Nonunique continuation for uniformly parabolic and elliptic equations in selfadjoint divergence form with Hölder continuous coefficients
Keith Miller · Bulletin of the American Mathematical Society · 1973
Consider the problem of backward uniqueness for the uniformly parabolic equationand the problem of unique continuation (and uniqueness for the Cauchy problem) for the uniformly elliptic equationwhere Q is a bounded domain in R n , v denotes the unit normal to 5Q, and the symmetric matrix sé has its eigenvalues in [a, a -1 ], with a > 0. We construct examples of nonuniqueness for (1) when n -2, and for (2) when n = 3; in each case a may be arbitrarily close to 1 and the coefficients are also Holder continuous.Backward uniqueness for (1) with %> l coefficients was shown by Lions-Malgrange [5]; probably the simplest proof is that of Agmon-Nirenberg [2] and Agmon [1] using the general method of logarithmic convexity.Carleman [4] long ago established unique continuation for (2) with ( € 1 coefficients when n = 2.For n ^ 3, unique continuation for (2) with l coefficients by Agmon [2].See [1] and [6] for references to other results by Holmgren, Cordes, Hörmander, Landis, Lees and Protter, Bers and Nirenberg, and others.An example of nonunique continuation was constructed by Plis [6] for a uniformly elliptic equation in the nondivergence form AMS (MOS) subject classifications (1969).