Critical perturbations of elliptic operators by lower order terms

In this work we study issues of existence and uniqueness of solutions of certain boundary value problems for elliptic equations in the upper half-space. More specifically we treat the Dirichlet, Neumann, and Regularity problems for the general second order, linear, elliptic operator under a smallness assumption on the coefficients in certain critical Lebesgue spaces. Our results are perturbative in nature, asserting that if a certain operator L[subscript 0] has good properties (as far as boundedness and invertibility of certain associated solution operators), then the same is true for L[subscript 1], whenever the coefficients of these two operators are close in certain L[subscript p] spaces. Our approach is through the theory of layer potentials, though the lack of good estimates for solutions of L [equals] 0 force us to use a more abstract construction of these objects, as opposed to the more classical definition through the fundamental solution. On the other hand, these more general objects suggest a wider range of applications for these techniques. The results contained in this thesis were obtained in collaboration with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi. The resulting publications can be found in [BHL+a] and [BHL+b].

Lower Bounds for Solutions of Parabolic Differential Inequalities

Let P be the parabolic differential operatorwhere E is a linear elliptic operator of second order on D × [0, ∞), D being a bounded domain in Rn. The asymptotic behaviour of solutions u(x, t) of differential inequalities of the form1has been investigated by Protter (4). He found conditions on the functions ƒ and g under which solutions of (1), vanishing on the boundary of D and tending to zero with sufficient rapidity as t → ∞, vanish identically for all t ⩾ 0. Similar results have been found by Lees (1) for parabolic differential inequalities in Hilbert space.