Boundary layers and corrector estimates

For a clean sufficient setting, take a bounded smooth domain, a smooth periodic symmetric uniformly positive definite coefficient matrix, a fixed smooth load, and zero Dirichlet data. Let u₀ solve the constant-coefficient homogenized problem, and let χj be the mean-zero periodic cell correctors. Smoothness ensures the derivatives and corrector products used below are controlled. The bare expansion u₀+εχj(x/ε)∂ju₀ has a gradient containing the order-one term ∇yχj(x/ε)∂ju₀. This term is essential; discarding it defeats the approximation precisely where the microscopic response appears.

On the boundary, u₀ vanishes but its normal derivative usually does not. Consequently, the bare corrector term generally has a nonzero trace. A cutoff ηε that vanishes within distance ε of the boundary and equals one beyond distance 2ε repairs the trace. Its gradient has size at most a constant divided by ε. Multiplication by ε therefore produces an order-one gradient contribution inside a thin strip, even though the correction itself is small. Boundary layers are an energy issue, not simply an amplitude issue.

In this smooth setting, the standard cutoff construction gives an H¹ error bounded by C√ε between the true solution and wε. A strip of thickness ε has volume of order ε; an order-one gradient supported there has L² size of order √ε. Interior residual estimates complete the argument. The constant depends on ellipticity, the domain, coefficient regularity, and norms of the data. It is not uniform over arbitrary rough domains, degenerate coefficients, or arbitrarily large loads. A more carefully chosen Dirichlet corrector can improve some estimates, but it is a different approximation.

The displacement error can obey the better L² bound Cε when the homogenized primal and adjoint problems have the necessary H² regularity; smooth domains and the smooth data assumed here suffice. The proof uses duality as well as energy estimates. Neither bound says that ∇uε converges strongly to ∇u₀. The correct comparison for gradients contains the oscillating corrector. On rough Lipschitz domains, or with only the qualitative hypotheses of earlier lessons, one must consult the applicable theorem instead of importing these rates unchanged.

Return to the equal-half conductivities one and four, with ε=1/N on the unit interval and endpoint values zero and one. This coefficient is piecewise constant, so the following computation is independent of the smooth-coefficient statement. The mean-zero corrector from the cell calculation has value −3/20 at both ends of a cell and 3/20 at its midpoint. Conservation determines the exact solution, including its additive constant. The boundary adjustment subtracts χ(0), which is essential even though it changes the potential by only order ε.

The resulting potential stays within 3ε/10 of the affine limit. Its derivative differs from one by plus or minus 3/5 throughout alternating half-cells. Thus uniform convergence of potentials coexists with a fixed nonzero L² gradient error. In more dimensions, the boundary correction is generally not a single constant and interacts with geometry and the oscillatory trace. The one-dimensional example isolates the mechanism without pretending to describe every multidimensional boundary layer or its optimal rate.