论文标题
最佳空间中具有较低阶系的二阶椭圆方程的尺度规律性估计值
Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces
论文作者
论文摘要
我们展示了对等式$ - {\ rm div}（a \ nabla u++++++++++++c \ nabla u+du = - {\ rm div} f+g $的本地和全球规模不变性估计值和等式$ - {\ rm div}（\ nabla u+bu）+c \ nabla u+bu）+c \ nabla u+bu）的不变性估计。在Lorentz空间的设置中，在假设$ b，f \ in l^{n，1} $，$ d，$ d，g \ in l^{\ frac {n} {n} {2} {2}，1} $ and $ c \ in l^{n，q} $ for $ q \ leq \ leq \ leq \ lim的估计中，概述了，这是一个惊人的估计。使用“良好”常数（即，仅取决于系数的规范）一般不会成立。另一方面，假设在$ b，d $或$ c，d $上存在必要的较小条件，我们显示了最高原理，并且摩泽尔对具有“良好”常数的亚物业的估计。我们还显示了具有“良好”常数的非负超扫描的反向Moser估计值，在$ q <\ infty $时，在很小的假设下，导致非负解决方案和局部解决方案的局部连续性导致Harnack不平等。最后，我们表明，在洛伦兹空间的环境中，我们的假设是保证这些估计值的敏锐假设。
We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-{\rm div}(A\nabla u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$, $d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser's estimate for subsolutions with "good" constants. We also show the reverse Moser estimate for nonnegative supersolutions with "good" constants, under no smallness assumptions when $q<\infty$, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.
