September  2014, 19(7): 1969-1985. doi: 10.3934/dcdsb.2014.19.1969

Uniform weighted estimates on pre-fractal domains

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma "Sapienza", Via A. Scarpa 16, 00161 Roma, Italy, Italy

Received  April 2013 Revised  January 2014 Published  August 2014

We establish uniform estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake pre-fractal domains.
Citation: Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969
References:
[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. doi: 10.1016/j.matpur.2011.07.002. Google Scholar

[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. doi: 10.1051/m2an:2006027. Google Scholar

[3]

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

[4]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[10]

R. Capitanelli, Transfer across scale irregular domains,, Applied and industrial mathematics in Italy III, 82 (2010), 165. doi: 10.1142/9789814280303_0015. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983). Google Scholar

[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). doi: 10.1103/PhysRevE.73.021103. Google Scholar

[20]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

J. Kigami, Analysis on Fractals,, Cambridge Tracts in Mathematics, (2001). doi: 10.1017/CBO9780511470943. Google Scholar

[26]

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

[27]

S. Kusuoka, A diffusion Process on a Fractal,, Probabilistic Methods in Mathematical Physics, (1987), 251. Google Scholar

[28]

S. Kusuoka, Diffusion Processes in Nested Fractals,, Lect. Notes in Math., (1567). Google Scholar

[29]

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

[30]

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

[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. doi: 10.1016/j.jmaa.2008.06.011. Google Scholar

[32]

T. Lindstrøm, Brownian motion penetrating the Sierpinski gasket,, Asymptotic Problems in Probability Theory, (1993), 248. Google Scholar

[33]

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

[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, (2000). Google Scholar

[35]

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

[36]

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

[37]

U. Mosco, An elementary introduction to fractal analysis,, Nonlinear Analysis and Applications to Physical Sciences, (2004), 51. Google Scholar

[38]

K. Nyström, Smoothness Properties of Dirichlet Problems in Domains with a Fractal Boundary,, Ph. D. Dissertation, (1994). Google Scholar

[39]

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

[40]

C. Pommerenke, Boundary Behaviour of Conformal Maps,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1992). doi: 10.1007/978-3-662-02770-7. Google Scholar

[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. doi: 10.1016/S0021-7824(02)01256-4. Google Scholar

[42]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[43]

R. Strichartz, Differential Equations on Fractals,, Princeton University Press, (2006). Google Scholar

[44]

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

show all references

References:
[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. doi: 10.1016/j.matpur.2011.07.002. Google Scholar

[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. doi: 10.1051/m2an:2006027. Google Scholar

[3]

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

[4]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[10]

R. Capitanelli, Transfer across scale irregular domains,, Applied and industrial mathematics in Italy III, 82 (2010), 165. doi: 10.1142/9789814280303_0015. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983). Google Scholar

[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). doi: 10.1103/PhysRevE.73.021103. Google Scholar

[20]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics, (1985). Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

J. Kigami, Analysis on Fractals,, Cambridge Tracts in Mathematics, (2001). doi: 10.1017/CBO9780511470943. Google Scholar

[26]

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

[27]

S. Kusuoka, A diffusion Process on a Fractal,, Probabilistic Methods in Mathematical Physics, (1987), 251. Google Scholar

[28]

S. Kusuoka, Diffusion Processes in Nested Fractals,, Lect. Notes in Math., (1567). Google Scholar

[29]

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

[30]

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

[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. doi: 10.1016/j.jmaa.2008.06.011. Google Scholar

[32]

T. Lindstrøm, Brownian motion penetrating the Sierpinski gasket,, Asymptotic Problems in Probability Theory, (1993), 248. Google Scholar

[33]

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

[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, (2000). Google Scholar

[35]

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

[36]

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

[37]

U. Mosco, An elementary introduction to fractal analysis,, Nonlinear Analysis and Applications to Physical Sciences, (2004), 51. Google Scholar

[38]

K. Nyström, Smoothness Properties of Dirichlet Problems in Domains with a Fractal Boundary,, Ph. D. Dissertation, (1994). Google Scholar

[39]

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

[40]

C. Pommerenke, Boundary Behaviour of Conformal Maps,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1992). doi: 10.1007/978-3-662-02770-7. Google Scholar

[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. doi: 10.1016/S0021-7824(02)01256-4. Google Scholar

[42]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[43]

R. Strichartz, Differential Equations on Fractals,, Princeton University Press, (2006). Google Scholar

[44]

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

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