Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains

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October 2016, 9(5): 1493-1520. doi: 10.3934/dcdss.2016060

Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains

Maria Rosaria Lancia ^1, , Valerio Regis Durante ^2, and Paola Vernole ^3,

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

Received December 2014 Revised July 2015 Published October 2016

Abstract
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We study a Venttsel' problem in a three dimensional fractal domain for an operator in non divergence form. We prove existence, uniqueness and regularity results of the strict solution for both the fractal and prefractal problem, via a semigroup approach. In view of numerical approximations, we study the asymptotic behaviour of the solutions of the prefractal problems and we prove that the prefractal solutions converge in the Mosco-Kuwae-Shioya sense to the (limit) solution of the fractal one.

Keywords: non divergence form operator, energy forms, semigroups, fractal domains, Asymptotic behaviour, Venttsel' problems, varying Hilbert spaces..

Mathematics Subject Classification: Primary: 35K, 28A80; Secondary: 31C25, 47D0.

Citation: Maria Rosaria Lancia, Valerio Regis Durante, Paola Vernole. Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1493-1520. doi: 10.3934/dcdss.2016060

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]	H. Brezis, Analisi Funzionale, Liguori Ed., Napoli, 1986.
[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]	H. Brezis, Analisi Funzionale, Liguori Ed., Napoli, 1986.
[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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