A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities II. A local energy approximation result

Lemma 3.

The corrector z ϵ defined by ( 9 ) satisfies ‖ ∇ z ϵ ‖ L 2 ( Ω ˜ ∖ ω ϵ ‾ ) ⩽ C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) ϵ γ / 2 , for any γ < 1 .

By construction, the pair ( u test + z ϵ , v test ) belongs to V ϵ and u test + z ϵ = φ ˜ ev on ∂ Ω ˜ . Inserting the pair ( u test + z ϵ , v test ) as a “test function” in F ϵ , and using Lemma 3, we now calculate: F ϵ ( u test + z ϵ , v test ) = F ϵ ( u test , v test ) + 1 2 ∫ Ω ˜ ∖ ω ϵ ‾ | ∇ z ϵ | 2 + ∫ Ω ˜ ∖ ω ϵ ‾ ∇ z ϵ · ∇ u ev ⩽ E σ ϵ ( u ev ) + C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) 2 ϵ γ + C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) ϵ γ / 2 ‖ u ev ‖ H 1 ( Ω ˜ ∖ ω ϵ ‾ ) ⩽ E σ ϵ ( u ev ) + C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) 2 ϵ γ / 2 , and so by insertion into (8) we obtain E ϵ ( u ϵ ev ) − E σ ϵ ( u ev ) ⩽ F ϵ ( u test + z ϵ , v test ) − E σ ϵ ( u ev ) ⩽ C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) 2 ϵ γ / 2 , as desired. Remark.

We note that u test = u ev has singularities of the form r 1 / 2 sin θ / 2 near the endpoints of σ (see Proposition 1 of [1]), and that a ϵ ∫ r < ϵ [ ( ∂ ∂ r r 1 / 2 sin θ / 2 ) 2 + ( 1 r ∂ ∂ θ r 1 / 2 sin θ / 2 ) 2 ] r d r d θ = ϵ a ϵ π 2 . Since this appears to be of the same order as ϵ a ϵ 2 ∫ σ ( ( ∂ u e v + ∂ τ ) 2 + ( ∂ u e v − ∂ τ ) 2 ) d s = ϵ a ϵ ∫ σ ( ∂ u ∗ ∂ τ ) 2 d s , (see Proposition 1 of [1]) one might expect that the singularities would contribute to the “limiting energy” E σ , Ω ˜ ϵ . The analysis in this section has shown that, if instead of accounting for the boundary discrepancy in u test − v test ∘ Φ ϵ − 1 by adding a corrector inside ω ϵ , one adds a corrector z ϵ outside ω ϵ , one may indeed do so in a way that is of the order C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) ϵ γ / 2 , and thus doesn’t contribute to the “limiting energy”.

2.2. Proof of (7)

It is well known that E ϵ ( u ϵ ev ) = min u ∈ H ev 1 ( Ω ˜ ) u = φ ˜ ev on ∂ Ω ˜ E ϵ ( u ) (10) = min u ∈ H 1 ( Ω ˜ ) u = φ ˜ ev on ∂ Ω ˜ E ϵ ( u ) = max ξ ∈ L 2 ( Ω ˜ ) 2 − ∇ · ξ = 0 E c ϵ ( ξ ) , where the complementary energy expression E c ϵ is given by E c ϵ ( ξ ) = ∫ ∂ Ω ˜ ξ · n φ ˜ ev d s − 1 2 ∫ Ω ˜ γ ϵ − 1 | ξ | 2 d x d y . We may split the volume integral as ∫ Ω ˜ γ ϵ − 1 | ξ | 2 d x = ∫ Ω ˜ ∖ ω ϵ | ξ | 2 d x d y + a ϵ − 1 ∫ ω ϵ | ξ | 2 d x d y . Consistent with this splitting, the maximization in (10) may be rewritten as taking place over ξ + ∈ L 2 ( Ω ˜ ∖ ω ϵ ) 2 with − ∇ · ξ + = 0 in Ω ˜ ∖ ω ϵ , ξ − ∈ L 2 ( ω ϵ ) 2 with − ∇ · ξ − = 0 in ω ϵ , subject to the boundary condition ξ + · n = ξ − · n on ∂ ω ϵ . From now on n denotes the outward normal to ω ϵ at ∂ ω ϵ (or to ω 1 at ∂ ω 1 ). Let Φ ϵ denote the previously defined 1-to-1 mapping of ω 1 onto ω ϵ . As ξ ∗ ranges over the set { ξ ∈ L 2 ( ω ϵ ) 2 : ∇ · ξ = 0 in ω ϵ and ξ · n = ξ + · n on ∂ ω ϵ } , for a fixed ξ + , then the vector field η ∗ = | det ∇ Φ ϵ | ( ∇ Φ ϵ ) − 1 ξ ∗ ∘ Φ ϵ ranges over the set { η ∈ L 2 ( ω 1 ) 2 : ∇ · η = 0 in ω 1 and η · n = | ∂ ∂ τ Φ ϵ | ( ξ + · n ) ∘ Φ ϵ on ∂ ω 1 } , and ∫ ω ϵ | ξ ∗ | 2 d x d y = ∫ ω 1 1 | det ∇ Φ ϵ | ( ∇ Φ ϵ T ∇ Φ ϵ ) η ∗ · η ∗ d x d y . From these calculations it follows that (11) E ϵ ( u ϵ ev ) = max ( ξ , η ) ∈ W ϵ F c ϵ ( ξ , η ) , where F c ϵ is given by F c ϵ ( ξ , η ) = ∫ ∂ Ω ˜ ξ · n φ ˜ ev d s − 1 2 ∫ Ω ˜ ∖ ω ϵ | ξ | 2 d x d y (12) − 1 2 a ϵ ∫ ω 1 1 | det ∇ Φ ϵ | ( ∇ Φ ϵ T ∇ Φ ϵ ) η · η d x d y , and the set W ϵ is given by W ϵ ≡ { ( ξ , η ) ∈ L 2 ( Ω ˜ ∖ ω ϵ ) 2 × L 2 ( ω 1 ) 2 : ∇ · ξ = 0 in Ω ˜ ∖ ω ϵ , ∇ · η = 0 in ω 1 , (13) and η · n = | ∂ ∂ τ Φ ϵ | ( ξ · n ) ∘ Φ ϵ on ∂ ω 1 } . We now divide ω 1 into two parts as before ω 1 int = σ × ( − 1 , 1 ) = ( − 1 , 1 ) × ( − 1 , 1 ) and ω 1 ends = ω 1 ∖ ω 1 int . and compute (14) ∇ Φ ϵ = 1 0 0 ϵ in ω 1 int , | ∂ ∂ τ Φ ϵ | = 1 on ∂ ω 1 ∩ ∂ ω 1 int , and (15) ∇ Φ ϵ = ϵ I in ω 1 ends , | ∂ ∂ τ Φ ϵ | = ϵ on ∂ ω 1 ∩ ∂ ω 1 ends . It thus follows that F c ϵ ( ξ , η ) = ∫ ∂ Ω ˜ ξ · n φ ˜ ev d s − 1 2 ∫ Ω ˜ ∖ ω ϵ | ξ | 2 d x d y − 1 2 ϵ a ϵ ∫ ω 1 int ( η 1 ) 2 d x d y − ϵ 2 a ϵ ∫ ω 1 int ( η 2 ) 2 d x d y − 1 2 a ϵ ∫ ω 1 ends | η | 2 d x d y and that W ϵ alternatively may be characterized as W ϵ = { ( ξ , η ) ∈ L 2 ( Ω ˜ ∖ ω ϵ ) 2 × L 2 ( ω 1 ) 2 : ∇ · ξ = 0 in Ω ˜ ∖ ω ϵ , ∇ · η = 0 in ω 1 , and η 2 ( x , y ) = ξ 2 ( x , ϵ y ) for − 1 ⩽ x ⩽ 1 , y = ± 1 , η · n = ϵ ( ξ · n ) ∘ Φ ϵ on ∂ ω 1 ∩ ∂ ω 1 ends } . From (11) we have that (16) E σ ϵ ( u ev ) − E ϵ ( u ϵ ev ) ⩽ E σ ϵ ( u ev ) − F c ϵ ( ξ , η ) , for any pair ( ξ , η ) ∈ W ϵ . We now proceed to construct a test pair from W ϵ , so that we may establish (7), in other words a test pair which satisfies F c ϵ ( ξ , η ) ⩾ E σ ϵ ( u ev ) − C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) 2 ϵ γ / 4 . From the representation formula for u ev in Proposition 1 of Appendix A in [1], and the remark immediately following that proposition, we know that (17) u ev = b 1 r 1 / 2 sin ( θ / 2 ) ϕ ( r ) + b 1 ′ r ′ 1 / 2 sin ( θ ′ / 2 ) ϕ ( r ′ ) + u ∗ , where ( r , θ ) and ( r ′ , θ ′ ) are polar coordinates around the two endpoints of σ, and u ∗ is bounded in C ev 1 , γ / 2 ( K ∖ σ ) , up to the curve σ, by C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) . Here we make use of the fact that ϵ a ϵ > m > 0 . If we differentiate this representation with respect to y and evaluate it on σ, then we obtain ∂ u ev ∂ y ( x , 0 ± ) = ± ( b 1 2 ( x + 1 ) − 1 / 2 ϕ ( x + 1 ) + b 1 ′ 2 ( 1 − x ) − 1 / 2 ϕ ( 1 − x ) + ∂ u ∗ ∂ y ( x , 0 + ) ) , for − 1 < x < 1 . Notice that these two functions are not in L 2 ( − 1 , 1 ) . However, we may define approximations that are in L 2 , as follows D U ± ( x ) = ± D U + ( x ) = ± ( b 1 2 max { x + 1 , ϵ } − 1 / 2 ϕ ( x + 1 ) + b 1 ′ 2 max { 1 − x , ϵ } − 1 / 2 ϕ ( 1 − x ) + ∂ u ∗ ∂ y ( x , 0 + ) + d ϵ ) , with the constant d ϵ chosen so that ∫ − 1 1 D U ± ( x ) d x = ∫ − 1 1 ∂ u ev ∂ y ( x , 0 ± ) d x . This requires that d ϵ = b 1 4 ∫ − 1 − 1 + ϵ [ ( x + 1 ) − 1 / 2 − ϵ − 1 / 2 ] d x + b 1 ′ 4 ∫ 1 − ϵ 1 [ ( 1 − x ) − 1 / 2 − ϵ − 1 / 2 ] d x = ϵ ( b 1 4 + b 1 ′ 4 ) . Since ϵ a ϵ > m , we observe that | d ϵ | ⩽ C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) ϵ . We also calculate (18) ‖ D U + ‖ L 2 ( − 1 , 1 ) ⩽ C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) | log ϵ | 1 / 2 . Define the function R by R ( x ) = ∫ − 1 x [ ∂ u ev ∂ y ( s , 0 + ) − D U + ( s ) ] d s , − 1 < x < 1 . A simple calculation shows that (19) ‖ R ‖ L 2 ( − 1 , 1 ) ⩽ ϵ C ( | b 1 | + | b 1 ′ | ) + | d ϵ | ⩽ C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) ϵ . The function R is actually continuous with the same C 0 -bound of C ‖ φ ˜ ev ‖ H 1 / 2 ( ∂ Ω ˜ ) ϵ , and (20) R ( − 1 ) = R ( 1 ) = 0 . To construct a test pair for (16) we select ξ ∗ = ∇ u ev and define η ∗ by η 1 ∗ ( x , y ) η 2 ∗ ( x , y ) = ϵ a ϵ ∂ ∂ x u ev ( x , 0 ) + R ( x ) y D U + ( x ) for ( x , y ) ∈ ω 1 int , and, η 1 ∗ ( x , y ) η 2 ∗ ( x , y ) = 0 0 for ( x , y ) ∈ ω 1 ends . The pair ( ξ ∗ , η ∗ ) clearly lies in L 2 ( Ω ∖ ω ϵ ) 2 × L 2 ( ω 1 ) 2 , and it satisfies the two divergence constraints prescribed in W ϵ : ∇ · ξ ∗ = 0 in Ω ˜ ∖ ω ϵ , and ∇ · η ∗ = 0 in ω 1 . Here we use the fact that ∂ ∂ x ( ϵ a ϵ ∂ ∂ x u ev ( x , 0 ) + R ( x ) ) = − ∂ u ev ∂ y ( x , 0 + ) + ( ∂ u ev ∂ y ( x , 0 + ) − D U + ( x ) ) = − ∂ ∂ y ( y D U + ( x ) ) in ω 1 int . We also use (20) and the fact that ∂ ∂ x u ev ( − 1 + , 0 ) = ∂ ∂ x u ∗ ( − 1 , 0 ) = 0 , and ∂ ∂ x u ev ( 1 − , 0 ) = ∂ ∂ x u ∗ ( 1 , 0 ) = 0 , which follows from the decomposition (17) and Lemma 2 in Appendix A of [1]. In general, this pair ( ξ ∗ , η ∗ ) fails to satisfy the “interface” conditions, instead it satisfies (21) η 2 ∗ ( x , y ) = ± D U + ( x ) = D U ± ( x ) , for − 1 < x < 1 , y = ± 1 , and η ∗ · n = 0 on ∂ ω 1 ∩ { ( x , y ) : | x | > 1 } . To remedy this situation we seek ξ ϵ ∈ ( L 2 ( Ω ˜ ∖ ω ϵ ) ) 2 so that ∇ · ξ ϵ = 0 in Ω ˜ ∖ ω ϵ , ξ ϵ · n = 0 on ∂ Ω ˜ , (22) ( ξ ϵ ) 2 ( x , ± ϵ ) = D U ± ( x ) − ∂ ∂ y u ev ( x , ± ϵ ) , ξ ϵ · n = − ∂ ∂ n u ev on ∂ ω ϵ ∩ { ( x , y ) : | x | > 1 } . In Appendix B it is shown that