On asymptotics of a solution to the Dirichlet problem for an elliptic equation in the half-space (Q2713896)

Under consideration is the Dirichlet problem for the elliptic equation NEWLINE\[NEWLINE p(x)u_{yy} + \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \Bigl( a_{i,j}(x,y) \frac{\partial}{\partial x_j} u\Bigr)=0 \;\;((x,y)\in {\mathbb R}_{n+1}^+={\mathbb R}^n\times (0,\infty)). \tag{1} NEWLINE\]NEWLINE The coefficients of this equation are assumed to be measurable and bounded. Moreover, the ellipticity implies the inequalities NEWLINE\[NEWLINE \lambda^{-1}\leq p(x)\leq \lambda,\;\lambda^{-1}|\xi|^2\leq \sum_{i,j=1}^{n}a_{i,j}\xi_i \xi_j\leq \lambda |\xi|^2 \tag{2} NEWLINE\]NEWLINE valid for all \((x,y)\in \{y>0\}\) and some constant \(\lambda>0\). By a solution to equation (1) the author means a generalized solution \(u(x,y)\in W_{2,\text{loc}}^1({\mathbb R}_{n+1}^+) \cap C(\overline{{\mathbb R}_{n+1}^+})\) satisfying the Dirichlet boundary condition \(u(x,0)=\varphi(x)\). The article contains necessary and sufficient conditions ensuring the existence of the pointwise (and uniform in \(x\)) limit NEWLINE\[NEWLINE \lim_{y\to\infty}u(x,y)=A=\text{const}.\tag{2} NEWLINE\]NEWLINE In particular, if \(p(Rx) \to \overline{p}=\text{const}\) weakly in \(L_{2,loc}({\mathbb R}^n)\) as \(R\to \infty\) then the pointwise limit (2) exists if and only if NEWLINE\[NEWLINE \lim_{R\to \infty} \int_{|x|<R} p(x)\varphi(x) dx/ \int_{|x|<R} p(x) dx =A.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].