Abstract
Let Ω be a subset of ℝn, and let S be the set of bounded superharmonic functions on Ω. If f is denned and bounded in Ω outside a polar set, then there is a best approximation g0 to f from S that differs from the least superharmonic majorant of f by a constant. Least superharmonic majorants are fundamental in Potential Theory, where they are called réduites, and they occur in the the theory of elliptic PDEs as solutions of variational inequalities related to the obstacle problem. By applying results from these two areas one may give sufficient conditions on f and Ω which guarantee that g0 has certain continuity or smoothness properties.
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Zwick, D. (1990). The Obstacle Problem and Best Superharmonic Approximation. In: Haußmann, W., Jetter, K. (eds) Multivariate Approximation and Interpolation. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 94. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5685-0_24
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DOI: https://doi.org/10.1007/978-3-0348-5685-0_24
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