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Uniform weighted estimates on pre-fractal domains

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  • We establish uniform estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake pre-fractal domains.
    Mathematics Subject Classification: Primary: 28A80; Secondary: 35J25, 35D35.


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  • [1]

    Y. Achdou, T. Deheuvels and N. Tchou, JLip versus Sobolev spaces on a class of self-similar fractal foliages, J. Math. Pures Appl. (9), 97 (2012), 142-172.doi: 10.1016/j.matpur.2011.07.002.


    Y. Achdou, C. Sabot and N. Tchou, Diffusion and propagation problems in some ramified domains with a fractal boundary, M2AN Math. Model. Numer. Anal., 40 (2006), 623-652.doi: 10.1051/m2an:2006027.


    Y. Achdou and N. Tchou, Neumann conditions on fractal boundaries, Asymptot. Anal., 53 (2007), 61-82.


    R. Adams, Sobolev Spaces, Academic Press, New York, 1975.


    B. Bennewitz and J. L. Lewis, On the dimension of p-harmonic measure, Ann. Acad. Sci. Fenn. Math., 30 (2005), 459-505.


    M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory. Related Fields, 79 (1988), 543-623.doi: 10.1007/BF00318785.


    R. F. Bass, K. Burdzy and Z.Chen, On the Robin problem in fractal domains, Proc. Lond. Math. Soc. (3), 96 (2008), 273-311.doi: 10.1112/plms/pdm045.


    M. T. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpinski carpets, J. Eur. Math. Soc. (JEMS), 12 (2010), 655-701.


    M. Borsuk and V. A. Kondratiev, Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, North-Holland Mathematical Library, 69. Elsevier Science B.V., Amsterdam, 2006.doi: 10.1016/S0924-6509(06)80026-7.


    R. Capitanelli, Transfer across scale irregular domains, Applied and industrial mathematics in Italy III, Ser. Adv. Math. Appl. Sci., World Sci. Publ., Hackensack, NJ, 82 (2010), 165-174.doi: 10.1142/9789814280303_0015.


    R. Capitanelli, Robin boundary condition on scale irregular fractals, Commun. Pure Appl. Anal., 9 (2010), 1221-1234.doi: 10.3934/cpaa.2010.9.1221.


    R. Capitanelli, Asymptotics for mixed Dirichlet-Robin problems in irregular domains, J. Math. Anal. Appl., 362 (2010), 450-459.doi: 10.1016/j.jmaa.2009.09.042.


    R. Capitanelli and M. A. Vivaldi, Insulating layers and Robin problems on Koch mixtures, J. Differential Equations, 251 (2011), 1332-1353.doi: 10.1016/j.jde.2011.02.003.


    R. Capitanelli and M. A. Vivaldi, On the Laplacean transfer across fractal mixtures, Asymptot. Anal., 83 (2013), 1-33.


    R. Capitanelli, M. R. Lancia and M. A. Vivaldi, Insulating layers of fractal type, Differential and Integral Equations, 26 (2013), 1055-1076.


    B. E. J. Dahlberg, $L^q$-estimates for Green potentials in Lipschitz domains, Math. Scand., 44 (1979), 149-170.


    M. Filoche and B. Sapoval, Transfer across random versus Deterministic Fractal Interfaces, Phys. Rev. Lett., 84 (2000), 5776-5779.doi: 10.1103/PhysRevLett.84.5776.


    D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983.


    D. S. Grebenkov, M. Filoche and B. Sapoval, Mathematical Basis for a General Theory of Laplacian Transport towards Irregular Interfaces, Phys. Rev. E, 73 (2006), 021103.doi: 10.1103/PhysRevE.73.021103.


    P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp.


    J. E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J., 30 (1981), 713-747.doi: 10.1512/iumj.1981.30.30055.


    D. S. Jerison and C. E. Kenig, Boundary behaviour of harmonic functions in non-tangentially accessible domains, Adv. in Math., 46 (1982), 80-147.doi: 10.1016/0001-8708(82)90055-X.


    A. Jonsson and H. Wallin, Function spaces on subsets of $\mathbbR^n$, Math. Rep., 2 (1984), xiv+221.


    A. Jonsson and H. Wallin, Boundary value problems and Brownian motion on fractals, Chaos Solitons Fractals, 8 (1997), 191-205.doi: 10.1016/S0960-0779(96)00048-3.


    J. Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143. Cambridge University Press, Cambridge, 2001.doi: 10.1017/CBO9780511470943.


    V. A. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Ob., 16 (1967), 209-292.


    S. Kusuoka, A diffusion Process on a Fractal, Probabilistic Methods in Mathematical Physics, Academic Press, 1987, 251-274.


    S. Kusuoka, Diffusion Processes in Nested Fractals, Lect. Notes in Math., 1567, Springer, 1993.


    M. R. Lancia and M. A. Vivaldi, On the regularity of the solutions for transmission problems, Adv. Math. Sci. Appl., 12 (2002), 455-466.


    M. R. Lancia and M. A. Vivaldi, Asymptotic convergence of transmission energy forms, Adv. Math. Sc. Appl., 13 (2003), 315-341.


    M. R. Lancia, U. Mosco and M. A. Vivaldi, Homogenization for conductive thin layers of pre-fractal type, J. Math. Anal. Appl., 347 (2008), 354-369.doi: 10.1016/j.jmaa.2008.06.011.


    T. Lindstrøm, Brownian motion penetrating the Sierpinski gasket, Asymptotic Problems in Probability Theory, Stochastic Models and Diffusions on Fractals, Longman Scientific, (1993), 248-278.


    V. G. Maz'ya, Sobolev Spaces, {Springer-Verlag}, {Berlin}, 1985.doi: 10.1007/978-3-662-09922-3.


    V. G. Maz'ya, S. Nazarov and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. I. Operator Theory: Advances and Applications, 111. Birkh\ae user Verlag, Basel, 2000.


    U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.doi: 10.1016/0001-8708(69)90009-7.


    U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.doi: 10.1006/jfan.1994.1093.


    U. Mosco, An elementary introduction to fractal analysis, Nonlinear Analysis and Applications to Physical Sciences, Springer Italia, Milan, (2004), 51-90.


    K. Nyström, Smoothness Properties of Dirichlet Problems in Domains with a Fractal Boundary, Ph. D. Dissertation, Ume$\dota$, 1994.


    K. Nyström, Integrability of Green potentials in fractal domains, Ark. Mat., 34 (1996), 335-381.doi: 10.1007/BF02559551.


    C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 299. Springer-Verlag, Berlin, 1992.doi: 10.1007/978-3-662-02770-7.


    G. Savaré and G. Schimperna, Domain perturbations and estimates for the solutions of second order elliptic equations, J. Math. Pures Appl. (9), 81 (2002), 1071-1112.doi: 10.1016/S0021-7824(02)01256-4.


    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.


    R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006.


    A. Wannebo, Hardy inequalities, Proc. Amer. Math. Soc., 109 (1990), 85-95.doi: 10.1090/S0002-9939-1990-1010807-1.

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