity of Laplace’s equations in $\bbfR^n$. Concerning this property Maz’ya was the first who gave a condition for boundary regularity of nonlinear equations. The last chapter “Nonlinear potential theory”, an extension of the classical potential theory, puts in evidence the contributions of Maz’ya to the theory and ranges him through the most significant creators. Maz’ya introduced general $(\alpha-p)$-capacities, applied them in many of his papers and gave them another equivalent definition, more advantageous in applications. This presentation of the scientific activity of a known mathematician is at the same time a homage brought to a valuable scientist and a useful source of information for those interested in mathematical history. For the entire collection see [Zbl 0923.00034].
MSC:
31-03Historical (potential theory)
46-03Historical (functional analysis)
01A70Biographies, obituaries, personalia, bibliographies
35-03Historical (partial differential equations)
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
01A65Development of contemporary mathematics
31C45Nonlinear potential theory, etc.