\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Uniform weighted estimates on pre-fractal domains

Abstract Related Papers Cited by
  • 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.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

    [2]

    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.

    [3]

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

    [4]

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

    [5]

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

    [6]

    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.

    [7]

    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.

    [8]

    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.

    [9]

    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.

    [10]

    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.

    [11]

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

    [12]

    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.

    [13]

    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.

    [14]

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

    [15]

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

    [16]

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

    [17]

    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.

    [18]

    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.

    [19]

    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.

    [20]

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

    [21]

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

    [22]

    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.

    [23]

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

    [24]

    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.

    [25]

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

    [26]

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

    [27]

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

    [28]

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

    [29]

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

    [30]

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

    [31]

    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.

    [32]

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

    [33]

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

    [34]

    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.

    [35]

    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.

    [36]

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

    [37]

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

    [38]

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

    [39]

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

    [40]

    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.

    [41]

    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.

    [42]

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

    [43]

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

    [44]

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

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(176) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return