Maximum principles for Laplacian and fractional Laplacian with critical integrability

In this paper, we study maximum principles for Laplacian and fractional Laplacian with critical integrability. We first consider the critical cases for Laplacian with zero order term and first order term. It is well known that for the Laplacian with zero order term $-\Delta +c(x)$ in $B_1$, $c(x)\in L^p(B_1)$($B_1\subset \mathbf{R}^n$), the critical case for the maximum principle is $p=\frac{n}{2}$. We show that the critical condition $c(x)\in {L^{\frac{n}{2}}(B_1)}$ is not enough to guarantee the strong maximum principle. For the Laplacian with first order term $-\Delta +\vec{b}(x)$($\vec{b}(x)\in L^p(B_1)$), the critical case is $p=n$. In this case, we establish the maximum principle and strong maximum principle for Laplacian with first order term. We also extend some of the maximum principles above to the fractional Laplacian. We replace the classical lower semi-continuous condition on solutions for the fractional Laplacian with some integrability condition. Then we establish a series of maximum principles for fractional Laplacian under some integrability condition on the coefficients. These conditions are weaker than the previous regularity conditions. The weakened conditions on the coefficients and the non-locality of the fractional Laplacian bring in some new difficulties. Some new techniques are developed.