The Growth And Unique Continuation Of Bi-harmonic Operator

The Bi-harmonic equation corresponding to the Bi-harmonic operator ?2 is ?2u=0.This equation is the simplest higher order equation and also one of the most important equations.In this paper,we mainly study the growth of Bi-harmonic operators and the unique continuity of solutions of corresponding equations.Firstly,we define the frequency function of Bi-harmonic function and calculate the base of homogeneous bi-harmonic polynomial space and the zero properties of homogeneous Bi-harmonic polynomials.Using them,we prove the growth of Bi-harmonic functions.Then we study a class of fourth-order partial differential equations with singular potential terms and Bi-harmonic operators.With a method similar to Harmonic Function,we proved the almost monotonicity of the Almgren's type frequency function and the doubling conditions for the solution.Finally,we prove the unique continuity of the solution.