Geometry of Prime End Boundary and the Dirichlet Problem for Bounded Domains in Metric Measure Spaces
Tóm tắt
In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincaré inequality. We show the resolutivity of functions that are continuous on the prime end boundary and are Lipschitz regular when restricted to the subset of all prime ends whose impressions are singleton sets. We also consider a new notion of capacity adapted to the prime end boundary, and show that bounded perturbations of such functions on subsets of the prime end boundary with zero capacity are resolutive and that their Perron solutions coincide with the Perron solution of the original functions. We also describe some examples which demonstrate the efficacy of the prime end boundary approach in obtaining new results even for the classical Dirichlet problem for some Euclidean domains.
Tài liệu tham khảo
Adamowicz, T., Björn, A., Björn, J., Shanmugalingam, N.: Prime ends for domains in metric spaces. Adv. Math. 238, 459–505 (2013)
Björn, A.: The Dirichlet problem for p-harmonic functions on the topologist’s comb. Preprint, http://lanl.arxiv.org/abs/1304.1681. 16 pp. (2013)
Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, vol. 17. European Math. Soc., Zurich (2011)
Björn, A., Björn, J., Shanmugalingam, N.: The Perron method for p-harmonic functions in metric spaces. J. Diff. Equ. 195, 398–429 (2003)
Björn, A., Björn, J., Shanmugalingam, N.: Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions in metric measure spaces Houston. J. Math. 34, 1197–1211 (2008)
Björn, A., Björn, J., Shanmugalingam, N.: The Dirichelt problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities. Preprint, http://lanl.arxiv.org/abs/1302.3887. 34 pp. (2013)
Björn, A., Björn, J., Shanmugalingam, N.: The Mazurkiewicz distance and sets that are finitely connected at the boundary Preprint, http://arxiv.org/abs/1311.5122. 22 pp (2013)
Dalrymple, K., Strichartz, R., Vinson, J.: Fractal differential equations on the Sierpiński gasket. J. Fourier Anal. Appl. 5, 203–284 (1999)
Farnana, Z.: Convergence results for obstacle problems on metric spaces. J. Math. Anal. Appl. 371, 436–446 (2010)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré Mem. Amer. Math. Soc. 145, 688 (2000)
Hajłasz, P., Martio, O.: Traces of Sobolev functions on fractal type sets and characterization of extension domains. J. Funct. Anal. 143, 221–246 (1997)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, p. 416. Dover Publications. Reprint Edition (2006)
Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)
Kinnunen, J., Martio, O.: Nonlinear potential theory on metric spaces. Ill. Math. J. 46, 857–883 (2002)
Kinnunen, J., Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105, 401–423 (2001)
Lucia, M., Puls, M.: The p-harmonic boundary for metric measure spaces Preprint, http://lanl.arxiv.org/abs/1309.3596. 1–12 (2013)
Maz’ya, V.G.: On the extension of functions belonging to S.L. Sobolev spaces. Investigations on linear operators and the theory of functions. XI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113, 231–236, 269 (1981)
Munkres, J.R.: Topology: A First Course, p. 413. Prentice-Hall (1975)
Näkki, R.: Prime ends and quasiconformal mappings. J. Anal. Math. 35, 13–40 (1979)
Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45, 1021–1050 (2001)
Shanmugalingam, N.: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279 (2000)
Shanmugalingam, N.: Some convergence results for p-harmonic functions on metric measure spaces. Proc. Lond. Math. Soc. 87, 226–246 (2003)