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Communications on Pure and Applied Analysis (CPAA)
 

Approximation of a nonlinear fractal energy functional on varying Hilbert spaces
Pages: 647 - 669, Issue 2, March 2018

doi:doi:10.3934/cpaa.2018035      Abstract        References        Full text (580.8K)           Related Articles

Simone Creo - Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università degli studi di Roma Sapienza, Via A. Scarpa 16, 00161 Roma, Italy (email)
Maria Rosaria Lancia - Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy (email)
Alejandro Vélez-Santiago - Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Puerto Rico, 00681, United States (email)
Paola Vernole - Dipartimento di Matematica, Università degli Studi di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy (email)

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 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.       
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, Università 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, Théorèmes de traces relatifs à un polyèdre, 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 $\mathbbR^n$, 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 $\mathbbR^d$, 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 Mèthodes Directes en Thèorie des Èquationes Elliptiques, Masson, Paris, 1967.       
36 J. M. Tölle, Variational Convergence of Nonlinear Partial Differential Operators on Varying Banach Spaces, Ph.D thesis, Universität 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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