Abstract
Let (X, d) be a pathwise connected metric space equipped with an Ahlfors Q-regular measure μ, Q ∈ [1, ∞ ). Suppose that (X, d, μ) supports a 2-Poincaré inequality and a Sobolev–Poincaré type inequality for the corresponding “Gaussian measure”. The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation Δu = f, where \(f\in L^{p}_{\rm{loc}}\). When p > Q, the local Lipschitz continuity of u is established.
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References
Bakry, D.: On Sobolev and Logarithmic Inequalities for Markov Semigroups, New Trends in Stochastic Analysis (Charingworth, 1994), pp. 43–75. World Scientific Publishing, River Edge, NJ (1997)
Bakry, D., Emery, M.: Diffusions hypercontractives. Sémin. Probab. XIX, 177–206 (1983/84)
Biroli, M., Mosco, U.: Sobolev inequalities for Dirichlet forms on homogeneous spaces. Boundary value problems for partial differential equations and applications. RMA Res. Notes Appl. Math., vol. 29, pp. 305–311. Masson, Paris (1993)
Biroli, M., Mosco, U.: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. 169, 125–181 (1995)
Caffarelli, L.A., Kenig, C.E.: Gradient estimates for variable coefficient parabolic equations and singular perturbation problems. Am. J. Math. 120, 391–439 (1998)
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)
Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. In: de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter & Co., Berlin (1994)
Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)
Gross, L.: Hypercontractivity over complex manifolds. Acta Math. 182, 159–206 (1999)
Hajłasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 320(10), 1211–1215 (1995)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688) (2000)
Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)
Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3), 233–247 (2000)
Koskela, P., Rajala, K., Shanmugalingam, N.: Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces. J. Funct. Anal. 202, 147–173 (2003)
Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 2, 27–38 (1992)
Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16, 243–279 (2000)
Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
Sturm, K.T.: Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1996)
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Renjin Jiang was partially supported by the Academy of Finland grants 120972 and 131477.
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Jiang, R. Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces. Potential Anal 37, 281–301 (2012). https://doi.org/10.1007/s11118-011-9256-7
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DOI: https://doi.org/10.1007/s11118-011-9256-7