Counting dimensions of L-harmonic functions with exponential growth
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Counting dimensions of L-harmonic functions with exponential growth Xian-Tao Huang1 Received: 12 January 2020 / Accepted: 5 March 2020 © Springer Nature B.V. 2020
Abstract Let ⊂ Rn−1 be a bounded open set, X = × R ⊆ Rn be the infinite strip. Let L be a 1,2 second order uniformly elliptic operator of divergence form acting on a function f ∈ Wloc (X ) n ∂f ∂ i j given by L f = i, j=1 ∂ xi a (x) ∂ x j . It is natural to consider the solutions of Lu = 0 with ˜ d|xn | for boundary value u|∂×R = 0 and exponential growth at most d: |u(x , xn )| ≤ Ce some C˜ > 0. Denote by Ad the solution space. In (Acta Math Sin (Engl Ser)15:525–534, 1999), Hang and Lin proved that dimAd ≤ Cd n−1 . The power n − 1 is sharp, but one may wonder whether there are more precise estimates for the constant C. In this note, we consider some natural subspaces of Ad and obtain some estimates of dimensions of these subspaces. Compared with the case L = X , when d is sufficiently large, the estimates obtained in this note are sharp both on the power n − 1 and the constant C. Keywords Second order elliptic equation · Exponential growth function 2010 Mathematics Subject Classification 35J15 · 58J05
1 Introduction Suppose ⊂ Rn−1 is a bounded open set with C 2 boundary, denote by V0 its volume. In this note, 0 = η0 < η1 ≤ η2 ≤ · · · denote the Dirichlet eigenvalues (counted with multiplicity) of and each ϕi is the eigenfunctions corresponding to ηi such that ϕi ϕ j d x = δi j . Let N (s) = max{i|ηi ≤ s} be the counting function. Denote by X = × R ⊆ Rn the infinite strip. We will use the rectangular coordinate system {x1 , . . . , xn }. Sometimes we use x = (x , xn ) to denote a point with x the -factor and xn the R-factor. We also use d x and d x to denote the volume forms induced by the Euclidean metric on X and respectively. In this note, we always assume (a i j (x))1≤i, j≤n is a (n × n)-symmetric matrix for every x ∈ X , and a i j (x) ∈ Lip(×[r1 , r2 ]) for every r1 < r2 , and there exists constants 0 < λ ≤ satisfying
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Xian-Tao Huang [email protected] School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
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Geometriae Dedicata
λ|ξ |2 ≤
n
a i j (x)ξi ξ j ≤ |ξ |2
(1.1)
i, j=1
for any ξ ∈ Rn and x ∈ X . Denote the operator of divergence form, L, acting on a function 1,2 f ∈ Wloc (X ) defined by n ∂ ∂f ij Lf = a (x) . ∂ xi ∂x j
(1.2)
i, j=1
We will consider the solutions of the following problem: ⎧ ⎨ Lu = 0, ∈ W 1,2 ( × (r1 , r2 )) for every r1 < r2 , u| ⎩ ×(r1 ,r2 ) u|∂×R = 0.
(1.3)
There have been plenty of works on counting the dimensions of solution spaces of some elliptic operators with certain growth rates on Rn or even on manifolds. See e.g. [2,3,7,9,10] for results on harmonic functions with polynomial growth on manifolds, and see e.g. [1,11,12] for results on L-harmonic functions with polynomial growth on Rn . 1,2 In [6], Hang and Lin proved that every nonzero function u ∈ Wloc (X ) solving (1.3) must satisfy lim inf r −1 log u 2 d x ≥ C(n, , λ, V0 ) > 0
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