We show: ∫_(B_1)▒〖e^(2(P_0 ) ̃u) dx〗≤1/4 (∫_(∂B_1)▒〖e^u(x) dσ〗)^2 holds for all u(x)∈L^1 (∂B_1 ), ，where (P_0 ) ̃u is the harmonic extension of u(x) to the unit ball B_1⊂R^2. Moreover, the equality holds if and only if u=c or u=-2 ln⁡〖|x-x_0 |+c〗 where c is a constant and x_0∈R^...

Liouville type theorems, classification of solutions and maximum principles are very interesting problems and are also very useful tools in PDE analysis. We give a brief introduction on the problems, the related methods, and some results with applications. The following are three examples to illust...