We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

我们证明了对散度形式的椭圆型和抛物型方程组的弱解的局部正则性的度量理论方法。在第一部分中，我们显示了在对一组度量零进行重新定义之后，弱上解变得较低的半连续。证明依赖于一般原理，即De Giorgi型引理，该引理为各种椭圆和抛物线方程提供了统一的方法，包括各向异性椭圆方程，抛物线p-拉普拉斯方程和多孔介质方程。在第二部分中，我们将表明，对于抛物线问题，可以从过去的“ ess lim inf”中逐点恢复下半连续的代表。我们还表明，可以通过过去时间的某些整数平均值的限制来恢复它。证明依赖于弱弱解的正性扩展。我们的结果是偏微分方程的结构性质，与任何一种比较原理无关。