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In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of $Lu(\mathbf{x})=a_{ij}(\mathbf{x})D_{ij}u(\mathbf{x})+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ or $Lu(\mathbf{x})=D_{i}(a_{ij}(\mathbf{x})$ $D_{j}u(\mathbf{x}))+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term $f$ of the equation.
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참고문헌 (10)
참고문헌 신청
J. Bao / 2016 / Positive solutions to elliptic equations in unbounded cylinder / Discrete Contin. Dyn. Syst. Ser. B 21(5):1389~1400
M. Benedicks / 1980 / Positive harmonic functions vanishing on the boundary of certain domains in Rn / Ark. Mat 18(1):53~72
S. Agmon / 1982 / Lectures on exponential decay of solutions of second-order elliptic equations:bounds on eigenfunctions of N-body Schrodinger operators / Princeton University Press
X. Cabre / 1995 / On the Alexandro-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations / Comm. Pure Appl. Math 48(5):539~570
S. J. Gardiner / 1990 / The Martin boundary of NTA strips / Bull. Lond. Math. Soc 22(2):163~166
M. Benedicks / 1980 / Positive harmonic functions vanishing on the boundary of certain domains in Rn / Ark. Mat 18(1):53~72
S. Agmon / 1982 / Lectures on exponential decay of solutions of second-order elliptic equations:bounds on eigenfunctions of N-body Schrodinger operators / Princeton University Press
X. Cabre / 1995 / On the Alexandro-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations / Comm. Pure Appl. Math 48(5):539~570
S. J. Gardiner / 1990 / The Martin boundary of NTA strips / Bull. Lond. Math. Soc 22(2):163~166
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