Boundary Homogenization Of Two Classes Of Obstacle Problems

In this paper,we mainly study the asymptotic behavior of solutions as ∈→0+of the following variational problem:min{∫D|γ▽v|pdx:v∈W1,p(D),v|Γ=φ(x),v|sε≥φ(x)}where 1<p≤n and D(?)Rn is a bounded open subset satisfying(?)D=r∪∑,r∩∑二(?)且(?)D∈C1,αand for any ε>0,Sε(?)∑ and φ(x),φ(x)∈C∞(D).Under suitable assumptions,we proved the following two results:If γ(x)is a identity matrix and p=n,then the minimizer uε of energy functional fD|▽u|n dx converges weakly in W1,n(D)to a limit u which minimizes the energy functional ∫D |▽v|ndx+cn∫∑(v-φ)n-μ(x)dSx;If γ(x)=(rij(x))n×n is a symmetric matric-valued function and p=2<n,then the minimizer uε of energy functional ∫D|γ▽u|2dx converges weakly in H1(D)to a limit u0 which minimizes the energy functional ∫D|γ▽u|2+cn ∫∑(u-φ)2-μ(x)dSx.Here cn and cn are positive constants depending only on n and μ(x)and μ(x)are some density functions on ∑.