论文标题
关于对流的变性椭圆问题的强大比较原理
On the Strong Comparison Principle for Degenerate Elliptic Problems with Convection
论文作者
论文摘要
分别研究了$ p $ -laplacian在一个空间维度中的准线性椭圆边界值问题分别研究了弱和强的比较原则。我们将$ 2 <p <\ infty $的退化案例处理，并允许在基础域$（-1,1）中的非平凡对流速度$ b \ colon [-1,1] \ to \ Mathbb {r} $。我们在对流速度的相当普遍，自然的条件下建立了弱比较原理，$ b（x）$和反应函数$φ（x，u）$。此外，我们还根据许多其他假设建立了强大的比较原理。相比之下，由于违反了这些假设，我们还构建了一些相当自然的反示例，以进行强有力的比较原则，并讨论了它们在多孔培养基中流体流动的有趣的经典问题，在倾斜床上流体的渗流流动。我们的方法基于古典技术和新技术的混合。
The weak and strong comparison principles, respectively, are investigated for quasi-linear elliptic boundary value problems with the $p$-Laplacian in one space dimension. We treat the degenerate case of $2 < p < \infty$ and allow also for the nontrivial convection velocity $b\colon [-1,1]\to \mathbb{R}$ in the underlying domain $(-1,1)$. We establish the weak comparison principle under a rather general, natural sufficient condition on the convection velocity, $b(x)$, and the reaction function, $φ(x,u)$. Furthermore, we establish also the strong comparison principle under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the strong comparison principle and discuss their applications to an interesting classical problem of fluid flow in porous medium, seepage flow of fluids in inclined bed. Our methods are based on a mixture of classical and new techniques.
