Abstract

Given a bounded open set UU in Rd\real^d, the space of all continuous real functions on U‾\ov U which are harmonic on UU is denoted by H(U)H(U). Further, a lower bounded, Borel measurable numerical function ss on U‾\ov U is said to be H(U)H(U)-concave if ∫s dμ≤s(x)\int s\,d\mu\le s(x) for every x∈U‾x\in\ov U and every measure μ\mu on U‾\ov U satisfying ∫h dμ=h(x)\int h\,d\mu= h(x) for all h∈H(U)h\in H(U). It is shown that every H(U)H(U)-concave function is continuous on UU and, under additional assumptions on UU, several characterizations of H(U)H(U)-concave functions are given. For compact sets KK in Rd\real^d, continuity properties of H0(K)\hk-concave functions are studied, where H0(K)\hk is the space of all functions on KK which can be extended to be harmonic in some neighborhood of KK (depending on the given function). We prove that these functions are finely upper semicontinuous on the fine interior of KK, but not necessarily finely continuous there. Most of the results are established in the context of harmonic spaces, covering solutions of elliptic and parabolic second order partial differential equations. For example, it is shown that H(U)H(U)-concave functions are always continuous on UU if and only if the underlying harmonic space has the Brelot convergence property.

Author information

Contact details are reproduced from the original publication and may be historical.

Ivan Netuka

Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 86, 186 75 Praha 8, Czech Republic

netuka@karlin.mff.cuni.cz

Keywords

  • Harmonic functions
  • Choquet theory
  • concave functions
  • balayage
  • fine topology
  • function spaces

Mathematics Subject Classification

31B05, 31C40, 31D05, 46A55

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Suggested citation

W. Hansen, I. Netuka. “Continuity Properties of Concave Functions in Potential Theory.” Journal of Convex Analysis 15 (2008), No. 1, 39–53. https://doi.org/10.68381/jca15003

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.