| In this dissertation,the Calderon-Zygmund estimates are studied for two asymp-totic regular elliptic problems with Lp log L-growths.The first problem is to consider asymptotic regular elliptic equations with Lp log L-growths,and we get a global Calderon-Zygmund estimate under some weak regular assumptions on the given datum.The second problem is to study an asymptotic regular elliptic problem confined by the double barri-er functions,and we also conclude a global Calderon-Zygmund estimate under the same regular assumptions as the first question imposed on the coefficients and the boundary of underlying domain.More details are listed as follows:Chapter 1 is to introduce some basic knowledge and recall recent related references,the boundedness of the Hardy-Littlewood maximal function in Lq and the maximum func-tion level estimation are given.Finally,we state our main results.In Chapter 2,we are to consider the zero-Dirichlet problem for the following asymp-totic regular elliptic problem with Lp log L-growths:(?)where a=a(x,ξ):Ω×Rn is asymptotically regular to the regular operator b(x,ξ)with Lp log L-growth as |ξ| approaches infinity,and F=(f1,f2,…,fn)is a given vectorial-value function.Let D(|ξ|)=|ξ|p log(e+|ξ|).If the inhomogeneous term Φ(|F|)∈ Lq(Ω),1<q<∞,under the weak regular assumptions of the regular operator b(x,ξ)we conclude that Φ(|Du|)∈Lq(Ω)with the following estimate(?)where C=C(n,v,A,p,q,R,ω(·),Ω).Our argument is to make use of a series of the perturbed comparison methods to translate our asymptotic problem to the regular one on the basis of the gradients approaches infinity.A global Calderon-Zygmund estimate is established by using the estimate of decay for the upper level set of gradients.In fact,this series of the upper level set is equivalent to the q-power of Lq-norm of gradients.Chapter 3 is to study an asymptotic regular elliptic problem with double barriers.Forψ1,ψ2 being two given obstacle function,let A(Ω)={φ∈W1,Φ(Ω):ψ1≤φ≤ψ2} be an admissible set.Then the above-mentioned problem with double barriers is described by the following variational inequality:(?)where u,φ∈A(Ω).The nonlinearity a=a(x,ξ):ΩxRn is asymptotically regular with the same structural and regular assumptions as shown in chapter 2.If Φ(|F|),Φ(|Dψ1|),Φ(|Dψ2|)∈Lq(Ω)for 1<q<∞,then we conclude that Φ(|Du|)∈Lq(Ω)with the following estimate(?)where Ψ(x)=Φ(|F|)+Φ(|Dψ1})+Φ(|Dψ2|)+1 and C=C(n,v,Λ,P,q,R,ω(·),Ω)>0.Here,by using poisson transformation to change the asymptotic regular double barrier problem into a regular one.Then the estimate of asymptotic regular problem is derived based on the given conclusion of a recent reference for the regular double barrier problem. |
