# American Institute of Mathematical Sciences

February  2019, 12(1): 65-90. doi: 10.3934/dcdss.2019005

## Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains

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

* Corresponding author: Simone Creo

Received  April 2017 Revised  November 2017 Published  July 2018

In this paper we study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a three dimensional fractal cylindrical domain $Q$, whose lateral boundary is a fractal surface $S$. We consider suitable approximating pre-fractal problems in the corresponding pre-fractal varying domains. After proving existence and uniqueness results via standard semigroup approach, we prove density results for the domains of energy functionals defined on $Q$ and $S$. Then we prove that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals.

Citation: Simone Creo, Valerio Regis Durante. Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 65-90. doi: 10.3934/dcdss.2019005
##### References:
 [1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4. [2] D. E. Apushkinskaya and A. I. Nazarov, The Venttsel' problem for nonlinear elliptic equations, J. Math. Sci. (New York), 101 (2000), 2861-2880.  doi: 10.1007/BF02672175. [3] H. Attouch, Variational Convergence for Functions and Operators, Eds. Pitman Advanced Publishing Program, London, 1984. [4] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Value Problems, Wiley, New York, 1984. [5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Translated from the Romanian, Noordhoff International Publishing, Leiden, 1976. [6] H. Brézis, Propriétés régularisantes de certains semi-groupes non linéaires,, Israel J. Math., 9 (1971), 513-534.  doi: 10.1007/BF02771467. [7] F. Brezzi and G. Gilardi, Fundamentals of P. D. E. for Numerical Analysis, in: Finite Element Handbook (ed.: H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987. [8] R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235. [9] R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257. [10] M. Cefalo, G. Dell'Acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, Applied Mathematics and Computation, 218 (2012), 5453-5473.  doi: 10.1016/j.amc.2011.11.033. [11] M. Cefalo and M. R. Lancia, An optimal mesh generation algorithm for domains with Koch type boundaries, Math. Comput. Simulation, 106 (2014), 133-162.  doi: 10.1016/j.matcom.2014.04.009. [12] M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differ. Integral Equ., 26 (2013), 1027-1054. [13] P. Ciarlet, Basic Error Estimates for Elliptic Problems, in: Handbook of Numerical Analysis Ⅱ (ed.: P. Ciarlet and J. J. Lions), North-Holland, Amsterdam, 1991, 17-351. [14] S. Creo, M. R. Lancia, A. Vélez-Santiago and P. Vernole, Approximation of a nonlinear fractal energy functional on varying Hilbert spaces, Commun. Pure Appl. Anal., 17 (2018), 647-669.  doi: 10.3934/cpaa.2018035. [15] J. I. Díaz and L. Tello, On a climate model with a dynamic nonlinear diffusive boundary condition, Discrete Contin. Dyn. Syst., 1 (2009), 253-262.  doi: 10.3934/dcdss.2008.1.253. [16] L. C. Evans, Regularity properties for the heat equation subject to nonlinear boundary constraints, Nonlinear Analysis, 1 (1976/77), 593-602.  doi: 10.1016/0362-546X(77)90020-7. [17] K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986. [18] U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendingen., 23 (2004), 115-137.  doi: 10.4171/ZAA/1190. [19] C. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, Nonlinear Phenomena with Energy Dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 117-139. [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 subsets of $\mathbb{R}^n$, Math. Rep., 2 (1984), xiv+221 pp. [23] A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on ${\mathbb{R}^d}$, Forum Math., 17 (2005), 225-259.  doi: 10.1515/form.2005.17.2.225. [24] K. Kuwae and T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom., 11 (2003), 599-673.  doi: 10.4310/CAG.2003.v11.n4.a1. [25] M. R. Lancia, Second order transmission problems across a fractal surface, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 191-213. [26] M. R. Lancia, V. Regis Durante and P. Vernole, Density results for energy spaces on some fractafolds, Z. Anal. Anwend., 34 (2015), 357-372. [27] M. R. Lancia, V. Regis Durante and P. Vernole, Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1493-1520.  doi: 10.3934/dcdss.2016060. [28] M. R. Lancia, A. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions, Nonlinear Anal. Real World Appl., 35 (2017), 265-291.  doi: 10.1016/j.nonrwa.2016.11.002. [29] 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. [30] M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567.  doi: 10.1137/090761173. [31] M. R. Lancia and P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal., 75 (2012), 4222-4240.  doi: 10.1016/j.na.2012.03.011. [32] M. R. Lancia and P. Vernole, Semilinear evolution problems with Ventcel-type conditions on fractal boundaries, Nonlinear Anal., 80 (2013), 216-232.  doi: 10.1016/j.na.2012.08.020. [33] M. R. Lancia and P. Vernole, Semilinear evolution problems with Ventcel-type conditions on fractal boundaries, International Journal of Partial Differential Equations, 2014 (2014), Article ID 461046, 13 pages. doi: 10.1155/2014/461046. [34] M. R. Lancia and P. Vernole, Semilinear Venttsel' problems in fractal domains, Applied Mathematics, 5 (2014), 1820-1833.  doi: 10.1007/s00028-014-0233-7. [35] M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, J. Evol. Equ., 14 (2014), 681-712.  doi: 10.1007/s00028-014-0233-7. [36] 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. [37] J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications, Vol. 1, Berlin, Springer-Verlag, 1972. [38] V. Maz' ya and S. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. [39] U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.  doi: 10.1016/0001-8708(69)90009-7. [40] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093. [41] J. Necas, Les Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967. [42] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, London, 1992. [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, Water waves over periodic bottoms in three dimensions, J. Inst. Math. Appl., 25 (1980), 367-385.  doi: 10.1093/imamat/25.4.367. [45] J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces, Ph. D thesis, Universität Bielefeld, 2010. [46] H. Triebel, Fractals and Spectra Related to Fourier Analysis and Function Spaces, Monographs in Mathematics, vol. 91, Birkhäuser, Basel, 1997. doi: 10.1007/978-3-0348-0034-1. [47] A. D. Venttsel', On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185; English translation: Theor. Probability Appl., 4 (1959), 164-177. [48] H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125.  doi: 10.1007/BF02567633.

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##### References:
 [1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4. [2] D. E. Apushkinskaya and A. I. Nazarov, The Venttsel' problem for nonlinear elliptic equations, J. Math. Sci. (New York), 101 (2000), 2861-2880.  doi: 10.1007/BF02672175. [3] H. Attouch, Variational Convergence for Functions and Operators, Eds. Pitman Advanced Publishing Program, London, 1984. [4] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Value Problems, Wiley, New York, 1984. [5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Translated from the Romanian, Noordhoff International Publishing, Leiden, 1976. [6] H. Brézis, Propriétés régularisantes de certains semi-groupes non linéaires,, Israel J. Math., 9 (1971), 513-534.  doi: 10.1007/BF02771467. [7] F. Brezzi and G. Gilardi, Fundamentals of P. D. E. for Numerical Analysis, in: Finite Element Handbook (ed.: H. Kardestuncer and D. H. Norrie), McGraw-Hill Book Co., New York, 1987. [8] R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235. [9] R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257. [10] M. Cefalo, G. Dell'Acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, Applied Mathematics and Computation, 218 (2012), 5453-5473.  doi: 10.1016/j.amc.2011.11.033. [11] M. Cefalo and M. R. Lancia, An optimal mesh generation algorithm for domains with Koch type boundaries, Math. Comput. Simulation, 106 (2014), 133-162.  doi: 10.1016/j.matcom.2014.04.009. [12] M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differ. Integral Equ., 26 (2013), 1027-1054. [13] P. Ciarlet, Basic Error Estimates for Elliptic Problems, in: Handbook of Numerical Analysis Ⅱ (ed.: P. Ciarlet and J. J. Lions), North-Holland, Amsterdam, 1991, 17-351. [14] S. Creo, M. R. Lancia, A. Vélez-Santiago and P. Vernole, Approximation of a nonlinear fractal energy functional on varying Hilbert spaces, Commun. Pure Appl. Anal., 17 (2018), 647-669.  doi: 10.3934/cpaa.2018035. [15] J. I. Díaz and L. Tello, On a climate model with a dynamic nonlinear diffusive boundary condition, Discrete Contin. Dyn. Syst., 1 (2009), 253-262.  doi: 10.3934/dcdss.2008.1.253. [16] L. C. Evans, Regularity properties for the heat equation subject to nonlinear boundary constraints, Nonlinear Analysis, 1 (1976/77), 593-602.  doi: 10.1016/0362-546X(77)90020-7. [17] K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986. [18] U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendingen., 23 (2004), 115-137.  doi: 10.4171/ZAA/1190. [19] C. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, Nonlinear Phenomena with Energy Dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 117-139. [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 subsets of $\mathbb{R}^n$, Math. Rep., 2 (1984), xiv+221 pp. [23] A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on ${\mathbb{R}^d}$, Forum Math., 17 (2005), 225-259.  doi: 10.1515/form.2005.17.2.225. [24] K. Kuwae and T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom., 11 (2003), 599-673.  doi: 10.4310/CAG.2003.v11.n4.a1. [25] M. R. Lancia, Second order transmission problems across a fractal surface, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 191-213. [26] M. R. Lancia, V. Regis Durante and P. Vernole, Density results for energy spaces on some fractafolds, Z. Anal. Anwend., 34 (2015), 357-372. [27] M. R. Lancia, V. Regis Durante and P. Vernole, Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1493-1520.  doi: 10.3934/dcdss.2016060. [28] M. R. Lancia, A. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions, Nonlinear Anal. Real World Appl., 35 (2017), 265-291.  doi: 10.1016/j.nonrwa.2016.11.002. [29] 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. [30] M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567.  doi: 10.1137/090761173. [31] M. R. Lancia and P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal., 75 (2012), 4222-4240.  doi: 10.1016/j.na.2012.03.011. [32] M. R. Lancia and P. Vernole, Semilinear evolution problems with Ventcel-type conditions on fractal boundaries, Nonlinear Anal., 80 (2013), 216-232.  doi: 10.1016/j.na.2012.08.020. [33] M. R. Lancia and P. Vernole, Semilinear evolution problems with Ventcel-type conditions on fractal boundaries, International Journal of Partial Differential Equations, 2014 (2014), Article ID 461046, 13 pages. doi: 10.1155/2014/461046. [34] M. R. Lancia and P. Vernole, Semilinear Venttsel' problems in fractal domains, Applied Mathematics, 5 (2014), 1820-1833.  doi: 10.1007/s00028-014-0233-7. [35] M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, J. Evol. Equ., 14 (2014), 681-712.  doi: 10.1007/s00028-014-0233-7. [36] 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. [37] J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications, Vol. 1, Berlin, Springer-Verlag, 1972. [38] V. Maz' ya and S. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. [39] U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.  doi: 10.1016/0001-8708(69)90009-7. [40] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093. [41] J. Necas, Les Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967. [42] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, London, 1992. [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, Water waves over periodic bottoms in three dimensions, J. Inst. Math. Appl., 25 (1980), 367-385.  doi: 10.1093/imamat/25.4.367. [45] J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces, Ph. D thesis, Universität Bielefeld, 2010. [46] H. Triebel, Fractals and Spectra Related to Fourier Analysis and Function Spaces, Monographs in Mathematics, vol. 91, Birkhäuser, Basel, 1997. doi: 10.1007/978-3-0348-0034-1. [47] A. D. Venttsel', On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185; English translation: Theor. Probability Appl., 4 (1959), 164-177. [48] H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125.  doi: 10.1007/BF02567633.
The pre-fractal curve $F_h$ for $h = 3$.
The fractal domain $Q$.
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