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The Koch snowflake
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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 |
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.
Keywords:
Asymptotic behavior,
Venttsel' problems,
nonlinear energy forms,
fractal domains,
trace theorems,
varying Hilbert spaces,
p-Laplacian,
nonlinear semigroups.
Mathematics Subject Classification:
Primary:35K, 28A80;Secondary:37L05, 31C25.
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.
|
| [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. 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.
|
| [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. 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.
|
| [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.
|
| [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. 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.
|
| [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. 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.
|
| [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.
|
| [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.
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