On the uniqueness for the cauchy problem for elliptic operators

Abstract

Let P be an elliptic differential operators of order m with C{sup {infinity}} - coefficients in an open neighborhood {Omega} of the origin in R{sup n}. Let {var_phi} be a real valued C{sup {infinity}} - function on {Omega} with d{var_phi}(0) {plus_minus} 0,{var_phi}(0) = 0. It is said that uniqueness for the Cauchy problem for P with respect to a hypersurface {var_phi}(x) = 0 holds at the origin if for any small open neighborhood {omega} {contained_in} {Omega} of the origin there is an open neighborhood {omega}` {contained_in} {omega} of the origin such that every u {epsilon} C{sup {infinity}}({omega} with Pu = 0,u{parallel}{sub {var_phi}}{>=}0 = 0 vanishes in {omega}`. Uniqueness for the Cauchy problem for elliptic operators has been studied in connection with its unique continuation property. Let P(x,{nu}) = P{sub m}(x,{nu}) + ... + P{sub 0}(x,{nu}) be the symbol of P in a coordinates. Then the roots of the following equation in {tau} are called characteristic roots. P{sub m}(x,{nu} + {tau}d{var_phi}(x)) = 0 {zeta} {epsilon} R{sup n} {backslash} Rd{nu}(x)).