Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T21:20:14.485Z Has data issue: false hasContentIssue false
  • English
  • Français

WEIGHTED ESTIMATES ON FRACTAL DOMAINS

Part of: Elliptic equations and systems Classical measure theory

Published online by Cambridge University Press:  03 February 2015

Raffaela Capitanelli  and
Maria Agostina Vivaldi
Raffaela Capitanelli
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, “Sapienza Università di Roma”, Via A. Scarpa 16, 00161 Roma, Italy email [email protected]
Maria Agostina Vivaldi
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, “Sapienza Università di Roma”, Via A. Scarpa 16, 00161 Roma, Italy email [email protected]
Get access

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.

MSC classification

Primary: 28A80: Fractals 35J08: Green's functions 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 370 - 384
Copyright
Copyright © University College London 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bennewitz, B. and Lewis, J. L., On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. 30(2) 2005, 459–505.Google Scholar
Brennan, J. E., The integrability of the derivative in conformal mapping. J. Lond. Math. Soc. 18(2) 1978, 261–272.CrossRefGoogle Scholar
Capitanelli, R., Robin boundary condition on scale irregular fractals. Commun. Pure Appl. Anal. 9(5) 2010, 1221–1234 ; doi: 10.3934/cpaa.2010.9.1221.CrossRefGoogle Scholar
Capitanelli, R. and Vivaldi, M. A., On the Laplacean transfer across fractal mixtures. Asymptot. Anal. 83(1–2) 2013, 1–33 ; doi: 10.3233/ASY-2012-1149.Google Scholar
Capitanelli, R. and Vivaldi, M. A., Uniform weighted estimates on pre-fractal domains. Discrete Contin. Dyn. Syst. Ser. B 19(7) 2014, 1969–1985 ; doi: 10.3934/dcdsb.2014.19.1969.Google Scholar
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).Google Scholar
Gol’dshtein, V. and Ukhlov, A., Brennan’s Conjecture and universal Sobolev inequalities. Bull. Sci. Math. 138(2) 2014, 253–269.CrossRefGoogle Scholar
Grisvard, P., Elliptic Problems in Non-Smooth Domains, Pitman (Boston, 1985).Google Scholar
Hedenmalm, H., The dual of a Bergman space on simply connected domains. J. Anal. Math. 88 2002, 311–335.CrossRefGoogle Scholar
Hedenmalm, H. and Shimorin, S., Weighted Bergman spaces and the integral means spectrum of conformal mappings. Duke Math. J. 127(2) 2005, 341–393.CrossRefGoogle Scholar
Hurri-Syrjänen, R. and Staples, S. G., A quasiconformal analogue of Brennan’s conjecture. Complex Var. Elliptic Equ. 35 1998, 27–32.Google Scholar
Hutchinson, J. E., Fractals and selfsimilarity. Indiana Univ. Math. J. 30 1981, 713–747.CrossRefGoogle Scholar
Jerison, D. S. and Kenig, C. E., Boundary behaviour of harmonic functions in non-tangentially accessible domains. Adv. Math. 46 1982, 80–147.CrossRefGoogle Scholar
Kondratiev, V. A., Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. Obs. 16 1967, 209–292.Google Scholar
Mosco, U., Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 1969, 510–585.CrossRefGoogle Scholar
Mosco, U., Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2) 1994, 368–421.CrossRefGoogle Scholar
Nyström, K., Smoothness properties of Dirichlet problems in domains with a fractal boundary. PhD Dissertation, Umeȧ, 1994.Google Scholar
Nyström, K., Integrability of Green potentials in fractal domains. Ark. Mat. 34 1996, 335–381.CrossRefGoogle Scholar
Pommerenke, Ch., On the integral means of the derivative of a univalent function. J. Lond. Math. Soc. 32(2) 1985, 254–258.CrossRefGoogle Scholar
Stein, E. M., Singular Integrals and Differentiability Properties of Functions (Princeton Mathematical Series 30), Princeton University Press (Princeton, NJ, 1970).Google Scholar
Troianiello, G. M., Elliptic Partial Differential Equations and Obstacle Problems (The University Series in Mathematics), Plenum Press (New York, 1987).CrossRefGoogle Scholar
Wannebo, A., Hardy inequalities. Proc. Amer. Math. Soc. 109(1) 1990, 85–95.CrossRefGoogle Scholar