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

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

Simone Creo , and  Valerio Regis Durante

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.

Keywords: Fractal surfaces, density results, asymptotic behavior, Venttsel' problems, nonlinear energy forms, trace theorems, varying Hilbert spaces, $p$-Laplacian, nonlinear semigroups.
Mathematics Subject Classification: Primary: 35K, 28A80; Secondary: 35K90, 46E35, 31C25.
Citation: 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
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.  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.  doi: 10.1007/BF02672175.  Google Scholar

[3]

H. Attouch, Variational Convergence for Functions and Operators, Eds. Pitman Advanced Publishing Program, London, 1984.  Google Scholar

[4]

C. Baiocchi and A. Capelo, 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.  doi: 10.1007/BF02771467.  Google Scholar

[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.  Google Scholar

[8]

R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235.   Google Scholar

[9]

R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88.  doi: 10.1007/BF02392869.  Google Scholar

[21]

A. Jonsson, Besov spaces on closed subsets of $ {\mathbb{R}^n} $, Trans. Amer. Math. Soc., 341 (1994), 355-370.  doi: 10.2307/2154626.  Google Scholar

[22]

A. Jonsson and H. Wallin, Function spaces on subsets of $\mathbb{R}^n$, Math. Rep., 2 (1984), xiv+221 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. R. Lancia, Second order transmission problems across a fractal surface, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 191-213.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[30]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567.  doi: 10.1137/090761173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[37]

J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications, Vol. 1, Berlin, Springer-Verlag, 1972.  Google Scholar

[38]

V. Maz' ya and S. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997.  Google Scholar

[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.  Google Scholar

[40]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[41]

J. Necas, Les Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967.  Google Scholar

[42]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, London, 1992.  Google Scholar

[43]

B. Sapoval, General formulation of Laplacian transfer across irregular surfaces, Phys. Rev. Lett., 73 (1994), 3314-3316.  doi: 10.1103/PhysRevLett.73.3314.  Google Scholar

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[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.  Google Scholar

[48]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125.  doi: 10.1007/BF02567633.  Google Scholar

show all references

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.  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.  doi: 10.1007/BF02672175.  Google Scholar

[3]

H. Attouch, Variational Convergence for Functions and Operators, Eds. Pitman Advanced Publishing Program, London, 1984.  Google Scholar

[4]

C. Baiocchi and A. Capelo, 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.  doi: 10.1007/BF02771467.  Google Scholar

[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.  Google Scholar

[8]

R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235.   Google Scholar

[9]

R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1986.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88.  doi: 10.1007/BF02392869.  Google Scholar

[21]

A. Jonsson, Besov spaces on closed subsets of $ {\mathbb{R}^n} $, Trans. Amer. Math. Soc., 341 (1994), 355-370.  doi: 10.2307/2154626.  Google Scholar

[22]

A. Jonsson and H. Wallin, Function spaces on subsets of $\mathbb{R}^n$, Math. Rep., 2 (1984), xiv+221 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. R. Lancia, Second order transmission problems across a fractal surface, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 191-213.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[30]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. on Mathematical Analysis, 42 (2010), 1539-1567.  doi: 10.1137/090761173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[37]

J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications, Vol. 1, Berlin, Springer-Verlag, 1972.  Google Scholar

[38]

V. Maz' ya and S. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997.  Google Scholar

[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.  Google Scholar

[40]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[41]

J. Necas, Les Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967.  Google Scholar

[42]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, London, 1992.  Google Scholar

[43]

B. Sapoval, General formulation of Laplacian transfer across irregular surfaces, Phys. Rev. Lett., 73 (1994), 3314-3316.  doi: 10.1103/PhysRevLett.73.3314.  Google Scholar

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[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.  Google Scholar

[48]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125.  doi: 10.1007/BF02567633.  Google Scholar

Figure 1.  The pre-fractal curve $F_h$ for $h = 3$.
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Figure 2.  The fractal domain $Q$.
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