论文标题
椭圆形和抛物线方程的弱超溶液的规律性：较低的半持续性和指尖行为
Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior
论文作者
论文摘要
我们证明了一种理论方法，即以差异形式对椭圆形和抛物线方程弱的局部规律性的局部规律性。在第一部分中，我们表明弱超溶液在一组零度零上重新定义后的半连续性变得较低。证明依赖于一般原则，即De Giorgi型引理，该原理为广泛的椭圆形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 $ p $ laplace方程以及多孔培养基方程。在第二部分中，我们将表明，对于抛物线问题，下一个半连续的代表可以从过去时期的“ Essliminf”中点恢复。我们还表明，可以通过过去时间的某些积分平均值的限制来恢复它。证据取决于弱超溶液的阳性扩大。我们的结果是部分微分方程的结构特性，与任何类型的比较原理无关。
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.
