各向异性障碍问题弱解的局部正则性和局部有界性.pdf

河北大学 硕士学位论文 各向异性障碍问题弱解的局部正则性和局部有界性 姓名：黄秋花 申请学位级别：硕士 专业：基础数学 指导教师：高红亚 2011-05 ?? ?? A-?????Æ????????????????????????A-?? ???????????? ????????????A-??????????? ??????????????Æ???????A-?????????????? ?????Æ????????A-?????????????????????? ???????Æ????????????????Æ????Sobolev????Æ ??????????A-??????????????? ????????????????????????? I Abstract Abstract A-harmonic equation is an important kind of quasilinear ellipitc equation.The local regularity and the local boundedness for weak solution of A-harmonic equation is the classical results for the thoery of A-harmonic equation. This thesis mainly studies the properties for weak solutions of A-harmonic equation.This dissertation focuses on two sides: one is the local regularity for weak solutions of Kqi ψ,θ-obstacle problems to the homogeneous equation, the other is the local boundedness for weak solutions of Kqi ψ,θ-obstacle problems to the non-homogeneous equation. By means of changing some conditions that satisfi es the equation appropriately, we proved the local regularity and the local boundedness for weak solution of A-harmonic equation by using a special kind of Sobolev inequality, constructing a test function and combining with some basic inequalities. KeywordsLocal regularityLocal boundednessAntisotropic spaceOb- stacle problemsWeak solution II Chapter 1Introduction Chapter 1Introduction Sobolev theory on Riemannian manifl ods has come into widespread usage in modern geometry ang topology. It aslo continues to be of great importance in nonlinear partial diff erential equations (PDE’s for short), variational problems, like those in the theory of harmonic maps or quasiconformal derformations,nonlinear elasticity, continuum mechan- ics, and much more. In sobolev space, many interesting results of A-harmonic tensors and their applications have been found. We have learned that the p-harmonic functions are H¨ older continuous. In fact, much more regularity is valid, even the gradients are locally H¨ older continuous. In symbols, the functions of class C1,α loc(Ω). More precisely, if u is p-harmonic in Ω and if D ⊂⊂ Ω, then |∇u(x) − ∇u(y)| ≤ LD|x − y| where x,y ∈ D. Here α = α(n,p) and LDdepends on n,p,dist(D,∂Ω) and kuk∞. This was proved in 1968 by N.Uraltseva. The weak solutions of the p-harmonic equation are by defi nition, members of the Sobolev space W 1,p loc(Ω). In fact, they are aslo of class Cα loc (Ω). More precisely, a weak solution can be redefi ned in a set of Lebesgue measure zero, so that the new function is locally H¨ older continuous with exponent α = α(n,p). Actualy, a deeper and stronger regularity result holds. In 1968 N.Uraltseva proved that even the gradient is locally H¨ older continuous. In 1923 O.Perron published a method for solving the Dirichlet boundary value problem ( ∇h(x) = 0,x ∈ Ω, h(x) = g(x),x ∈ ∂Ω and it is of interest, especially if ∂Ω or g are irregular. The same method works with virtually no essential modifi cations for many other partial diff erential equations obeying a comparison principle. We will treat it for the p-Laplace equation. The p-superharmonic and p-subharmonic functions are the building blocks. 1 ???????????? In recent years, there have been remarkable advances made in the investigations of A-harmonic equations. In 1990 and 1991, B.Stroff olini and Nicola Fusco-Carlo Sbordone establish the relations between local ang glodbal boundedness of solutions of anisotri- onal problems.In 1994, Li Gongbao and Olli Martio gives some local ang glodbal higher higher integrability results for the derivatives of the solutions to obstacle prob- lems associated with the second order edegenerate elliptic partial diff erential equation −divA(x,∇u) = 0, where |A(x,ξ)| ≈ |ξ|p−1,p 1. In 2008, Shenzhou Zheng establishs an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coeffi cient matrix A(x,u) satisfi es a VMO condition in the variable x uniformly with respect to all u, and the lower order item B(x,u,∇u) satisfi es the subcritical growth B(x,u,∇u) ≤ µ(M)(|∇u|p−δ+ f(x)),∀0 0. If t ∈ R, we denote by Btthe ball of radius t centered at x0. For k 0, let Ak= {x ∈ Ω : |u(x)| k}, Ak,t= Ak∩ Bt. 3 ???????????? Tk(u) = max{−k,min{k,u}} ψ+ k = max{ψ,Tk(u)} Moreover, if m k Since p k0, we have |Ak,t| ≤ 1 2|Bt|. For such a constant k0 0, we suppose R Ak0,t |u|q ∗dx ≤ 1. For such values of k′, we have Z Ak,t |u|α1dx ≤ C(n,p)|Ω|α1−p+ p n Z Ak,t n X i=1 ? ? ? ? ∂u ∂xi ? ? ? ? qi dx = C1 Z Ak,t n X i=1 ? ? ? ? ∂u ∂xi ? ? ? ? qi dx (4.12) 12 Chapter 4Local Boundedness for Weak Solutions of Antisotropic Obstacle Problems where C1depend only on α1,n,p and |Ω|. Combining the inequalities (4.8) and (4.12) yields, we have Z Ak,t hA(x,u,∇u),φp∇uidx ≥ Z Ak,τ a0 n X i=1 ? ? ? ? ∂u ∂xi ? ? ? ? qi dx + b0C1 Z Ak,t n X i=1 ? ? ? ? ∂u ∂xi ? ? ? ? qi dx − b1b0|Ak,τ| = Z Ak,τ (a0+ b0C1) n X i=1 ? ? ? ? ∂u ∂xi ? ? ? ? qi dx − b1b0|Ak,τ|. (4.13) Next, we estimate the right-hand side of inequal。