Research On Nodal Sets And Singular Sets Of Solutions Of Some Elliptic Equations

The subject of nodal sets and singular sets are important research topics for solutions of partial differential equations. In some cases, properties of nodal sets and singular sets of solu-tions are themselves the primary concern. In other cases, nodal sets and singular sets provide important tools in the study of properties of solutions. The study of nodal sets and singular sets refers to a lot of fields in mathematics, including the P.D.E theory, complex analysis, geometric analysis and geometric measure theory, etc. So the study of nodal sets and singular sets has great significance in theory and application.In study of nodal sets and singular sets, the definition "frequency" plays an important role. Frequency was introduced first in order to describe the growth property of harmonic functions. For harmonic functions, the frequencies have the classical monotonicity formula. From this property, many important properties can be got, including the doubling conditions of harmonic functions and the control of vanishing order of harmonic functions. By using frequency, mea-sure estimates of nodal sets of harmonic functions can be got. In this paper, we study the nodal sets of bi-harmonic functions,k-harmonic functions and harmonic functions on the Heisenberg group(H-harmonic functions). We also study the growth property of k-harmonic functions and the geometric structure of H-harmonic functions.Bi-Laplacian equation is the simplest, and very important forth order equation. And k-harmonic functions are the further extension of harmonic functions and bi-harmonic func-tions. So the study of nodal sets of bi-harmonic functions and k-harmonic functions is very significant. Referring to the frequency of harmonic functions, we define the frequencies of bi-harmonic functions and k-harmonic functions. For a bi-harmonic function u, we define the frequency of u is For a k-harmonic function u, we denote by u1=u, u-i=△i-1u,i=2,3,…,k. Then we define the frequency of u is Then we give some useful properties of these frequencies. First is that the frequencies are bounded from below. For k-harmonic functions(k≥2), the frequencies satisfy where C is a positive constant depending only on the dimension n. Second is the property of frequencies of polynomials. For a homogeneous k-harmonic(k≥2) polynomial of degree l, its frequency is between l and max(0,l-2k+2}. Third is the monotonicity formula. If u is a k-harmonic function(k≥2) defined in B(0,1) Rn(k≥3), n≥3and the frequency of u satisfies that N(0, r≥Co, then the inequality holds, where C and Co are positive constants depending only on the dimension n. We also prove the doubling conditions of bi-harmonic functions and k-harmonic functions. By using the above conclusions, we give the measure estimates of nodal sets of bi-harmonic functions and k-harmonic functions. Moreover, we also study the growth property of k-harmonic functions, and show that a k-harmonic function(k≥2) on the whole space is a polynomial if and only if the frequency of the k-harmonic function is bounded.A sub-Riemannian manifold, roughly speaking, is a smooth manifold associated with a distribution and a fibre-inner product on it. When the distribution is the whole tangent bundle, then the sub-Riemannian manifold reduces to be a Riemannian manifold. In recent years, there are a lot of investigations on sub-Riemannian manifolds which have strong relationships with many fields such as analysis, PDE, algebra and geometry. The Heisenberg group is the most important and simplest model of sub-Riemannian manifolds. There are essential differences between this manifold and the Euclidean spaces. H-harmonic functions are solutions of degen-erate elliptic equations in Euclidean spaces. And these equations are degenerate at every point. These bring us a lot of difficulty in study of H-harmonic functions. By using frequencies of H-harmonic functions, we give the measure estimates of nodal sets of H-harmonic functions. Then we define the horizontal singular sets and j-horizontal singular sets of functions on the Heisenberg group, show the geometric structure of j-horizontal singular sets of Heisenberg homogeneous polynomial of degree j. In order to get the geometric structure of horizontal singular set of some Hharmonic function, we first write the horizontal singular set a union of j-horizontal singular sets. Then for j=1,2,…, by using the structure of j-horizontal singu-lar sets of Heisenberg homogeneous polynomial of degree j, we give the geometric structure of j-horizontal singular sets of the H-harmonic function, and thus get the geometric structure of horizontal singular set of the H-harmonic function. From the geometric structure of horizontal singular sets of H-harmonic functions, we give the measure estimates of horizontal singular sets of H-harmonic functions on H1.