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

Abstract: 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 $|u_m|<em>{L<sup>\infty(B_2)}\leq1$, but [ \lim{m\to \infty}|Du_m|{L^1(B_1)}=\infty. ] Thus, there is no interior $W<sup>{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W<sup>{1,p}$ 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 $u\notin \text{BV}</em>{\rm loc}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L<sup>1$ limit of the $A_m$.