We consider the Dirichlet problem for uniformly elliptic operators L=∑ aᵢⱼDᵢⱼ with measurable coefficients aij in the unit ball B₁⊂ 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 \,ᵢⱼ0,k\\, and \,ᵢⱼ1,k\, which have the same ellipticity constant \,ν > 0\, and converge to the same functions aij almost everywhere (a.e.) in \,B₁\, as k→∞, while the corresponding sequences of solutions {u0,k}, and {u1,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 Λ > 0, we construct two sequences of linear elliptic operators with the same ellipticity constant \,ν=ν(Λ) > 0\, and the additional restriction \;|aᵢⱼ0,k- aᵢⱼ1,k|≤ Λ\, for all i, j, k, which define two different "weak" solutions tothe Dirichlet problem [N. S. Nadirashvili, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), pp. 537--550].
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M. V. Safonov (1999) studied this question.