The Dirichlet problem for $m$-subharmonic functions on compact sets
DOI:
https://doi.org/10.7146/math.scand.a-119708Abstract
We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.
References
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Åhag, P., Czyż, R., and Hed, L., Extension and approximation of $m$-subharmonic functions, Complex Var. Elliptic Equ. 63 (2018), no. 6, 783–801. https://doi.org/10.1080/17476933.2017.1345888
Åhag, P., Czyż, R., and Hed, L., The geometry of $m$-hyperconvex domains, J. Geom. Anal. 28 (2018), no. 4, 3196–3222. https://doi.org/10.1007/s12220-017-9957-2
Bedford, E., The Dirichlet problem for some overdetermined systems on the unit ball in $C^n$, Pacific J. Math. 51 (1974), 19–25.
Bliedtner, J. and Hansen, W., Potential theory: an analytic and probabilistic approach to balayage, Universitext, Springer-Verlag, Berlin, 1986. https://doi.org/10.1007/978-3-642-71131-2
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Klimek, M., Pluripotential theory, London Mathematical Society Monographs. New Series, no. 6, The Clarendon Press, Oxford University Press, New York, 1991.
Lu, H.-C., Complex Hessian equations, Ph.D. thesis, University of Toulouse III Paul Sabatier, 2012.
Perkins, T. L., The Dirichlet problem for harmonic functions on compact sets, Pacific J. Math. 254 (2011), no. 1, 211–226. https://doi.org/10.2140/pjm.2011.254.211
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Poletsky, E. A., Approximation by harmonic functions, Trans. Amer. Math. Soc. 349 (1997), no. 11, 4415–4427. https://doi.org/10.1090/S0002-9947-97-02041-2
Poletsky, E. A. and Sigurdsson, R., Dirichlet problems for plurisubharmonic functions on compact sets, eprint arXiv:1005.0248 [math.CV], 2010.
Poletsky, E. A. and Sigurdsson, R., Dirichlet problems for plurisubharmonic functions on compact sets, Math. Z. 271 (2012), no. 3-4, 877–892. https://doi.org/10.1007/s00209-011-0894-7
Sadullaev, A. and Abdullaev, B., Potential theory in the class of $m$-subharmonic functions, Tr. Mat. Inst. Steklova 279 (2012), 166–192. https://doi.org/10.1134/s0081543812080111
Sibony, N., Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987), no. 2, 299–319. https://doi.org/10.1215/S0012-7094-87-05516-5
Åhag, P., Czyż, R., and Hed, L., Extension and approximation of $m$-subharmonic functions, Complex Var. Elliptic Equ. 63 (2018), no. 6, 783–801. https://doi.org/10.1080/17476933.2017.1345888
Åhag, P., Czyż, R., and Hed, L., The geometry of $m$-hyperconvex domains, J. Geom. Anal. 28 (2018), no. 4, 3196–3222. https://doi.org/10.1007/s12220-017-9957-2
Bedford, E., The Dirichlet problem for some overdetermined systems on the unit ball in $C^n$, Pacific J. Math. 51 (1974), 19–25.
Bliedtner, J. and Hansen, W., Potential theory: an analytic and probabilistic approach to balayage, Universitext, Springer-Verlag, Berlin, 1986. https://doi.org/10.1007/978-3-642-71131-2
Gamelin, T. W., Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, no. 32, Cambridge University Press, Cambridge-New York, 1978.
Hansen, W., Harmonic and superharmonic functions on compact sets, Illinois J. Math. 29 (1985), no. 1, 103–107.
Klimek, M., Pluripotential theory, London Mathematical Society Monographs. New Series, no. 6, The Clarendon Press, Oxford University Press, New York, 1991.
Lu, H.-C., Complex Hessian equations, Ph.D. thesis, University of Toulouse III Paul Sabatier, 2012.
Perkins, T. L., The Dirichlet problem for harmonic functions on compact sets, Pacific J. Math. 254 (2011), no. 1, 211–226. https://doi.org/10.2140/pjm.2011.254.211
Poletsky, E. A., Analytic geometry on compacta in $\bf C^n$, Math. Z. 222 (1996), no. 3, 407–424. https://doi.org/10.1007/PL00004541
Poletsky, E. A., Approximation by harmonic functions, Trans. Amer. Math. Soc. 349 (1997), no. 11, 4415–4427. https://doi.org/10.1090/S0002-9947-97-02041-2
Poletsky, E. A. and Sigurdsson, R., Dirichlet problems for plurisubharmonic functions on compact sets, eprint arXiv:1005.0248 [math.CV], 2010.
Poletsky, E. A. and Sigurdsson, R., Dirichlet problems for plurisubharmonic functions on compact sets, Math. Z. 271 (2012), no. 3-4, 877–892. https://doi.org/10.1007/s00209-011-0894-7
Sadullaev, A. and Abdullaev, B., Potential theory in the class of $m$-subharmonic functions, Tr. Mat. Inst. Steklova 279 (2012), 166–192. https://doi.org/10.1134/s0081543812080111
Sibony, N., Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987), no. 2, 299–319. https://doi.org/10.1215/S0012-7094-87-05516-5
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Published
2020-09-03
How to Cite
Åhag, P., Czyż, R., & Hed, L. (2020). The Dirichlet problem for $m$-subharmonic functions on compact sets. MATHEMATICA SCANDINAVICA, 126(3), 497–512. https://doi.org/10.7146/math.scand.a-119708
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