Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles

3.2.1 Laplacian case

Continuing to the aforementioned argument, we can define a radius a ¯ σ and an approximated homogeneous solution W σ . Note that for the Laplacian case, we have α ∗ = n − 2 and so the normalized homogeneous solution is given by Φ ( x ) = ∣ x ∣ 2 − n . On the other hand, we see that the radius of hole a ε is assumed to be comparable to ε n / ( n − 2 ) . Since the corrector w ε will be constructed so that w ε ≈ 1 near ∂ T ε (see Section 5), we require the homogeneous solution Φ ( x ) ≈ ε − 2 near ∣ x ∣ = a ¯ ε = a ε / ε . Here we need to distinguish scale ε and scale 1.

Therefore, we let a ¯ σ ≔ σ 2 n − 2 and determine a quadratic polynomial W σ which is rotationally symmetric and satisfies

if ∣ x ∣ = a ¯ σ . Indeed, for σ > 0 , we set

where m σ = n − 2 2 σ − 2 n n − 2 and k σ = n 2 σ − 2 . Then Φ σ ∈ C 1 , 1 ( R n ) and it follows that

On the other hand, by its construction, we immediately have that

locally uniformly on R n ⧹ { 0 } . Moreover, for 0 < σ 1 ≤ σ 2 ,

and

Note that this approximation is related to the Dirac-delta measure:

in distribution sense, i.e.,

for any η ∈ C c ∞ ( R n ) .

3.2.2 Fully nonlinear case

In this case, the radius of hole a ε is comparable to ε α ∗ + 2 α ∗ . Since the normalized homogeneous solution is given by

in spherical coordinates, we let a ¯ σ = σ 2 α ∗ . Moreover, we consider a (strict) superlevel set of Φ :

Note that we have C a ¯ σ = B a ¯ σ as before, if we let ϕ ( θ ) ≡ 1 , i.e., F is rotationally symmetric. Then for σ > 0 , we set

where s (which is independent of σ > 0 ), m σ , and k σ will be determined.

1. ( Φ σ ∈ C 1 , 1 ) We only need to check this property on ∂ C a ¯ σ . In fact, for ( r , θ ) ∈ ∂ C a ¯ σ , we have

   Φ ( r , θ ) = a ¯ σ − α ∗ , W σ ( r , θ ) = − m σ a ¯ σ α ∗ s + k σ , ∂ r Φ ( r , θ ) = − α ∗ r a ¯ σ − α ∗ , ∂ r W σ ( r , θ ) = − m σ α ∗ s r a ¯ σ α ∗ s ,

   and

   ∇ θ Φ ( r , θ ) = r − α ∗ ∇ θ ϕ , ∇ θ W σ ( r , θ ) = m σ s r α ∗ s ϕ ( θ ) − s − 1 ∇ θ ϕ = m σ s r − α ∗ a ¯ σ α ∗ ( s + 1 ) ∇ θ ϕ .

   Therefore, we conclude W σ ∈ C 1 , 1 provided that

   m σ = 1 s σ − 2 ( s + 1 ) and k σ = 1 + 1 s σ − 2 ,

   for some constant s > 0 .

2. ( F ( D 2 W σ ) ≕ − ν σ ≤ 0 ) To verify this property, it is enough to show that there exists a sufficiently large s such that

   P − ( − D 2 W σ ) ≥ 0 .

   For this purpose, we claim that for sufficiently large s = s ( λ , Λ , f ) > 0 , we have

   P − ( D 2 w ) ≥ 0 ,

   where w ( r , θ ) ≔ r s f ( θ ) for a positive function f ∈ C 2 ( S n − 1 ) . Indeed, one can calculate the Hessian of w as follows:

   Hess ( w ) ∼ ( a i j ( s , θ ) ) r s − 2 ,

   where

   a i j ( s , θ ) = s ( s − 1 ) f ( θ ) if ( i , j ) = ( 1 , 1 ) , o ( s 2 ) g i j ( θ ) otherwise ,

   for g i j ∈ C ( S n − 1 ) , 1 ≤ i , j ≤ n . See Appendix in [24] for the computation of the Hessian matrix in spherical coordinates. In short, we have the dominant s 2 -order only in ( 1 , 1 ) -component of Hess ( w ) , since the power of s is added if and only if we take a radial derivative with respect to w .

   Moreover, since det ( t I − A ) = ( t − λ 1 ( A ) ) ⋯ ( t − λ n ( A ) ) and the determinant function is smooth, one can easily check that the eigenvalues of Hess ( w ) are given by

   ( s 2 f ( θ ) + o ( s 2 ) b 1 ( θ ) , o ( s 2 ) b 2 ( θ ) , … , o ( s 2 ) b n ( θ ) ) ⋅ r s − 2 ,

   for some functions b i ∈ C ( S n − 1 ) . Hence, for sufficiently large s = s ( λ , Λ , f ) > 0 , we have

   P − ( D 2 w ) ≥ [ λ s 2 f ( θ ) − Λ b ( θ ) o ( s 2 ) ] ⋅ r s − 2 ≥ 0 ,

   as claimed.

locally uniformly on R n ⧹ { 0 } . Moreover, by its construction, we have

and

for 0 < σ 1 ≤ σ 2 .