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

The authors construct smooth $A_m$ and smooth solutions $u_m$ in $B_2\subset\mathbb R^3$ with one fixed ellipticity bound $$I\leq A_m\leq2^{81}I,$$ common boundary data and $\|u_m\|_{L^\infty(B_2)}\leq1$, but $$\|Du_m\|_{L^1(B_1)}\to\infty.$$ Thus no interior $W^{1,1}$ estimate can depend only on dimension and ellipticity. Consequently, no such $W^{1,p}$ estimate exists for any $p\geq1$. They further obtain a uniformly convergent limit $u$ with measurable uniformly elliptic coefficient matrix $A$, where $$u\notin BV_{\rm loc}(B_1).$$ The construction even rules out coefficient-independent weak-$L^1$ gradient estimates.

The authors construct smooth $A_m$ and smooth solutions $u_m$ in $B_2\subset\mathbb R^3$ with one fixed ellipticity bound $$I\leq A_m\leq2^{81}I,$$ common boundary data and $\|u_m\|_{L^\infty(B_2)}\leq1$, but $$\|Du_m\|_{L^1(B_1)}\to\infty.$$ Thus no interior $W^{1,1}$ estimate can depend only on dimension and ellipticity. Consequently, no such $W^{1,p}$ estimate exists for any $p\geq1$. They further obtain a uniformly convergent limit $u$ with measurable uniformly elliptic coefficient matrix $A$, where $$u\notin BV_{\rm loc}(B_1).$$ The construction even rules out coefficient-independent weak-$L^1$ gradient estimates.