By Bruce P. Palka

ISBN-10: 038797427X

ISBN-13: 9780387974279

This publication offers a rigorous but effortless creation to the speculation of analytic services of a unmarried advanced variable. whereas presupposing in its readership a level of mathematical adulthood, it insists on no formal must haves past a valid wisdom of calculus. ranging from uncomplicated definitions, the textual content slowly and thoroughly develops the guidelines of complicated research to the purpose the place such landmarks of the topic as Cauchy's theorem, the Riemann mapping theorem, and the concept of Mittag-Leffler should be taken care of with out sidestepping any problems with rigor. The emphasis all through is a geometrical one, so much suggested within the vast bankruptcy facing conformal mapping, which quantities primarily to a "short path" in that vital zone of complicated functionality concept. each one bankruptcy concludes with a big variety of routines, starting from straight forward computations to difficulties of a extra conceptual and thought-provoking nature.

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And Diff. Equ. Sci. 1980 Science Press, Beijing, China. Ivert, On quasilinear systems in diagonal form. Math. Z. 170 {1980), 283-286. Landes, On the existence of weak solutions of perturbed systems with critical growth. To appear in Crelle's Journal. [Mei 1] M. Meier, Reguliire und singulare Losungen quasilinearer elliptischer Gleichungen und Systeme. Diss. Bonn 1978. , Multiple integrals in the calculus of variations. Springer-Verlag, Berlin-Heidelberg -New York, (1968). Stampacchia, Le probleme de Dirichlet pour les equations elliptiques du second ordre acoefficients discontinus, Ann.

5) which holds for a minimizing sequence (or subsequence) yield at once the existence of minimizers under various boundary conditions. The next result which is reasonable to expect and one would like to prove is a partial regularity theorem for minimizers u in Cartm(n,RN), saying that, under certain smoothness and growth conditions on the integrand F a minimizer is actually a C 1 map, except for a closed singular set of measure zero. It is the aim of this paper to prove such a theorem, but unfortunately only for a class of integrands which satisfy a sharp growth condition.

T) = l{x E Qo: Mf(x) > t}l. 1} t<'•) Ap(f(x))dx] Ap(sf(x)) dS. 2 S Now, using iv), the last integral is controlled by Reverse Jensen inequalities with different supports ~ c I -IQ o 1- 1 1 o Ap(t) 2 dt t Qon{f~1} + cIQol Q 0 n{f~1} 1 549 Ap(f(x))dx f(x)dx. 2) we get Finally and this gives the result. 3. Iff is a nonnegative, bounded measurable function in the cube Q 0 and A is a function verifying i) and ii), then PROOF. Let to = }qo A(f)dx. { Ap(f)dx = - 1 JQ 0 { Then A(f)(A(f))p- 1 dx f IQoi }{A(f)~to} A_P(f) dx = I+ II.

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