The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $H^2$-regularity nor $ L^{\infty} $-error estimation, but only $ H^1_0 $-error estimation. In [J. Comput. Appl. Math, Vol. 370, (2020) 112647], we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $ L^{\infty} $-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.

翻译：本文的目的是制定一种统一的后继方法,通过假设既不是H$2美元(常规值),也不是L ⁇ infty}美元(error)估算,而只是H$1_0美元(error)估算,来核实椭圆边界值问题解决办法的假设性。在[J.Comput. Appl. Math,vol.370,(202020年)112647]中,我们提出了两种方法,以核实若干半线性椭圆边界值问题解决办法的假设性。然而,有些情况需要美元(infty)美元(error)估算,因此是狭义的可适用性。在本文件中,我们扩展了一种方法,并将其与Laplacian eigenvalies的先验误差结合,以获得一种具有广泛应用的统一方法。我们描述了如何评价一些必要的常数,以核实所希望的解决办法的假设性。我们将我们的方法应用于若干问题,包括以前的方法不适用的问题。