UNIQUENESS PRINCIPLE FOR FRACTIONAL (NON)-COERCIVE ANISOTROPIC POLYHARMONIC OPERATORS AND APPLICATIONS TO INVERSE PROBLEMS

In this work, we are concerned with inverse problems involving anisotropic poly-fractional operators, where the poly-fractional operator is of the form P((-Delta(g))(s))u: = Sigma(M)(i=1) alpha(i)(-Delta g(i))(si)u for s = (s(1), ..., s(M)), 0 < s(1) < center dot center dot center dot < s(M) < infinity, s(M) is an element of R+\Z, g = (g(1), ..., g(M)), and sufficiently regular coefficients alpha(i)(x). There are three major contributions in this work that are new to the literature. First, we propose equations involving such anisotropic poly-fractional operators P, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Levy flights. Secondly, we give novel results for the unique continuation properties for u when it is fractional polyharmonic, in the sense that u satisfies (P) over tilde((-Delta((g) over tilde))((s) over tilde))u = 0 in a bounded Lipschitz domain Omega for some (P) over tilde. Our unique continuation property holds for relatively general (P) over tilde, which is also anisotropic and in addition may not necessarily be coercive. With these results in hand, we consider the inverse problems for P, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator.