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Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains

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  • We study a Venttsel' problem in a three dimensional fractal domain for an operator in non divergence form. We prove existence, uniqueness and regularity results of the strict solution for both the fractal and prefractal problem, via a semigroup approach. In view of numerical approximations, we study the asymptotic behaviour of the solutions of the prefractal problems and we prove that the prefractal solutions converge in the Mosco-Kuwae-Shioya sense to the (limit) solution of the fractal one.
    Mathematics Subject Classification: Primary: 35K, 28A80; Secondary: 31C25, 47D06.


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  • [1]

    D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1966.doi: 10.1007/978-3-662-03282-4.


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


    C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Value Problems, Wiley, New York, 1984.


    F. Brezzi and G. Gilardi, Fundamentals of P.D.E. for Numerical Analysis, In Finite Element Handbook (ed: H. Kardenstuncer and D.H. Norrie), McGraw-Hill Book Co., New York, 1987.


    H. Brezis, Analisi Funzionale, Liguori Ed., Napoli, 1986.


    A. Buffa and P. Ciarlet, On traces for functional spaces related to Maxwell's equations, part 1: An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appli. Sci., 24 (2001), 9-30.doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2.


    J. R. Cannon and G. H. Meyer, On a diffusion in a fractured medium, SIAM J. Appl. Math., 20 (1971), 434-448.doi: 10.1137/0120047.


    R. Capitanelli, Asymptotics for mixed Dirichlet-Robin problems in irregular domains, J. Math. Anal. Appl., 362 (2010), 450-459.doi: 10.1016/j.jmaa.2009.09.042.


    M. Cefalo, G. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473.doi: 10.1016/j.amc.2011.11.033.


    M. Cefalo, M. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation Differential and Integral equations, Differential Integral Equations, 26 (2013), 1027-1054.


    R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, volume 5: Evolution problem 1, Springer-Verlag, Berlin, 1992.doi: 10.1007/978-3-642-58090-1.


    K. Falconer, The Geometry of Fractal Sets, $2^{nd}$ ed. Cambridge Univ. Press, Cambridge, 1986.


    A. Favini, R. Labbas, K. Lemrabet and S. Maingot, Study of limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. mat. complut., 18 (2005), 143-176.


    A. Favini , J. A. Goldstein, G. Ruiz and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.doi: 10.1007/s00028-002-8077-y.


    U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137.doi: 10.4171/ZAA/1190.


    P. Grisvard, Elliptic Problems in non Smooth Domains, Pitman, Boston, 1985.


    W. Hackbush, Elliptic Partial Differential Equations, Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, Springer-Verlag, Berlin, 1992.


    M. Hino, Convergence of non-symmetric forms, J. Math. Kyoto Univ., 38 (1998), 329-341.


    D. Jerison and C. E. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc., 4 (1981), 203-207.doi: 10.1090/S0273-0979-1981-14884-9.


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


    A. Jonsson, Besov spaces on closed subsets of $\mathbbR^n$, Trans. Amer. math. Soc., 341 (1994), 355-370.doi: 10.2307/2154626.


    A. Jonsson and H. Wallin, Function spaces on subset of $\mathbbR^n$, Part 1., Math. Reports, 2 (1984), xiv+221 pp.


    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.


    A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on $\mathbbR^d$, Forum Math., 17 (2005), 225-259.doi: 10.1515/form.2005.17.2.225.


    P. Korman, Existence of periodic solutions for a class of non linear problems, Non linear Anal., 7 (1983), 873-879.doi: 10.1016/0362-546X(83)90063-9.


    K. Kuwae and T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Communications in analysis and geometry, 11 (2003), 599-673.doi: 10.4310/CAG.2003.v11.n4.a1.


    M. R. Lancia, A transmission problem with a fractal interface, Z. Anal. Anwendungen, 21 (2002), 113-133.doi: 10.4171/ZAA/1067.


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


    M. R. Lancia, V. Regis Durante and P. Vernole, Density results for energy spaces on some fractafolds, Z. Anal. Anwendungen, 34 (2015), 357-372.doi: 10.4171/ZAA/1544.


    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.


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


    M. R. Lancia and P. Vernole, Venttsel' problems in fractal domains, Jour. of Evol. Eq., 14 (2014), 681-712.doi: 10.1007/s00028-014-0233-7.


    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.


    M. R. Lancia and M. A. Vivaldi, Asymptotic convergence of transmission energy forms, Adv. Math. Sc. Appl., 13 (2003), 315-341.


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


    A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in non linear differential equations and their applications, 16, Birkhauser-Verlag, Basel, 1995.doi: 10.1007/978-3-0348-9234-6.


    S. Mataloni, On a type of convergence for non-symmetric Dirichlet forms, Adv. Math. Sci. Appl.,9 (1999), 749-773.


    Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992.


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


    J. Necas, Les Methodes Directes en Theorie des Équationes Elliptiques, Masson, Paris, 1967.


    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.doi: 10.1007/978-1-4612-5561-1.


    M. Röckner and T. S. Zhang, Convergence of operators semigroups generated by elliptic operators, Osaka J. Math., 34 (1997), 923-932.


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


    M. Shinbrot, Watern waves over periodic bottoms in three dimensions, J. Inst. Math. Appl., 25 (1980), 367-385.doi: 10.1093/imamat/25.4.367.


    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press., Princeton, 1970.


    M. Tőlle, Convergence of Non-Symmetric Forms with Changing Reference Measures, Thesis, University of Bielefeld August, 2006.


    A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen., 4 (1959), 172-185; (English) English transl. Theor. Probability Appl., 4 (1959), 164-177.doi: 10.1137/1104014.


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

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