Lemma 18 ([Natural-growth scalar estimate]). Let n≥3 and let Q be a Euclidean coordinate ball, or a smooth flattened half-ball with constant Dirichlet value on its face. Suppose a bounded classical function Z satisfies weakly -∂_i(A^ij …

If a Dirichlet face is present, first subtract its constant boundary value. This changes neither the equation nor its source bounds. The reflection creates no measure on that face. Flatten it as x_n=0 and write J=diag(1,…,1,-1) . For x_n<0 extend the four quantities by Z^e(x)=-Z(Jx), A^e(x)=JA(Jx)J, B^e(x)=-JB(Jx), S^e(x)=-S(Jx). The zero trace makes Z^e a W^ 1,2 function, and ellipticity is preserved. To verify the equation across the face, test the original equation on the upper half with φ(x)-φ(Jx) , which has zero trace. Changing variables in the reflected half gives precisely Equation for the extended quantities. Equivalently, the normal component of the full flux ADZ+B agrees in the weak flux sense on the two sides. Thus there is no boundary flux jump to estimate. A different constant Dirichlet value is handled by subtracting that constant first. Reflect the nonnegative density 1+F_0 evenly. The ball-mass bound changes by at most a fixed multiplicative factor. The reflected coefficients need not be smooth. absorption and potentials. Use ν=(1+F_0)\,dx , with the even extension just described where needed. We give the two-sign exponential argument for Equation. Put Q_ε=e^ εLZ for ε∈-1,1 , and let L=-∂_i(A^ ij ∂_j) . The weak chain rule gives L Q_ε , =∂_i(εLQ_εB^i) -L^2Q_εB·DZ +εLQ_εS -L^2Q_εA DZ·DZ. Indeed it follows directly by testing the equation for Z with εLQ_εφ . Young's inequality bounds the second term on the right by 12λL^2Q_ε|DZ|^2+C L^2Q_ε . Taking L≥2C/λ absorbs also the quadratic part of εLQ_εS . Since Z is bounded and L is now fixed, we obtain, for both signs, L Q_ε ≤ B_ε+Cν, B_ε=εLQ_εB, \|B_ε\|_∞≤C. This uses no derivative of B , A , or the density of ν . On a ball B_r in the resulting full patch, let v_ε solve L v_ε= B_ε+Cν, v_ε|_ ∂B_r =0. The density of ν is bounded for each individual classical solution, so this solve and the following tests can first be made in W^ 1,2 . Only the uniform mass bound in Equation enters the estimate. The part with right side B_ε has absolute value at most Cr\|B_ε\|_∞ . This is the scaled bounded-divergence-source estimate: testing by positive and negative level truncations gives ∫|D(v-k)_+|^2≤C\|B_ε\|_∞^2 | v>k | , and Sobolev level iteration on the unit ball proves the asserted bound after rescaling. For the measure part, the scalar Dirichlet Green function of a uniformly elliptic divergence operator in dimension n≥3 satisfies 0≤G_r(x,y)≤C|x-y|^ 2-n . For bounded measurable coefficients this follows from the whole-space Green-function comparison : extend A uniformly elliptically off the ball, and compare the whole-space and zero-Dirichlet Green functions by the maximum principle. Dividing B_r into annuli about any x∈B_r gives ∫_ B_r G_r(x,y)\,dν(y) ≤C∑_ j≥0 (2^ -j 2r)^ 2-n ν(B_ 2^ -j 2r (x)) ≤C∑_ j≥0 (2^ -j 2r)^a≤Cr^a. We work sufficiently far inside the coordinate patch that the mass bound applies to the balls in this sum. Consequently \|v_ε\|_ L^∞(B_r) ≤E_r, E_r=C(r+r^a). decay. Here is the oscillation reduction, including the role of the two signs. Let m=_ B_r Z , M=_ B_r Z , and ω=M-m . By Equations and, the functions W_+=e^ LM -Q_++v_++E_r, W_-=e^ -Lm -Q_-+v_-+E_r are nonnegative weak supersolutions of L W≥0 on B_r . At least one of the sets Z≤(m+M)/2 ∩B_ r/2 , Z≥(m+M)/2∩B_ r/2 has at least half the measure of B_ r/2 . On the first set e^ LM -Q_+≥cω , and on the second e^ -Lm -Q_-≥cω , because the exponential and its inverse are uniformly Lipschitz on the fixed bounded range of Z . For the corresponding sign the weak Harnack inequality, with its fixed exponent q>0 , yields cω≤ C(1 |B_ r/2 | ∫_ B_ r/2 W_ε^q)^ 1/q ≤C_ B_ r/4 W_ε. Subtracting v_ε+E_r , whose absolute value is at most 2E_r , and using the same Lipschitz comparison with Z , improves one of its two extremes. Hence, for a fixed θ∈(0,1) , osc_ B_ r/4 Z ≤(1-θ)osc_ B_r Z+C(r+r^a). Iteration gives a uniform C^b bound for any fixed 0<b<,(-(1-θ))/(4). Applying the same conclusion to the reflected equation proves the estimate up to the constant Dirichlet face.