Three Spherical Inequality Elliptic Equations

The three-spheres inequality for elliptic equations is an important property for the solutions of elliptic equations. It has several representations. In this paper, three-spheres inequality for homogeneous elliptic equations in two dimension and higher dimension, for homogeneous elliptic equations with lower order terms, for inhomogeneous elliptic equations, and for elliptic equations with weight or singular potentials are introduced. Four chapters are divided in this paper. In chapter one the background and significance of three-spheres inequality for elliptic equations are given. It is mainly about some definitions of elliptic equations, some theorems and results, and then the basic knowledge is simply introduced in chapter two. The various representations of three-spheres inequality and their proofs are specifically provided in chapter three. At the same time, the improvement of some results are given. The condition of strong solutions u in Hloc2for elliptic equations with singular potential is weaken to the one of weak solutions in Hloc1. The methods of the proofs on three-spheres inequality for elliptic equations contains difference method, variational methods, some methods in Functional Analysis, and in Integral Equation and so on. The important application to elliptic and parabolic equations is involved in three-spheres inequality, such as the stability, the propagation of smallness, the uniqueness of solutions, and the inverse problems. The great value of three-spheres inequality in the partial differential equation urge us to study it.