# American Institute of Mathematical Sciences

• Previous Article
Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature
• CPAA Home
• This Issue
• Next Article
A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates
March  2018, 17(2): 647-669. doi: 10.3934/cpaa.2018035

## Approximation of a nonlinear fractal energy functional on varying Hilbert spaces

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Universitá degli studi di Roma Sapienza, Via A. Scarpa 16,00161 Roma, Italy 2 Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Puerto Rico, 00681, USA 3 Dipartimento di Matematica, Universitá degli Studi di Roma Sapienza, Piazzale Aldo Moro 2,00185 Roma, Italy

* Corresponding author: Maria Rosaria Lancia

Received  November 2016 Revised  July 2017 Published  March 2018

We study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a two dimensional domain with Koch-type fractal boundary. 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 that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals adapted by Tölle to the nonlinear framework in varying Hilbert spaces.

Citation: Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. Approximation of a nonlinear fractal energy functional on varying Hilbert spaces. Communications on Pure & Applied Analysis, 2018, 17 (2) : 647-669. doi: 10.3934/cpaa.2018035
##### References:
 [1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1966. Google Scholar [2] D. E. Apushkinskaya and A. I. Nazarov, The Venttsel' problem for nonlinear elliptic equations, J. Math. Sci. (New York), 101 (2000), 2861-2880. Google Scholar [3] H. Attouch, Familles d'oprateurs maximaux monotones et mesurabilité, Ann. Mat. Pura e Applicata, 120 (1979), 35-111. Google Scholar [4] C. Baiocchi and C. Baiocchi, Variational and Quasivariational Inequalities: Applications to Free{Boundary Value Problems, Wiley, New York, 1984. Google Scholar [5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces Translated from the Romanian, Noordhoff International Publishing, Leiden, 1976. Google Scholar [6] H. Brézis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math., 9 (1971), 513-534. Google Scholar [7] H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal., 9 (1972), 63-74. Google Scholar [8] 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. Google Scholar [9] R. Capitanelli, Lagrangians on Homogeneous Spaces Ph. D thesis, Universitá degli Studi di Roma "La Sapienza", 2002.Google Scholar [10] R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80. Google Scholar [11] 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. Google Scholar [12] 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, 16-351. Google Scholar [13] 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. Google Scholar [14] K. Falconer, The Geometry of Fractal Sets, 2nd edition, Cambridge University Press, 1990. Google Scholar [15] U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendingen., 23 (2004), 115-135. Google Scholar [16] C. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. Google Scholar [17] P. Grisvard, Théorémes de traces relatifs á un polyédre, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 581-1583. Google Scholar [18] D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207. Google Scholar [19] P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. Google Scholar [20] A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$ Math. Reports, vol. 2, Harwood Acad. Publ., London, 1984. Google Scholar [21] A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math., 112 (1995), 285-300. Google Scholar [22] A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbb{R}^d$, Forum Math., 17 (2005), 225-259. Google Scholar [23] S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153 (1993), 339-357. Google Scholar [24] S. Kusuoka, Lecture on Diffusion Processes on Nested Fractals In: Statistical Mechanics and Fractals, Lecture Notes in Mathematics, vol 1567, Springer, Berlin, Heidelberg, 1993. Google Scholar [25] 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. Google Scholar [26] 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. Google Scholar [27] 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. Google Scholar [28] M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567. Google Scholar [29] M. R. Lancia and P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal., 75 (2012), 4222-4240. Google Scholar [30] M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, J. Evol. Equ., 14 (2014), 681-712. Google Scholar [31] V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. Google Scholar [32] U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585. Google Scholar [33] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421. Google Scholar [34] U. Mosco, Analysis and numerics of some fractal boundary value problems, Analysis and numerics of partial differential equations, Springer INdAM Ser., 4 (2013), 237-255. Google Scholar [35] J. Necas, Les Méthodes Directes en Théorie des Èquationes Elliptiques Masson, Paris, 1967. Google Scholar [36] J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Ph. D thesis, Universit ät Bielefeld, 2010.Google Scholar [37] H. Triebel, Fractals and Spectra Related to Fourier Analysis and Function Spaces Monographs in Mathematics, vol. 91, Birkhäuser, Basel, 1997. Google Scholar [38] A. Vélez-Santiago, Quasi-linear variable exponent boundary value problems with Wentzell-Robin and Wentzell boundary conditions, J. Functional Analysis, 266 (2014), 560-615. Google Scholar [39] A. Vélez-Santiago, On the well-posedness of first-order variable exponent Cauchy problems with Robin and Wentzell-Robin boundary conditions on arbitrary domains, J. Abstr. Differ. Equ. Appl., 6 (2015), 1-20. Google Scholar [40] 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. Google Scholar [41] H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. Google Scholar [42] M. Warma, Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains, Nonlinear Analysis, 14 (2012), 5561-5588. Google Scholar

show all references

##### References:
 [1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1966. Google Scholar [2] D. E. Apushkinskaya and A. I. Nazarov, The Venttsel' problem for nonlinear elliptic equations, J. Math. Sci. (New York), 101 (2000), 2861-2880. Google Scholar [3] H. Attouch, Familles d'oprateurs maximaux monotones et mesurabilité, Ann. Mat. Pura e Applicata, 120 (1979), 35-111. Google Scholar [4] C. Baiocchi and C. Baiocchi, Variational and Quasivariational Inequalities: Applications to Free{Boundary Value Problems, Wiley, New York, 1984. Google Scholar [5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces Translated from the Romanian, Noordhoff International Publishing, Leiden, 1976. Google Scholar [6] H. Brézis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math., 9 (1971), 513-534. Google Scholar [7] H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal., 9 (1972), 63-74. Google Scholar [8] 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. Google Scholar [9] R. Capitanelli, Lagrangians on Homogeneous Spaces Ph. D thesis, Universitá degli Studi di Roma "La Sapienza", 2002.Google Scholar [10] R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80. Google Scholar [11] 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. Google Scholar [12] 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, 16-351. Google Scholar [13] 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. Google Scholar [14] K. Falconer, The Geometry of Fractal Sets, 2nd edition, Cambridge University Press, 1990. Google Scholar [15] U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendingen., 23 (2004), 115-135. Google Scholar [16] C. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. Google Scholar [17] P. Grisvard, Théorémes de traces relatifs á un polyédre, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 581-1583. Google Scholar [18] D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207. Google Scholar [19] P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. Google Scholar [20] A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$ Math. Reports, vol. 2, Harwood Acad. Publ., London, 1984. Google Scholar [21] A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math., 112 (1995), 285-300. Google Scholar [22] A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbb{R}^d$, Forum Math., 17 (2005), 225-259. Google Scholar [23] S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153 (1993), 339-357. Google Scholar [24] S. Kusuoka, Lecture on Diffusion Processes on Nested Fractals In: Statistical Mechanics and Fractals, Lecture Notes in Mathematics, vol 1567, Springer, Berlin, Heidelberg, 1993. Google Scholar [25] 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. Google Scholar [26] 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. Google Scholar [27] 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. Google Scholar [28] M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567. Google Scholar [29] M. R. Lancia and P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal., 75 (2012), 4222-4240. Google Scholar [30] M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, J. Evol. Equ., 14 (2014), 681-712. Google Scholar [31] V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. Google Scholar [32] U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585. Google Scholar [33] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421. Google Scholar [34] U. Mosco, Analysis and numerics of some fractal boundary value problems, Analysis and numerics of partial differential equations, Springer INdAM Ser., 4 (2013), 237-255. Google Scholar [35] J. Necas, Les Méthodes Directes en Théorie des Èquationes Elliptiques Masson, Paris, 1967. Google Scholar [36] J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Ph. D thesis, Universit ät Bielefeld, 2010.Google Scholar [37] H. Triebel, Fractals and Spectra Related to Fourier Analysis and Function Spaces Monographs in Mathematics, vol. 91, Birkhäuser, Basel, 1997. Google Scholar [38] A. Vélez-Santiago, Quasi-linear variable exponent boundary value problems with Wentzell-Robin and Wentzell boundary conditions, J. Functional Analysis, 266 (2014), 560-615. Google Scholar [39] A. Vélez-Santiago, On the well-posedness of first-order variable exponent Cauchy problems with Robin and Wentzell-Robin boundary conditions on arbitrary domains, J. Abstr. Differ. Equ. Appl., 6 (2015), 1-20. Google Scholar [40] 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. Google Scholar [41] H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. Google Scholar [42] M. Warma, Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains, Nonlinear Analysis, 14 (2012), 5561-5588. Google Scholar
The Koch snowflake
 [1] Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475 [2] Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 [3] Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055 [4] Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 [5] Simone Creo, Valerio Regis Durante. Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 65-90. doi: 10.3934/dcdss.2019005 [6] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [7] Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 [8] Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure & Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359 [9] Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 [10] Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743 [11] Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219 [12] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [13] Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103 [14] Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 [15] C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545 [16] Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173 [17] Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729 [18] E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209 [19] Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143 [20] Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184

2018 Impact Factor: 0.925