Regularity For The Gradients Of Solutions To Double Phase Elliptic Obstacle Problems Of Divergence Form

As is well known,nonlinear elliptic partial differential equation is an important part of partial differential equation,which is usually the description of various mechanical and physical problems in equilibrium state(or steady state).Therefore,it is significant theory and practical values to study elliptic partial differential equations.The double phase variational integrals and elliptic equations,as a typical situation of non-uniformly elliptic problems,can be derived from the description of nonlinear elasticity,anisotropic phase transition materials and Lavrentiev’s phenomenon.Actually,it has become an interesting recent research subject regarding anisotropic variational problems and non-uniform elliptic equations.We devote this thesis to the regularity of gradients for the solutions of non-uniformly elliptic obstacle problems:one is to consider a class of non-uniformly elliptic obstacle problems with borderline growth,and we are to prove a global gradient estimate in the scale of variable Lebesgue spaces to their solutions.The other is to establish the gradient estimates in the scale of Besov spaces to the solutions of elliptic obstacle problems with double phase growth.To state our main results more details,let Ω(?)Rn be a bounded domain for n≥2.Our main results are listed as follows:1.Let ψ∈W1,H(Ω)be an obstacle function satisfying ψ∈W1,H(Ω)and ψ≤0 on (?)Ω.An admissible set A0(Ω)is introduced as follows:A0(Ω)={v∈W01,H(Ω):v≥ψ a.e.x ∈Ω}.We first consider the following double phase obstacle problems defined in the Reifenberg domain Ω:where A(x,z),B(x,z)satisfies the following borderline growths:The problem under consideration is essentially characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position,which describes a feature of strongly anisotropic materials.We establish global Calderón-Zygmund type estimates in the scale of variable Lebesgue spaces via the so-called large-M-inequality principle under main assumptions that the associated nonlinearity A(x,z)has a small BMO with respect to x∈Ω and the function a(x)is a strong log-H(?)lder continuity.2.Let ψ be an obstacle function satisfying ψ∈W1,G(Ω)and ψ≤0 on (?)Ω.Here the admissible set A(Ω)is defined by:A(Ω)={v∈W1,G(Ω):v≥ψ a.e.x ∈Ω).Next,we are to consider the double phase problems with unilateral obstacle:∫Ω<A(x,Du),D(u-v)>dx ≤∫Ω<|F|p-2F+a(x)|F|q-2F,D(u-v)>dx (?)v∈A(Ω),where the nonlinearity A(x,Du)satisfies the following(p,q)-growth condition:there exist positive constants v,,μ such that for any x,x1,x2 ∈Ω and ξ,η∈Rn,where 2 ≤ p<q,q/p<1+a/n and 0≤a(·)∈C0,α(Ω),α∈(0,1].If F ∈ Bq,st,locβ(Ω)and Dψ∈Bq,t,locβ(Ω),then we have a local gradient estimate in the Besov space to the solutions.We prove it mainly based on the finite difference technique.