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 and 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), 142-172. 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 $\mathbb{R}^{N}$, 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.

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), 142-172. 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 $\mathbb{R}^{N}$, 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.

[1]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596

[2]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[3]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061

[4]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[5]

Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024

[6]

Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28 (2) : 821-836. doi: 10.3934/era.2020042

[7]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1

[8]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control and Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

[9]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

[10]

Joachim Escher, Christina Lienstromberg. A survey on second order free boundary value problems modelling MEMS with general permittivity profile. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 745-771. doi: 10.3934/dcdss.2017038

[11]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053

[12]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[13]

Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations and Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101

[14]

Roberto Triggiani. Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space. Evolution Equations and Control Theory, 2016, 5 (4) : 489-514. doi: 10.3934/eect.2016016

[15]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209

[16]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[17]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609

[18]

Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022

[19]

M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181

[20]

Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations. Electronic Research Archive, 2021, 29 (6) : 4159-4175. doi: 10.3934/era.2021077

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (150)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]