Harmonic Functions

Luis T. Magalhães
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Luis T. Magalhães: University of Lisbon, Instituto Superior Técnico

Chapter Chapter 9 in Complex Analysis and Dynamics in One Variable with Applications, 2025, pp 189-217 from Springer

Abstract: Abstract Relationships of harmonic and holomorphic functions, mean value property of harmonic functions, solution of the Dirichlet problem in disks, Poisson kernel and integral, characterization of harmonic functions by mean value property, Harnack inequality, maximum principle for harmonic functions, uniqueness of solution of Dirichlet problem in bounded sets, Harnack theorem, subharmonic and superharmonic functions, characterization of subharmonic functions by a submean value property and by the maximum principle, Perron solution in open set, barrier function at an open set boundary point, regular boundary point, existence of solution of Dirichlet problem in an open set with all boundary points regular by the Perron method. Exercises, including on mean spatial property, conjugate harmonic functions, conjugate harmonic differentials and Hodge operator, Poisson–Jensen formula, Hadamard factorization theorem, and with applications to elasticity.

Date: 2025
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DOI: 10.1007/978-3-031-64999-8_9

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