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Asymptotic analysis of a nonsimple thermoelastic rod
Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains
1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy |
2. | Università degli studi Roma 3, Dipartimento di Matematica, Largo San Leonardo Murialdo 1, 00146 Roma, Italy |
3. | Dipartimento di Matematica, Universita degli Studi di Roma "La Sapienza", P.zale Aldo Moro 2, 00185 Roma, Italy |
References:
[1] |
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1966.
doi: 10.1007/978-3-662-03282-4. |
[2] |
H. Attouch, Variational Convergence for Function and Operators, Eds. Pitman Advanced Publishing Program, London, 1984. |
[3] |
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Value Problems, Wiley, New York, 1984. |
[4] |
F. Brezzi and G. Gilardi, Fundamentals of P.D.E. for Numerical Analysis, In Finite Element Handbook (ed: H. Kardenstuncer and D.H. Norrie), McGraw-Hill Book Co., New York, 1987. |
[5] | |
[6] |
A. Buffa and P. Ciarlet, On traces for functional spaces related to Maxwell's equations, part 1: An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appli. Sci., 24 (2001), 9-30.
doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. |
[7] |
J. R. Cannon and G. H. Meyer, On a diffusion in a fractured medium, SIAM J. Appl. Math., 20 (1971), 434-448.
doi: 10.1137/0120047. |
[8] |
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. |
[9] |
M. Cefalo, G. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473.
doi: 10.1016/j.amc.2011.11.033. |
[10] |
M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation Differential and Integral equations, Differential Integral Equations, 26 (2013), 1027-1054. |
[11] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, volume 5: Evolution problem 1, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58090-1. |
[12] |
K. Falconer, The Geometry of Fractal Sets, $2^{nd}$ ed. Cambridge Univ. Press, Cambridge, 1986. |
[13] |
A. Favini, R. Labbas, K. Lemrabet and S. Maingot, Study of limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. mat. complut., 18 (2005), 143-176. |
[14] |
A. Favini , J. A. Goldstein, G. Ruiz and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.
doi: 10.1007/s00028-002-8077-y. |
[15] |
U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137.
doi: 10.4171/ZAA/1190. |
[16] |
P. Grisvard, Elliptic Problems in non Smooth Domains, Pitman, Boston, 1985. |
[17] |
W. Hackbush, Elliptic Partial Differential Equations, Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, Springer-Verlag, Berlin, 1992. |
[18] |
M. Hino, Convergence of non-symmetric forms, J. Math. Kyoto Univ., 38 (1998), 329-341. |
[19] |
D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207.
doi: 10.1090/S0273-0979-1981-14884-9. |
[20] |
P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces}, Acta Math., 147 (1981), 71-88.
doi: 10.1007/BF02392869. |
[21] |
A. Jonsson, Besov spaces on closed subsets of $\mathbb{R}^{N}$, Trans. Amer. math. Soc., 341 (1994), 355-370.
doi: 10.2307/2154626. |
[22] |
A. Jonsson and H. Wallin, Function spaces on subset of $\mathbb{R}^{N}$, Part 1., Math. Reports, 2 (1984), xiv+221 pp. |
[23] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. |
[24] |
A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbbmathbb{R}^{d}$, Forum Math., 17 (2005), 225-259.
doi: 10.1515/form.2005.17.2.225. |
[25] |
P. Korman, Existence of periodic solutions for a class of non linear problems, Non linear Anal., 7 (1983), 873-879.
doi: 10.1016/0362-546X(83)90063-9. |
[26] |
K. Kuwae and T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Communications in analysis and geometry, 11 (2003), 599-673.
doi: 10.4310/CAG.2003.v11.n4.a1. |
[27] |
M. R. Lancia, A transmission problem with a fractal interface, Z. Anal. Anwendungen, 21 (2002), 113-133.
doi: 10.4171/ZAA/1067. |
[28] |
M. R. Lancia, Second order transmission problems across a fractal surface, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 191-213. |
[29] |
M. R. Lancia, V. Regis Durante and P. Vernole, Density results for energy spaces on some fractafolds, Z. Anal. Anwendungen, 34 (2015), 357-372.
doi: 10.4171/ZAA/1544. |
[30] |
M. R. Lancia and P. Vernole, Convergence results for parabolic transmission problems across higly conductive layers with small capacity, Adv. Math. Sci. Appl., 16 (2006), 411-445. |
[31] |
M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM Journal on Mathematical Analysis, 42 (2010), 1539-1567.
doi: 10.1137/090761173. |
[32] |
M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, Jour. of Evol. Eq., 14 (2014), 681-712.
doi: 10.1007/s00028-014-0233-7. |
[33] |
M. R. Lancia and M. A. Vivaldi, Lipschitz spaces and Besov traces on self similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 101-116. |
[34] |
M. R. Lancia and M. A. Vivaldi, Asymptotic convergence of transmission energy forms, Adv. Math. Sc. Appl., 13 (2003), 315-341. |
[35] |
J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications,, Berlin, Springer-Verlag, 1972. |
[36] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in non linear differential equations and their applications, 16, Birkhauser-Verlag, Basel, 1995.
doi: 10.1007/978-3-0348-9234-6. |
[37] |
S. Mataloni, On a type of convergence for non-symmetric Dirichlet forms, Adv. Math. Sci. Appl.,9 (1999), 749-773. |
[38] |
Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992. |
[39] |
U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.
doi: 10.1006/jfan.1994.1093. |
[40] |
J. Necas, Les Methodes Directes en Theorie des Équationes Elliptiques, Masson, Paris, 1967. |
[41] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[42] |
M. Röckner and T. S. Zhang, Convergence of operators semigroups generated by elliptic operators, Osaka J. Math., 34 (1997), 923-932. |
[43] |
B. Sapoval, General formulation of Laplacian transfer across irregular surfaces, Phys. Rev. Lett., 73 (1994), 3314-3316.
doi: 10.1103/PhysRevLett.73.3314. |
[44] |
M. Shinbrot, Watern waves over periodic bottoms in three dimensions, J. Inst. Math. Appl., 25 (1980), 367-385.
doi: 10.1093/imamat/25.4.367. |
[45] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press., Princeton, 1970. |
[46] |
M. Tőlle, Convergence of Non-Symmetric Forms with Changing Reference Measures, Thesis, University of Bielefeld August, 2006. |
[47] |
A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185; (English) English transl. Theor. Probability Appl., 4 (1959), 164-177.
doi: 10.1137/1104014. |
[48] |
H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Mskr. Math., 73 (1991), 117-125.
doi: 10.1007/BF02567633. |
show all references
References:
[1] |
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1966.
doi: 10.1007/978-3-662-03282-4. |
[2] |
H. Attouch, Variational Convergence for Function and Operators, Eds. Pitman Advanced Publishing Program, London, 1984. |
[3] |
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Value Problems, Wiley, New York, 1984. |
[4] |
F. Brezzi and G. Gilardi, Fundamentals of P.D.E. for Numerical Analysis, In Finite Element Handbook (ed: H. Kardenstuncer and D.H. Norrie), McGraw-Hill Book Co., New York, 1987. |
[5] | |
[6] |
A. Buffa and P. Ciarlet, On traces for functional spaces related to Maxwell's equations, part 1: An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appli. Sci., 24 (2001), 9-30.
doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. |
[7] |
J. R. Cannon and G. H. Meyer, On a diffusion in a fractured medium, SIAM J. Appl. Math., 20 (1971), 434-448.
doi: 10.1137/0120047. |
[8] |
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. |
[9] |
M. Cefalo, G. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473.
doi: 10.1016/j.amc.2011.11.033. |
[10] |
M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation Differential and Integral equations, Differential Integral Equations, 26 (2013), 1027-1054. |
[11] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, volume 5: Evolution problem 1, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58090-1. |
[12] |
K. Falconer, The Geometry of Fractal Sets, $2^{nd}$ ed. Cambridge Univ. Press, Cambridge, 1986. |
[13] |
A. Favini, R. Labbas, K. Lemrabet and S. Maingot, Study of limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. mat. complut., 18 (2005), 143-176. |
[14] |
A. Favini , J. A. Goldstein, G. Ruiz and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.
doi: 10.1007/s00028-002-8077-y. |
[15] |
U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137.
doi: 10.4171/ZAA/1190. |
[16] |
P. Grisvard, Elliptic Problems in non Smooth Domains, Pitman, Boston, 1985. |
[17] |
W. Hackbush, Elliptic Partial Differential Equations, Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, Springer-Verlag, Berlin, 1992. |
[18] |
M. Hino, Convergence of non-symmetric forms, J. Math. Kyoto Univ., 38 (1998), 329-341. |
[19] |
D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207.
doi: 10.1090/S0273-0979-1981-14884-9. |
[20] |
P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces}, Acta Math., 147 (1981), 71-88.
doi: 10.1007/BF02392869. |
[21] |
A. Jonsson, Besov spaces on closed subsets of $\mathbb{R}^{N}$, Trans. Amer. math. Soc., 341 (1994), 355-370.
doi: 10.2307/2154626. |
[22] |
A. Jonsson and H. Wallin, Function spaces on subset of $\mathbb{R}^{N}$, Part 1., Math. Reports, 2 (1984), xiv+221 pp. |
[23] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. |
[24] |
A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbbmathbb{R}^{d}$, Forum Math., 17 (2005), 225-259.
doi: 10.1515/form.2005.17.2.225. |
[25] |
P. Korman, Existence of periodic solutions for a class of non linear problems, Non linear Anal., 7 (1983), 873-879.
doi: 10.1016/0362-546X(83)90063-9. |
[26] |
K. Kuwae and T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Communications in analysis and geometry, 11 (2003), 599-673.
doi: 10.4310/CAG.2003.v11.n4.a1. |
[27] |
M. R. Lancia, A transmission problem with a fractal interface, Z. Anal. Anwendungen, 21 (2002), 113-133.
doi: 10.4171/ZAA/1067. |
[28] |
M. R. Lancia, Second order transmission problems across a fractal surface, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 191-213. |
[29] |
M. R. Lancia, V. Regis Durante and P. Vernole, Density results for energy spaces on some fractafolds, Z. Anal. Anwendungen, 34 (2015), 357-372.
doi: 10.4171/ZAA/1544. |
[30] |
M. R. Lancia and P. Vernole, Convergence results for parabolic transmission problems across higly conductive layers with small capacity, Adv. Math. Sci. Appl., 16 (2006), 411-445. |
[31] |
M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM Journal on Mathematical Analysis, 42 (2010), 1539-1567.
doi: 10.1137/090761173. |
[32] |
M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, Jour. of Evol. Eq., 14 (2014), 681-712.
doi: 10.1007/s00028-014-0233-7. |
[33] |
M. R. Lancia and M. A. Vivaldi, Lipschitz spaces and Besov traces on self similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 101-116. |
[34] |
M. R. Lancia and M. A. Vivaldi, Asymptotic convergence of transmission energy forms, Adv. Math. Sc. Appl., 13 (2003), 315-341. |
[35] |
J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications,, Berlin, Springer-Verlag, 1972. |
[36] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in non linear differential equations and their applications, 16, Birkhauser-Verlag, Basel, 1995.
doi: 10.1007/978-3-0348-9234-6. |
[37] |
S. Mataloni, On a type of convergence for non-symmetric Dirichlet forms, Adv. Math. Sci. Appl.,9 (1999), 749-773. |
[38] |
Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992. |
[39] |
U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.
doi: 10.1006/jfan.1994.1093. |
[40] |
J. Necas, Les Methodes Directes en Theorie des Équationes Elliptiques, Masson, Paris, 1967. |
[41] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[42] |
M. Röckner and T. S. Zhang, Convergence of operators semigroups generated by elliptic operators, Osaka J. Math., 34 (1997), 923-932. |
[43] |
B. Sapoval, General formulation of Laplacian transfer across irregular surfaces, Phys. Rev. Lett., 73 (1994), 3314-3316.
doi: 10.1103/PhysRevLett.73.3314. |
[44] |
M. Shinbrot, Watern waves over periodic bottoms in three dimensions, J. Inst. Math. Appl., 25 (1980), 367-385.
doi: 10.1093/imamat/25.4.367. |
[45] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press., Princeton, 1970. |
[46] |
M. Tőlle, Convergence of Non-Symmetric Forms with Changing Reference Measures, Thesis, University of Bielefeld August, 2006. |
[47] |
A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185; (English) English transl. Theor. Probability Appl., 4 (1959), 164-177.
doi: 10.1137/1104014. |
[48] |
H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Mskr. Math., 73 (1991), 117-125.
doi: 10.1007/BF02567633. |
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