| In the first part of the thesis, we address the strong unique continuation properties for 1D higher order parabolic partial differential equations with coefficients in the Gevrey class Gsigma for sigma > 1. We establish a quantitative estimate of unique continuation (observability estimate) under a mild assumption on the Gevrey exponent sigma. Also, we improve the existing upper bounds on the size of the level sets of solutions and remove the analyticity requirement on the coefficients. Next, we consider the strong unique continuation problem for elliptic and parabolic equations in higher space dimensions. As an application, we provide a polynomial upper bound on the Hausdorff measure of the nodal (zero) sets of solutions in terms of the coefficients. In particular, we cover the case of the Navier-Stokes equations with non-analytic forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and for the heat operator. |
