On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations

TLDR: In this paper, the authors studied the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered).

Abstract: We study the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞, where y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function. Introduction The phenomenon called stabilization is well known for parabolic equations both in linear (see e.g. [1] and references therein) and non-linear (see e.g. [2] and references therein) cases; it means the existence of a finite limit of the solution as t → ∞. However, there are well-posed non-isotropic elliptic boundary-value problems in unbounded domains (see e.g. [3]) for which we can talk about stabilization in the following sense: the solution has a finite limit as a selected spatial variable tends to infinity. This paper is devoted to the Dirichlet problem in half-space for elliptic equations. We present necessary and sufficient conditions for the stabilization of its solution; here the spatial variable, orthogonal to the hyperplane of boundary-value data, plays the role of time. In Section 1, the linear case is presented; Sections 2 and 3 are devoted to quasi-linear equations with the so-called Burgers-Kardar-ParisiZhang non-linearity type (see e.g. [4], [5]). Equations with such non-linearities arise, for example, in modeling of directed polymers and interface growth. They also present an independent theoretical interest because they contain second powers of the first derivatives (see e.g. [6] and references therein). Note that we deal with the stabilization problem in cylindrical domains with an unbounded base (in particular, here the base of the cylinder is the whole E ). As in the parabolic case, this problem is principally different (this refers both to the results and to the methods of research) from the stabilization problem in cylindrical domains with a bounded base. The latter problem has been investigated Received by the editors March 6, 2002. 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B40, 35J60.