A variational theory for some group invariant solutions

In this thesis, we are mainly concerned with the existence of periodic solutions for the equation uxx+u=a xu3 , and the related affine invariant nonlinear Hill's equation uxx+Qx u=ax u3. The first equation arises from the study of the generalized affine curve-shortening problem, which is connected to affine geometry and image process. Its solution is referred to a self-similar solution for the anisotropic affine curve-shortening problem. Self-similar solutions are important in describing the long time behavior of the curve shortening problems and the structure of the singularities. These equations admit a variational structure, that is to say, their solutions are essentially critical points of the corresponding functionals. If a(x) ≡ 1, these functionals are invariant under the action of the special affine group. Hence the study for these equations are very much like the famous Nirenberg problem.; This thesis is divided into three chapters. In the first Chapter, we establish some preliminary results, study the subcritical case of the first equation and present basic materials about the linear Hill's equation. In Chapter 2, we study the first equation. Similarity between the roles of the group SL(2, R) on the equation for self-similar solutions of the affine curve shortening problem and of the conformal group of S2 on the Nirenberg problem for prescribed scalar curvature is explored. A priori estimates for solutions under a nondegeneracy assumption are established. Using these estimates, we employ the continuity method to demonstrate the existence of solutions when a map associated to the given function has non-zero degree. In Chapter 3 the results are generalized to a class of affine invariant Hill's equation.