椭圆型和抛物型方程弱上解的正则性：下半连续性和逐点性
Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior
摘要：
我们证明了一个测度理论方法来研究散度型椭圆型和抛物型方程弱上解的局部正则性。在第一部分中，我们证明了在零测度集上重新定义弱上解后，弱上解变为下连续的。该证明依赖于一个一般原理，即De Giorgi型引理，它为广泛的一类椭圆和抛物型方程，包括各向异性椭圆型方程、抛物型$p$ -Laplace方程和多孔介质方程提供了一个统一的方法。在第二部分中，我们将表明，抛物问题的下连续代表在一个瞬间可以恢复逐点从“esslimif”的过去的时间。我们还证明了它可以通过过去时间的某个积分平均的极限来恢复。证明依赖于弱上解的正性展开。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。
英文摘要：
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 "essliminf" of past times. We also show that it can be recovered by the limit of certain integral average 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.
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