Nonuniqueness for second-order elliptic equations with measurable coefficients

We consider the Dirichlet problem for uniformly elliptic operators L = ∑a_ijD_ij with measurable coefficients a_ij in the unit ball B₁ ⊂ R^d. A recent sensational result of Nikolai Nadirashvili states that there is no uniqueness of "weak" solutions to this problem if d ≥ 3. He constructed two sequences of linear elliptic operators with smooth coefficients {a^{0, k}_ij} and {a^{1, k}_ij}, which have the same ellipticity constant v > 0 and converge to the same functions a_ij almost everywhere (a.e.) in B₁ as k → ∞, while the corresponding sequences of solutions {u^{o, k}} and {u^{1, k}} converge to two different functions; i.e., the Dirichlet problem has at least two "weak" solutions. In the present paper, we popularize and slightly generalize Nadirashvili's result: for an arbitrary constant ∧ > 0, we construct two sequences of linear elliptic operators with the same ellipticity constant v = v(∧) > 0 and the additional restriction \a^{0, k}_ij - a^{1, k}_ij\ ≤ ∧ for all i, j, k, which define two different "weak" solutions to the Dirichlet problem [N. S. Nadirashvili, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), pp. 537-550].