February  2019, 12(1): 1-26. doi: 10.3934/dcdss.2019001

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets

Laboratoire de Mathématiques et Informatique pour la Complexité et les Systèmes, CentralSupélec, Université Paris Saclay, Bȃtiment Bouygues, 3 rue Joliot-Curie, 91190 Gif-sur-Yvette, France

* Corresponding author: Anna Rozanova-Pierrat

Received  February 2017 Revised  June 2017 Published  July 2018

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for $n$ and $d$-sets.

Citation: Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001
References:
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R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

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G. Allaire, Analyse Numérique et Optimisation, École Polytechnique, 2012.

[3]

W. Arendt and A. F. M. T. Elst, Sectorial forms and degenerate differential operators, J. Operator Theory, 67 (2012), 33-72.

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W. Arendt and R. Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains, Ulmer Seminare, 12 (2007), 28-38.

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W. Arendt and R. Mazzeo, Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212. doi: 10.3934/cpaa.2012.11.2201.

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W. Arendt and A. ter Elst, The Dirichlet-to-Neumann operator on rough domains, J. Differential Equations, 251 (2011), 2100-2124. doi: 10.1016/j.jde.2011.06.017.

[7]

W. Arendt and A. F. M. ter Elst, The Dirichlet-to-Neumann operator on exterior domains, Potential Anal., 43 (2015), 313-340. doi: 10.1007/s11118-015-9473-6.

[8]

L. Banjai, Eigenfrequencies of fractal drums, J. of Comp. and Appl. Math., 198 (2007), 1-18. doi: 10.1016/j.cam.2005.11.015.

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J. Behrndt and A. ter Elst, Dirichlet-to-Neumann maps on bounded Lipschitz domains, J. Differential Equations, 259 (2015), 5903-5926. doi: 10.1016/j.jde.2015.07.012.

[10]

M. Bodin, Characterisations of Function Spaces on Fractals, Ph. D thesis, Ume$ \mathbb{R} aa$ University, 2005.

[11]

C. BardosD. Grebenkov and A. Rozanova-Pierrat, Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions, Math. Models Methods Appl. Sci., 26 (2016), 59-110. doi: 10.1142/S0218202516500032.

[12]

L. P. Bos and P. D. Milman, Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geom. Funct. Anal., 5 (1995), 853-923. doi: 10.1007/BF01902214.

[13]

A.-P. Calderon, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math., 4 (1961), 33-49.

[14]

R. Capitanelli, Mixed Dirichlet-Robin problems in irregular domains, Comm. to SIMAI Congress, 2 (2007).

[15]

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.

[16]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.

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M. Filoche and D. S. Grebenkov, The toposcopy, a new tool to probe the geometry of an irregular interface by measuring its transfer impedance, Europhys. Lett., 81 (2008), 40008. doi: 10.1209/0295-5075/81/40008.

[18]

A. GirouardR. S. Laugesen and B. A. Siudeja, Steklov eigenvalues and quasiconformal maps of simply connected planar domains, Arch. Ration. Mech. Anal., 219 (2016), 903-936. doi: 10.1007/s00205-015-0912-8.

[19]

A. GirouardL. ParnovskiI. Polterovich and D. A. Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc., 157 (2014), 379-389. doi: 10.1017/S030500411400036X.

[20]

A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, Shape Optimization and Spectral Theory, 120C148, De Gruyter Open, Warsaw, 2017., arXiv: 1411.6567.

[21]

D. S. Grebenkov, Transport Laplacien Aux Interfaces Irregulires: Étude Théorique, Numérique et Expérimentale, Ph. D thesis, Ecole Polytechnique, 2004.

[22]

D. S. Grebenkov, M. Filoche and B. Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces, Phys. Rev. E, 73(2006), 021103, 9pp. doi: 10.1103/PhysRevE.73.021103.

[23]

D. S. GrebenkovM. Filoche and B. Sapoval, A simplified analytical model for Laplacian transfer across deterministic prefractal interfaces, Fractals, 15 (2007), 27-39. doi: 10.1142/S0218348X0700340X.

[24]

P. HajlaszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, Journal of Functional Analysis, 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020.

[25]

D. A. Herron and P. Koskela, Uniform, Sobolev extension and quasiconformal circle domains, J. Anal. Math., 57 (1991), 172-202. doi: 10.1007/BF03041069.

[26]

L. Ihnatsyeva and A. V. Vähäkangas, Characterization of traces of smooth functions on Ahlfors regular sets, J. Funct. Anal., 265 (2013), 1870–1915, arXiv: 1109.2248v1. doi: 10.1016/j.jfa.2013.07.006.

[27]

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

[28]

A. JonssonP. Sjögren and H. Wallin, Hardy and Lipschitz spaces on subsets of $ \mathbb{R}^n$, Studia Math., 80 (1984), 141-166. doi: 10.4064/sm-80-2-141-166.

[29]

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

[30]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Mathematica, 112 (1995), 285-300. doi: 10.4064/sm-112-3-285-300.

[31]

A. Jonsson and H. Wallin, Boundary value problems and brownian motion on fractals, Chaos, Solitons & Fractals, 8 (1997), 191-205. doi: 10.1016/S0960-0779(96)00048-3.

[32]

M. R. Lancia, A transmission problem with a fractal interface, Zeitschrift für Analysis und ihre Anwendungen, 21 (2002), 113-133. doi: 10.4171/ZAA/1067.

[33]

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Berlin: Springer-Verlag, 1972.

[34]

G. Lu and B. Ou, A Poincaré inequality on $ \mathbb{R}^n$ and its application to potential fluid flows in space, Comm. Appl. Nonlinear Anal, 12 (2005), 1-24.

[35]

J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math, 58 (1987), 47-65. doi: 10.1007/BF01169082.

[36]

M. Martin and M. Putinar, Lectures on Hyponormal Operators, Vol. 39, Birkhauser, Basel, 1989. doi: 10.1007/978-3-0348-7466-3.

[37]

O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math., 4 (1979), 383-401. doi: 10.5186/aasfm.1978-79.0413.

[38]

V. N. Maslennikova, Partial Differential Equations, (in Russian) Moscow, Peoples Freindship University of Russia, 1997.

[39]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.

[40]

J. P. Pinasco and J. D. Rossi, Asymptotics of the spectral function for the Steklov problem in a family of sets with fractal boundaries, Appl. Maths. E-Notes, 5 (2005), 138-146.

[41]

P. Shvartsman, On the boundary values of Sobolev $ W^1_p$-functions, Adv. in Maths., 225 (2010), 2162-2221. doi: 10.1016/j.aim.2010.03.031.

[42]

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

[43]

M. Taylor, Partial Differential Equations II, Appl. Math. Sci., Vol. 116, Springer-Verlag, New-York, 1996. doi: 10.1007/978-1-4684-9320-7.

[44]

H. Triebel, Fractals and Spectra. Related to Fourier Analysis and Function Spaces, Birkhäuser, 1997. doi: 10.1007/978-3-0348-0034-1.

[45]

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

[46]

P. Wingren, Lipschitz spaces and interpolating polynomials on subsets of euclidean space, Function Spaces and Applications, Springer Science + Business Media, 1302 (1988), 424–435. doi: 10.1007/BFb0078893.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

G. Allaire, Analyse Numérique et Optimisation, École Polytechnique, 2012.

[3]

W. Arendt and A. F. M. T. Elst, Sectorial forms and degenerate differential operators, J. Operator Theory, 67 (2012), 33-72.

[4]

W. Arendt and R. Mazzeo, Spectral properties of the Dirichlet-to-Neumann operator on Lipschitz domains, Ulmer Seminare, 12 (2007), 28-38.

[5]

W. Arendt and R. Mazzeo, Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212. doi: 10.3934/cpaa.2012.11.2201.

[6]

W. Arendt and A. ter Elst, The Dirichlet-to-Neumann operator on rough domains, J. Differential Equations, 251 (2011), 2100-2124. doi: 10.1016/j.jde.2011.06.017.

[7]

W. Arendt and A. F. M. ter Elst, The Dirichlet-to-Neumann operator on exterior domains, Potential Anal., 43 (2015), 313-340. doi: 10.1007/s11118-015-9473-6.

[8]

L. Banjai, Eigenfrequencies of fractal drums, J. of Comp. and Appl. Math., 198 (2007), 1-18. doi: 10.1016/j.cam.2005.11.015.

[9]

J. Behrndt and A. ter Elst, Dirichlet-to-Neumann maps on bounded Lipschitz domains, J. Differential Equations, 259 (2015), 5903-5926. doi: 10.1016/j.jde.2015.07.012.

[10]

M. Bodin, Characterisations of Function Spaces on Fractals, Ph. D thesis, Ume$ \mathbb{R} aa$ University, 2005.

[11]

C. BardosD. Grebenkov and A. Rozanova-Pierrat, Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions, Math. Models Methods Appl. Sci., 26 (2016), 59-110. doi: 10.1142/S0218202516500032.

[12]

L. P. Bos and P. D. Milman, Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geom. Funct. Anal., 5 (1995), 853-923. doi: 10.1007/BF01902214.

[13]

A.-P. Calderon, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math., 4 (1961), 33-49.

[14]

R. Capitanelli, Mixed Dirichlet-Robin problems in irregular domains, Comm. to SIMAI Congress, 2 (2007).

[15]

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.

[16]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.

[17]

M. Filoche and D. S. Grebenkov, The toposcopy, a new tool to probe the geometry of an irregular interface by measuring its transfer impedance, Europhys. Lett., 81 (2008), 40008. doi: 10.1209/0295-5075/81/40008.

[18]

A. GirouardR. S. Laugesen and B. A. Siudeja, Steklov eigenvalues and quasiconformal maps of simply connected planar domains, Arch. Ration. Mech. Anal., 219 (2016), 903-936. doi: 10.1007/s00205-015-0912-8.

[19]

A. GirouardL. ParnovskiI. Polterovich and D. A. Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc., 157 (2014), 379-389. doi: 10.1017/S030500411400036X.

[20]

A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, Shape Optimization and Spectral Theory, 120C148, De Gruyter Open, Warsaw, 2017., arXiv: 1411.6567.

[21]

D. S. Grebenkov, Transport Laplacien Aux Interfaces Irregulires: Étude Théorique, Numérique et Expérimentale, Ph. D thesis, Ecole Polytechnique, 2004.

[22]

D. S. Grebenkov, M. Filoche and B. Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces, Phys. Rev. E, 73(2006), 021103, 9pp. doi: 10.1103/PhysRevE.73.021103.

[23]

D. S. GrebenkovM. Filoche and B. Sapoval, A simplified analytical model for Laplacian transfer across deterministic prefractal interfaces, Fractals, 15 (2007), 27-39. doi: 10.1142/S0218348X0700340X.

[24]

P. HajlaszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, Journal of Functional Analysis, 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020.

[25]

D. A. Herron and P. Koskela, Uniform, Sobolev extension and quasiconformal circle domains, J. Anal. Math., 57 (1991), 172-202. doi: 10.1007/BF03041069.

[26]

L. Ihnatsyeva and A. V. Vähäkangas, Characterization of traces of smooth functions on Ahlfors regular sets, J. Funct. Anal., 265 (2013), 1870–1915, arXiv: 1109.2248v1. doi: 10.1016/j.jfa.2013.07.006.

[27]

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

[28]

A. JonssonP. Sjögren and H. Wallin, Hardy and Lipschitz spaces on subsets of $ \mathbb{R}^n$, Studia Math., 80 (1984), 141-166. doi: 10.4064/sm-80-2-141-166.

[29]

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

[30]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia Mathematica, 112 (1995), 285-300. doi: 10.4064/sm-112-3-285-300.

[31]

A. Jonsson and H. Wallin, Boundary value problems and brownian motion on fractals, Chaos, Solitons & Fractals, 8 (1997), 191-205. doi: 10.1016/S0960-0779(96)00048-3.

[32]

M. R. Lancia, A transmission problem with a fractal interface, Zeitschrift für Analysis und ihre Anwendungen, 21 (2002), 113-133. doi: 10.4171/ZAA/1067.

[33]

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Berlin: Springer-Verlag, 1972.

[34]

G. Lu and B. Ou, A Poincaré inequality on $ \mathbb{R}^n$ and its application to potential fluid flows in space, Comm. Appl. Nonlinear Anal, 12 (2005), 1-24.

[35]

J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math, 58 (1987), 47-65. doi: 10.1007/BF01169082.

[36]

M. Martin and M. Putinar, Lectures on Hyponormal Operators, Vol. 39, Birkhauser, Basel, 1989. doi: 10.1007/978-3-0348-7466-3.

[37]

O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math., 4 (1979), 383-401. doi: 10.5186/aasfm.1978-79.0413.

[38]

V. N. Maslennikova, Partial Differential Equations, (in Russian) Moscow, Peoples Freindship University of Russia, 1997.

[39]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.

[40]

J. P. Pinasco and J. D. Rossi, Asymptotics of the spectral function for the Steklov problem in a family of sets with fractal boundaries, Appl. Maths. E-Notes, 5 (2005), 138-146.

[41]

P. Shvartsman, On the boundary values of Sobolev $ W^1_p$-functions, Adv. in Maths., 225 (2010), 2162-2221. doi: 10.1016/j.aim.2010.03.031.

[42]

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

[43]

M. Taylor, Partial Differential Equations II, Appl. Math. Sci., Vol. 116, Springer-Verlag, New-York, 1996. doi: 10.1007/978-1-4684-9320-7.

[44]

H. Triebel, Fractals and Spectra. Related to Fourier Analysis and Function Spaces, Birkhäuser, 1997. doi: 10.1007/978-3-0348-0034-1.

[45]

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

[46]

P. Wingren, Lipschitz spaces and interpolating polynomials on subsets of euclidean space, Function Spaces and Applications, Springer Science + Business Media, 1302 (1988), 424–435. doi: 10.1007/BFb0078893.

Figure 1.  Example of the considered domains: $\Omega_0$ (the von Koch snowflake) is the bounded domain, bounded by a compact boundary $\Gamma$, which is a $d$-set (see Definition 2.3) with $d = \log 4/ \log 3>n-1 = 1$. The truncated domain $\Omega_S$ is between the boundary $\Gamma$ and the boundary $S$ (presented by the same von Koch fractal as $\Gamma$). The boundaries $\Gamma$ and $S$ have no an intersection and here are separated by the boundary of a ball $B_r$ of a radius $r>0$. The domain, bounded by $S$, is called $\Omega_1 = \overline{\Omega}_0\cup \Omega_S$, and the exterior domain is $\Omega = \mathbb{R}^n\setminus \overline{\Omega}_0$
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