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

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VibeMathed record: Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three evidence for this event

This event attributed to Qi Sun

This event attributed to Hung V. Tran

Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three parent of this event

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

This event attributed to GPT-5.6 Sol

This event attributed to Nam Q. Le

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. parent of this event