# American Institute of Mathematical Sciences

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

## Harmonic functions in union of chambers

 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.
Citation: Laura Abatangelo, Susanna Terracini. Harmonic functions in union of chambers. Discrete and 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. [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. [3] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press 2003. [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. [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. [6] B. E. J. Dahlberg, Estimates for harmonic measure, Arch. Rational Mech. Anal., 65 (1977), 275-288. doi: 10.1007/BF00280445. [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. [8] T. Gilbarg, Elliptic Partial Differential Equations, Springer, 2001. [9] V. Isakov, Inverse problems for Partial Differential Equations, Springer, 2006. [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. [11] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1995. [12] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 1996. doi: 10.1007/978-1-4612-5338-9. [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. [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. [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. [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. [17] R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, 45, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511526244. [18] M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer, 1984. doi: 10.1007/978-1-4612-5282-5. [19] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

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. [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. [3] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press 2003. [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. [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. [6] B. E. J. Dahlberg, Estimates for harmonic measure, Arch. Rational Mech. Anal., 65 (1977), 275-288. doi: 10.1007/BF00280445. [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. [8] T. Gilbarg, Elliptic Partial Differential Equations, Springer, 2001. [9] V. Isakov, Inverse problems for Partial Differential Equations, Springer, 2006. [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. [11] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1995. [12] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 1996. doi: 10.1007/978-1-4612-5338-9. [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. [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. [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. [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. [17] R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, 45, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511526244. [18] M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer, 1984. doi: 10.1007/978-1-4612-5282-5. [19] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.
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