October  2016, 9(5): 1493-1520. doi: 10.3934/dcdss.2016060

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

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy

2. 

Università degli studi Roma 3, Dipartimento di Matematica, Largo San Leonardo Murialdo 1, 00146 Roma, Italy

3. 

Dipartimento di Matematica, Universita degli Studi di Roma "La Sapienza", P.zale Aldo Moro 2, 00185 Roma, Italy

Received  December 2014 Revised  July 2015 Published  October 2016

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.
Citation: Maria Rosaria Lancia, Valerio Regis Durante, Paola Vernole. Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1493-1520. doi: 10.3934/dcdss.2016060
References:
[1]

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

[2]

H. Attouch, Variational Convergence for Function and Operators,, Eds. Pitman Advanced Publishing Program, (1984).   Google Scholar

[3]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Value Problems,, Wiley, (1984).   Google Scholar

[4]

F. Brezzi and G. Gilardi, Fundamentals of P.D.E. for Numerical Analysis,, In Finite Element Handbook (ed: H. Kardenstuncer and D.H. Norrie), (1987).   Google Scholar

[5]

H. Brezis, Analisi Funzionale,, Liguori Ed., (1986).   Google Scholar

[6]

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.  doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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.   Google Scholar

[11]

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

[12]

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

[13]

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.   Google Scholar

[14]

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

[15]

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

[16]

P. Grisvard, Elliptic Problems in non Smooth Domains,, Pitman, (1985).   Google Scholar

[17]

W. Hackbush, Elliptic Partial Differential Equations, Theory and Numerical Treatment,, Springer Series in Computational Mathematics 18, (1992).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966).   Google Scholar

[24]

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

[25]

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

[26]

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.  doi: 10.4310/CAG.2003.v11.n4.a1.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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.   Google Scholar

[31]

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

[32]

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

[33]

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.   Google Scholar

[34]

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

[35]

J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications,,, Berlin, (1972).   Google Scholar

[36]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Progress in non linear differential equations and their applications, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

J. Necas, Les Methodes Directes en Theorie des Équationes Elliptiques,, Masson, (1967).   Google Scholar

[41]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Appl. Math. Sci. 44, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[42]

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

[43]

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

[44]

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

[45]

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

[46]

M. Tőlle, Convergence of Non-Symmetric Forms with Changing Reference Measures,, Thesis, (2006).   Google Scholar

[47]

A. D. Venttsel, On boundary conditions for multidimensional diffusion processes,, Teor. Veroyatnost. i Primenen., 4 (1959), 172.  doi: 10.1137/1104014.  Google Scholar

[48]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake,, Mskr. Math., 73 (1991), 117.  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, (1966).  doi: 10.1007/978-3-662-03282-4.  Google Scholar

[2]

H. Attouch, Variational Convergence for Function and Operators,, Eds. Pitman Advanced Publishing Program, (1984).   Google Scholar

[3]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Value Problems,, Wiley, (1984).   Google Scholar

[4]

F. Brezzi and G. Gilardi, Fundamentals of P.D.E. for Numerical Analysis,, In Finite Element Handbook (ed: H. Kardenstuncer and D.H. Norrie), (1987).   Google Scholar

[5]

H. Brezis, Analisi Funzionale,, Liguori Ed., (1986).   Google Scholar

[6]

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.  doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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.   Google Scholar

[11]

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

[12]

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

[13]

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.   Google Scholar

[14]

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

[15]

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

[16]

P. Grisvard, Elliptic Problems in non Smooth Domains,, Pitman, (1985).   Google Scholar

[17]

W. Hackbush, Elliptic Partial Differential Equations, Theory and Numerical Treatment,, Springer Series in Computational Mathematics 18, (1992).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966).   Google Scholar

[24]

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

[25]

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

[26]

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.  doi: 10.4310/CAG.2003.v11.n4.a1.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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.   Google Scholar

[31]

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

[32]

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

[33]

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.   Google Scholar

[34]

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

[35]

J. Lions and E. Magenes, Non-Homogeneous Boundary Valued Problems and Applications,,, Berlin, (1972).   Google Scholar

[36]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Progress in non linear differential equations and their applications, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

J. Necas, Les Methodes Directes en Theorie des Équationes Elliptiques,, Masson, (1967).   Google Scholar

[41]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Appl. Math. Sci. 44, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[42]

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

[43]

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

[44]

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

[45]

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

[46]

M. Tőlle, Convergence of Non-Symmetric Forms with Changing Reference Measures,, Thesis, (2006).   Google Scholar

[47]

A. D. Venttsel, On boundary conditions for multidimensional diffusion processes,, Teor. Veroyatnost. i Primenen., 4 (1959), 172.  doi: 10.1137/1104014.  Google Scholar

[48]

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

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