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 $$I\le A_m\le 2^{81}I$$ for which the smooth solutions of uniformly elliptic equations in nondivergence form $$\text{tr}(A_m(x)D^2{u}_m)=A_m(x):D^2{u}_m=0\text{in }{B}_2\subset {\mathbb{R}}^3$$ have common Dirichlet data, satisfy $\mathrm{\parallel }{u}_m{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(B_2)}\le 1$, but $$\underset{m\to \mathrm{\infty }}{\lim }\mathrm{\parallel }D{u}_m{\mathrm{\parallel }}_{L^1(B_1)}=\mathrm{\infty }\mathrm{.}$$ 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 $p\ge 1$. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit $u\mathrm{\notin }{\text{BV}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L^1$ limit of the $A_m$.