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Extremal problems and related Dirichlet problems. (English) Zbl 1201.31001

Summary: The paper considers extremal problems related to the Dirichlet problem. We give the solution of the Dirichlet problem on arbitrary compact sets in terms of the Fourier coefficients of the restriction-function.
We prove some related uniform convergence results for the solution. We show that the exact value of the minimum of the considered functional equals to the area of the image of the compact set on which we are working, through the related holomorphic function. In particular, this method can be used for problems involving holomorphic functions on the unit disc, for which the coefficients of the power series around the origin are known and the series converges. An example is given. Maximizing some convex operators on sets of subharmonic non-smooth functions and on sets of convex non-smooth functions is also a goal of this work. Here the solutions are extreme functions of the set. Finally, minimizing problems for some convex operators on a finite dimensional simplex are considered.

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30E25 Boundary value problems in the complex plane
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