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December  2015, 35(12): 5609-5629. doi: 10.3934/dcds.2015.35.5609

Harmonic functions in union of chambers

Laura Abatangelo 1, and  Susanna Terracini 2,

1. 

Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi, 55 - 20125 Milano, Italy
2. 

Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino

Received  March 2014 Published  May 2015

We characterize the set of harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of two different chambers. We analyse the asymptotic behavior of the solutions in connection with the changes in the domain's geometry; we classify all (possibly sign-changing) infinite energy solutions having given asymptotic frequency at the infinite ends of the domain; finally we sketch the case of several different chambers.
Keywords: Harmonic functions, unbounded domains, asymptotic estimates..
Mathematics Subject Classification: Primary: 35B40, 35J25, 35P15, 35B2.
Citation: Laura Abatangelo, Susanna Terracini. Harmonic functions in union of chambers. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5609-5629. doi: 10.3934/dcds.2015.35.5609
References:
[1]

L. Abatangelo, V. Felli and S. Terracini, On the sharp effect of attaching a thin handle on the spectral rate of convergence, Journal of Functional Analysis, 266 (2014), 3632-3684. doi: 10.1016/j.jfa.2013.11.019.  Google Scholar

[2]

L. Abatangelo, V. Felli and S. Terracini, Singularity of eigenfunctions at the junction of shrinking tubes. Part II, Journal of Differential Equations, 256 (2014), 3301-3334. doi: 10.1016/j.jde.2014.02.010.  Google Scholar

[3]

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G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761.  Google Scholar

[5]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769. doi: 10.1051/cocv:2006020.  Google Scholar

[6]

B. E. J. Dahlberg, Estimates for harmonic measure, Arch. Rational Mech. Anal., 65 (1977), 275-288. doi: 10.1007/BF00280445.  Google Scholar

[7]

V. Felli and S. Terracini, Singularity of eigenfunctions at the junction of shrinking tubes. Part I, J. Differential Equations, 255 (2013), 633-700. doi: 10.1016/j.jde.2013.04.017.  Google Scholar

[8]

T. Gilbarg, Elliptic Partial Differential Equations, Springer, 2001. Google Scholar

[9]

V. Isakov, Inverse problems for Partial Differential Equations, Springer, 2006.  Google Scholar

[10]

D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46 (1982), 80-147. doi: 10.1016/0001-8708(82)90055-X.  Google Scholar

[11]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1995.  Google Scholar

[12]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 1996. doi: 10.1007/978-1-4612-5338-9.  Google Scholar

[13]

M. Murata, On construction of Martin boundaries for second order elliptic equations, Publ. Res. Inst. Math. Sci., 26 (1990), 585-627. doi: 10.2977/prims/1195170848.  Google Scholar

[14]

Y. Pinchover, On positive solutions of second-order elliptic equations, stability results, and classification, Duke Math. J., 57 (1988), 955-980. doi: 10.1215/S0012-7094-88-05743-2.  Google Scholar

[15]

Y. Pinchover, On positive solutions of elliptic equations with periodic coefficients in unbounded domains, in Maximum Principles and Eigenvalue Problems in Partial Differential Equations (Knoxville, TN, 1987), Pitman Res. Notes Math. Ser., 175, Longman Sci. Tech., Harlow, 1988, 218-230.  Google Scholar

[16]

Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst.H. Poincaré Anal. Non Linéaire, 11 (1994), 313-341.  Google Scholar

[17]

R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, 45, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511526244.  Google Scholar

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M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[19]

W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.  Google Scholar

show all references

References:
[1]

L. Abatangelo, V. Felli and S. Terracini, On the sharp effect of attaching a thin handle on the spectral rate of convergence, Journal of Functional Analysis, 266 (2014), 3632-3684. doi: 10.1016/j.jfa.2013.11.019.  Google Scholar

[2]

L. Abatangelo, V. Felli and S. Terracini, Singularity of eigenfunctions at the junction of shrinking tubes. Part II, Journal of Differential Equations, 256 (2014), 3301-3334. doi: 10.1016/j.jde.2014.02.010.  Google Scholar

[3]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press 2003.  Google Scholar

[4]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761.  Google Scholar

[5]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optim. Calc. Var., 12 (2006), 752-769. doi: 10.1051/cocv:2006020.  Google Scholar

[6]

B. E. J. Dahlberg, Estimates for harmonic measure, Arch. Rational Mech. Anal., 65 (1977), 275-288. doi: 10.1007/BF00280445.  Google Scholar

[7]

V. Felli and S. Terracini, Singularity of eigenfunctions at the junction of shrinking tubes. Part I, J. Differential Equations, 255 (2013), 633-700. doi: 10.1016/j.jde.2013.04.017.  Google Scholar

[8]

T. Gilbarg, Elliptic Partial Differential Equations, Springer, 2001. Google Scholar

[9]

V. Isakov, Inverse problems for Partial Differential Equations, Springer, 2006.  Google Scholar

[10]

D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46 (1982), 80-147. doi: 10.1016/0001-8708(82)90055-X.  Google Scholar

[11]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1995.  Google Scholar

[12]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 1996. doi: 10.1007/978-1-4612-5338-9.  Google Scholar

[13]

M. Murata, On construction of Martin boundaries for second order elliptic equations, Publ. Res. Inst. Math. Sci., 26 (1990), 585-627. doi: 10.2977/prims/1195170848.  Google Scholar

[14]

Y. Pinchover, On positive solutions of second-order elliptic equations, stability results, and classification, Duke Math. J., 57 (1988), 955-980. doi: 10.1215/S0012-7094-88-05743-2.  Google Scholar

[15]

Y. Pinchover, On positive solutions of elliptic equations with periodic coefficients in unbounded domains, in Maximum Principles and Eigenvalue Problems in Partial Differential Equations (Knoxville, TN, 1987), Pitman Res. Notes Math. Ser., 175, Longman Sci. Tech., Harlow, 1988, 218-230.  Google Scholar

[16]

Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst.H. Poincaré Anal. Non Linéaire, 11 (1994), 313-341.  Google Scholar

[17]

R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, 45, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511526244.  Google Scholar

[18]

M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[19]

W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.  Google Scholar

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