By George F. Simmons

This article is a spin-off of Appendices A ("A number of extra Topics") and B ("Biographical Notes") of Simmons' winning CALCULUS WITH ANALYTIC GEOMETRY. The textual content is acceptable as a complement for a calculus path and/or background of arithmetic path. The textual content can also be acceptable for a liberal arts arithmetic direction for college students with minimum arithmetic heritage.

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**Calculus Gems: Brief Lives and Memorable Mathematics**

This article is a spin-off of Appendices A ("A number of extra Topics") and B ("Biographical Notes") of Simmons' profitable CALCULUS WITH ANALYTIC GEOMETRY. The textual content is acceptable as a complement for a calculus path and/or historical past of arithmetic direction. The textual content can also be acceptable for a liberal arts arithmetic direction for college students with minimum arithmetic history.

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For smoothly bounded domains Ω, it is known that for each fixed s there is no local obstruction in the boundary to continuity in W s (Ω) [Barrett 1986; Chen 1991b]. For all domains Ω where continuity in C ∞ (Ω) is known, one can actually prove continuity in W s (Ω) for all positive s. This intriguing phenomenon is not understood at present. ) Although regularity of the ∂-Neumann problem in C ∞ (Ω) is known in large classes of pseudoconvex domains (see sections 4–6), the example of the worm domains shows that regularity sometimes fails.

In Russian; translated in Functional Anal. Appl. 17 (1983), 285–294. [Varadarajan 1974] V. S. , 1974. Reprinted as Graduate Texts in Mathematics 102, Springer, New York, 1984. [Vitushkin 1985] A. G. Vitushkin, “Holomorphic mappings and the geometry of hypersurfaces”, pp. 159–214 in Several Complex Variables I, edited by A. G. Vitushkin, Encycl. Math. Sci. 7, Springer, Berlin, 1985. [Webster 1977a] S. M. Webster, “On the mapping problem for algebraic real hypersurfaces”, Invent. Math. 43:1 (1977), 53–68.

BOAS AND EMIL J. STRAUBE problem, in much the same way that the Dirichlet problem is the archetypal elliptic boundary-value problem. In this survey, we discuss global regularity of the ∂-Neumann problem in the L2 -Sobolev spaces W s (Ω) for all non-negative s and also in the space C ∞ (Ω). For estimates in other function spaces, such as H¨ older spaces and Lp -Sobolev spaces, see [Beals et al. 1987; Berndtsson 1994; Chang et al. 1992; Cho 1995; Christ 1991; Fefferman 1995; Fefferman and Kohn 1988; Fefferman et al.