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WEIGHTED ESTIMATES ON FRACTAL DOMAINS

  • Raffaela Capitanelli (a1) and Maria Agostina Vivaldi (a2)

Abstract

The aim of the paper is to establish estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake domains, as well as uniform estimates for the solutions of the Dirichlet problems on pre-fractal approximating domains.

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1.Bennewitz, B. and Lewis, J. L., On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. 30(2) 2005, 459505.
2.Brennan, J. E., The integrability of the derivative in conformal mapping. J. Lond. Math. Soc. 18(2) 1978, 261272.
3.Capitanelli, R., Robin boundary condition on scale irregular fractals. Commun. Pure Appl. Anal. 9(5) 2010, 12211234 ; doi: 10.3934/cpaa.2010.9.1221.
4.Capitanelli, R. and Vivaldi, M. A., On the Laplacean transfer across fractal mixtures. Asymptot. Anal. 83(1–2) 2013, 133 ; doi: 10.3233/ASY-2012-1149.
5.Capitanelli, R. and Vivaldi, M. A., Uniform weighted estimates on pre-fractal domains. Discrete Contin. Dyn. Syst. Ser. B 19(7) 2014, 19691985 ; doi: 10.3934/dcdsb.2014.19.1969.
6.Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224), 2nd edn edn., Springer (Berlin, 1983).
7.Gol’dshtein, V. and Ukhlov, A., Brennan’s Conjecture and universal Sobolev inequalities. Bull. Sci. Math. 138(2) 2014, 253269.
8.Grisvard, P., Elliptic Problems in Non-Smooth Domains, Pitman (Boston, 1985).
9.Hedenmalm, H., The dual of a Bergman space on simply connected domains. J. Anal. Math. 88 2002, 311335.
10.Hedenmalm, H. and Shimorin, S., Weighted Bergman spaces and the integral means spectrum of conformal mappings. Duke Math. J. 127(2) 2005, 341393.
11.Hurri-Syrjänen, R. and Staples, S. G., A quasiconformal analogue of Brennan’s conjecture. Complex Var. Elliptic Equ. 35 1998, 2732.
12.Hutchinson, J. E., Fractals and selfsimilarity. Indiana Univ. Math. J. 30 1981, 713747.
13.Jerison, D. S. and Kenig, C. E., Boundary behaviour of harmonic functions in non-tangentially accessible domains. Adv. Math. 46 1982, 80147.
14.Kondratiev, V. A., Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. Obs. 16 1967, 209292.
15.Mosco, U., Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 1969, 510585.
16.Mosco, U., Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2) 1994, 368421.
17.Nyström, K., Smoothness properties of Dirichlet problems in domains with a fractal boundary. PhD Dissertation, Umeȧ, 1994.
18.Nyström, K., Integrability of Green potentials in fractal domains. Ark. Mat. 34 1996, 335381.
19.Pommerenke, Ch., On the integral means of the derivative of a univalent function. J. Lond. Math. Soc. 32(2) 1985, 254258.
20.Stein, E. M., Singular Integrals and Differentiability Properties of Functions (Princeton Mathematical Series 30), Princeton University Press (Princeton, NJ, 1970).
21.Troianiello, G. M., Elliptic Partial Differential Equations and Obstacle Problems (The University Series in Mathematics), Plenum Press (New York, 1987).
22.Wannebo, A., Hardy inequalities. Proc. Amer. Math. Soc. 109(1) 1990, 8595.
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