July  2016, 21(5): 1389-1400. doi: 10.3934/dcdsb.2016001

Positive solutions to elliptic equations in unbounded cylinder

1. 

School of Mathematics and Information Science, Shanghai Lixin University of Commerce, Shanghai, 201620, China

2. 

Department of Mathematics, and MOE-LSC, Shanghai Jiaotong University, Shanghai 200240, China

Received  April 2014 Revised  August 2015 Published  April 2016

This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.
Citation: Jun Bao, Lihe Wang, Chunqin Zhou. Positive solutions to elliptic equations in unbounded cylinder. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1389-1400. doi: 10.3934/dcdsb.2016001
References:
[1]

M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in $R^n$, Ark. Mat., 18 (1980), 53-72. doi: 10.1007/BF02384681.

[2]

L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981), 621-640. doi: 10.1512/iumj.1981.30.30049.

[3]

M. C. Cranston and T. S. Salisbury, Martin boundaries of sectorial domains, Ark. Mat., 31 (1993), 27-49. doi: 10.1007/BF02559496.

[4]

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

[5]

E. B. Fabes, M. V. Safonov and Y. Yuan, Behavior near the boundary of positive solutions of second order parabolic equations. II, Trans. Amer. Math. Soc., 351 (1999), 4947-4961. doi: 10.1090/S0002-9947-99-02487-3.

[6]

S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc., 22 (1990), 163-166. doi: 10.1112/blms/22.2.163.

[7]

M. Ghergu and J. Pres, Positive harmonic functions that vanish on a subset of a cylindrical surface, Potential Anal., 31 (2009), 147-181. doi: 10.1007/s11118-009-9129-5.

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, second ed., 1983. doi: 10.1007/978-3-642-61798-0.

[9]

R. A. Hunt and R. L. Wheeden, Positive harmonic functions on lipschitz domains, Transactions of the American Mathematical Society, 147 (1970), 507-527. doi: 10.1090/S0002-9947-1970-0274787-0.

[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]

E. M. Landis and N. S. Nadirashvili, Positive solutions of second-order equations in unbounded domains, Mat. Sb. (N.S.), 126 (1985), 133-139, 144.

[12]

A. Lömker, Martin boundaries of quasi-sectorial domains, Potential Anal., 13 (2000), 11-67. doi: 10.1023/A:1008774010423.

[13]

R. S. Martin, Minimal positive harmonic functions, Transactions of the American Mathematical Society, 49 (1941), 137-172. doi: 10.1090/S0002-9947-1941-0003919-6.

[14]

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.

[15]

J. Pres, Positive harmonic functions on comb-like domains, Ann. Acad. Sci. Fenn. Math., 36 (2011), 577-591. doi: 10.5186/aasfm.2011.3630.

[16]

M. G. Shur, The martin boundary for a linear, elliptic, second-order operator, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 27 (1963), 45-60.

[17]

J. C. Taylor, On the martin compactification of a bounded lipschitz domain in a riemannian manifold, Annales de l'institut Fourier, 28 (1978), 25-52. doi: 10.5802/aif.688.

[18]

J. M. G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble), 28 (1978), 147-167. doi: 10.5802/aif.719.

show all references

References:
[1]

M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in $R^n$, Ark. Mat., 18 (1980), 53-72. doi: 10.1007/BF02384681.

[2]

L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981), 621-640. doi: 10.1512/iumj.1981.30.30049.

[3]

M. C. Cranston and T. S. Salisbury, Martin boundaries of sectorial domains, Ark. Mat., 31 (1993), 27-49. doi: 10.1007/BF02559496.

[4]

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

[5]

E. B. Fabes, M. V. Safonov and Y. Yuan, Behavior near the boundary of positive solutions of second order parabolic equations. II, Trans. Amer. Math. Soc., 351 (1999), 4947-4961. doi: 10.1090/S0002-9947-99-02487-3.

[6]

S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc., 22 (1990), 163-166. doi: 10.1112/blms/22.2.163.

[7]

M. Ghergu and J. Pres, Positive harmonic functions that vanish on a subset of a cylindrical surface, Potential Anal., 31 (2009), 147-181. doi: 10.1007/s11118-009-9129-5.

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, second ed., 1983. doi: 10.1007/978-3-642-61798-0.

[9]

R. A. Hunt and R. L. Wheeden, Positive harmonic functions on lipschitz domains, Transactions of the American Mathematical Society, 147 (1970), 507-527. doi: 10.1090/S0002-9947-1970-0274787-0.

[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]

E. M. Landis and N. S. Nadirashvili, Positive solutions of second-order equations in unbounded domains, Mat. Sb. (N.S.), 126 (1985), 133-139, 144.

[12]

A. Lömker, Martin boundaries of quasi-sectorial domains, Potential Anal., 13 (2000), 11-67. doi: 10.1023/A:1008774010423.

[13]

R. S. Martin, Minimal positive harmonic functions, Transactions of the American Mathematical Society, 49 (1941), 137-172. doi: 10.1090/S0002-9947-1941-0003919-6.

[14]

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.

[15]

J. Pres, Positive harmonic functions on comb-like domains, Ann. Acad. Sci. Fenn. Math., 36 (2011), 577-591. doi: 10.5186/aasfm.2011.3630.

[16]

M. G. Shur, The martin boundary for a linear, elliptic, second-order operator, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 27 (1963), 45-60.

[17]

J. C. Taylor, On the martin compactification of a bounded lipschitz domain in a riemannian manifold, Annales de l'institut Fourier, 28 (1978), 25-52. doi: 10.5802/aif.688.

[18]

J. M. G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains, Ann. Inst. Fourier (Grenoble), 28 (1978), 147-167. doi: 10.5802/aif.719.

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