Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three

For each nonnegative integer $m$, we construct smooth symmetric $3\times 3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound \[ I\leq A_m\leq 2^{81}I \] for which the smooth solutions of uniformly elliptic equations in nondivergence form \[ \text{tr}(A_m(x)D^2 u_m)=A_m(x):D^2u_m=0\qquad\text{in }B_2\subset {\mathbb R}^3 \] have common Dirichlet data, satisfy $\|u_m\|_{L^\infty(B_2)}\leq1$, but \[ \lim_{m\to \infty}\|Du_m\|_{L^1(B_1)}=\infty. \] Thus, there is no interior $W^{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W^{1,p}$ estimate for any $p\geq1$. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit $u\notin \text{BV}_{\rm loc}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L^1$ limit of the $A_m$.