The backward problems for the elliptic equation (BPEE) are widely used for modelling problems in many fields such as physics, geometry or engineering. This paper is the first investigation of BPEE in unbounded multiple-dimensional domain associated with locally Lipschitz source as follows $\frac{{\partial }^{2}u}{\partial x_1^2}+\frac{{\partial }^{2}u}{\partial x_2^2}+...\frac{{\partial }^{2}u}{\partial x_n^2}+\frac{{\partial }^{2}u}{\partial y^2}=S(x_1,x_2,\dots ,x_n,y,u(x_1,x_2,\dots ,x_n,y)),$in which $(x_1,x_2,\dots ,x_n,y)\in {\mathbb{R}}^n\times \left[0,L\right],$ where $L>0.$ Based on the idea of the truncation approach, we construct the regularized solution and obtain the H ölder type of its convergence rate under some assumptions on regularity of the exact solution. Eventually, a numerical experiment for the elliptic Allen–Cahn equation is proposed to illustrate the effectiveness and applicability of our method.

椭圆方程（BPEE）的后向问题已广泛用于许多领域的建模问题，例如物理，几何或工程。本文是对与局部Lipschitz源相关联的无界多维域中BPEE的首次研究，如下所示$\frac{{\partial }^2ü}{\partial X_{\mathrm{1个}}^2}+\frac{{\partial }^2ü}{\partial X_2^2}+。。。\frac{{\partial }^2ü}{\partial X_{ñ}^2}+\frac{{\partial }^2ü}{\partial {ÿ}^2}=\mathrm{小号}（X_{\mathrm{1个}}，X_2，\dots ，X_{ñ}，ÿ，ü（X_{\mathrm{1个}}，X_2，\dots ，X_{ñ}，ÿ）），$在其中 $（X_{\mathrm{1个}}，X_2，\dots ，X_{ñ}，ÿ）\in {\mathbb{[R}}^{ñ}\times \left[0，\mathrm{大号}\right]，$ 哪里 $\mathrm{大号}>0。$基于截断法的思想，我们构造了正则化解，并在一些精确解的正则性假设下获得了其收敛速度的Hölder类型。最后，提出了一个椭圆艾伦-卡恩方程的数值实验，以说明我们方法的有效性和适用性。