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 $\Vert u_m{\Vert }_{L^{\infty }(B_2)}\le 1$, 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 $p\ge 1$. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit $u\notin {\text{BV}}_{\mathrm{loc}}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L^1$ limit of the $A_m$.

14 pages. The main results of this paper were obtained through a series of chats with ChatGPT 5.6 Sol. The key strategies were obtained by ChatGPT. The authors reworked and rewrote the article entirely. All arguments have been checked and simplified by the authors. We take full responsibility for its correctness and content