Nonuniqueness for second-order elliptic equations with measurable coefficients

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Abstract

We consider the Dirichlet problem for uniformly elliptic operators L = ∑aijDij with measurable coefficients aij in the unit ball B1 ⊂ Rd. 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 {a0, kij} and {a1, kij}, which have the same ellipticity constant v > 0 and converge to the same functions aij almost everywhere (a.e.) in B1 as k → ∞, while the corresponding sequences of solutions {uo, 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 ∧ > 0, we construct two sequences of linear elliptic operators with the same ellipticity constant v = v(∧) > 0 and the additional restriction \a0, kij - a1, kij\ ≤ ∧ 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].

Original languageEnglish (US)
Pages (from-to)879-895
Number of pages17
JournalSIAM Journal on Mathematical Analysis
Volume30
Issue number4
DOIs
StatePublished - Jan 1 1999

Keywords

  • Elliptic PDE
  • Measurable coefficients
  • Nonuniqueness

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