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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 and 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. 
[2]

D. E. Apushkinskaya and A. I. Nazarov, The Venttsel' problem for nonlinear elliptic equations, J. Math. Sci. (New York), 101 (2000), 2861-2880. 

[3]

H. Attouch, Familles d'oprateurs maximaux monotones et mesurabilité, Ann. Mat. Pura e Applicata, 120 (1979), 35-111. 

[4] C. Baiocchi and C. Baiocchi, 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. 

[7]

H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal., 9 (1972), 63-74. 

[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.

[9]

R. Capitanelli, Lagrangians on Homogeneous Spaces Ph. D thesis, Universitá degli Studi di Roma "La Sapienza", 2002.

[10]

R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80. 

[11]

M. CefaloM. 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. 

[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.

[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. 

[14] K. Falconer, The Geometry of Fractal Sets, 2nd edition, Cambridge University Press, 1990. 
[15]

U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendingen., 23 (2004), 115-135. 

[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. 

[17]

P. Grisvard, Théorémes de traces relatifs á un polyédre, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 581-1583. 

[18]

D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207. 

[19]

P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. 

[20]

A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$ Math. Reports, vol. 2, Harwood Acad. Publ., London, 1984.

[21]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math., 112 (1995), 285-300. 

[22]

A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbb{R}^d$, Forum Math., 17 (2005), 225-259. 

[23]

S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153 (1993), 339-357. 

[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.

[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. 

[26]

M. R. LanciaV. 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. 

[27]

M. R. LanciaA. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions, Nonlinear Anal. Real World Appl., 35 (2017), 265-291. 

[28]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567. 

[29]

M. R. Lancia and P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal., 75 (2012), 4222-4240. 

[30]

M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, J. Evol. Equ., 14 (2014), 681-712. 

[31]

V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. 

[32]

U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585. 

[33]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421. 

[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. 

[35]

J. Necas, Les Méthodes Directes en Théorie des Èquationes Elliptiques Masson, Paris, 1967.

[36]

J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Ph. D thesis, Universit ät Bielefeld, 2010.

[37]

H. Triebel, Fractals and Spectra Related to Fourier Analysis and Function Spaces Monographs in Mathematics, vol. 91, Birkhäuser, Basel, 1997.

[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. 

[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. 

[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.

[41]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. 

[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. 

show all references

References:
[1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1966. 
[2]

D. E. Apushkinskaya and A. I. Nazarov, The Venttsel' problem for nonlinear elliptic equations, J. Math. Sci. (New York), 101 (2000), 2861-2880. 

[3]

H. Attouch, Familles d'oprateurs maximaux monotones et mesurabilité, Ann. Mat. Pura e Applicata, 120 (1979), 35-111. 

[4] C. Baiocchi and C. Baiocchi, 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. 

[7]

H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal., 9 (1972), 63-74. 

[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.

[9]

R. Capitanelli, Lagrangians on Homogeneous Spaces Ph. D thesis, Universitá degli Studi di Roma "La Sapienza", 2002.

[10]

R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80. 

[11]

M. CefaloM. 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. 

[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.

[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. 

[14] K. Falconer, The Geometry of Fractal Sets, 2nd edition, Cambridge University Press, 1990. 
[15]

U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendingen., 23 (2004), 115-135. 

[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. 

[17]

P. Grisvard, Théorémes de traces relatifs á un polyédre, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 581-1583. 

[18]

D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207. 

[19]

P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. 

[20]

A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$ Math. Reports, vol. 2, Harwood Acad. Publ., London, 1984.

[21]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Math., 112 (1995), 285-300. 

[22]

A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbb{R}^d$, Forum Math., 17 (2005), 225-259. 

[23]

S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153 (1993), 339-357. 

[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.

[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. 

[26]

M. R. LanciaV. 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. 

[27]

M. R. LanciaA. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions, Nonlinear Anal. Real World Appl., 35 (2017), 265-291. 

[28]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567. 

[29]

M. R. Lancia and P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal., 75 (2012), 4222-4240. 

[30]

M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, J. Evol. Equ., 14 (2014), 681-712. 

[31]

V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. 

[32]

U. Mosco, Convergence of convex sets and solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585. 

[33]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421. 

[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. 

[35]

J. Necas, Les Méthodes Directes en Théorie des Èquationes Elliptiques Masson, Paris, 1967.

[36]

J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces Ph. D thesis, Universit ät Bielefeld, 2010.

[37]

H. Triebel, Fractals and Spectra Related to Fourier Analysis and Function Spaces Monographs in Mathematics, vol. 91, Birkhäuser, Basel, 1997.

[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. 

[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. 

[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.

[41]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. 

[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. 

Figure 1.  The Koch snowflake
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