Existence of positive definite noncoercive sums of squares

Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.Comment: 27 page

Optimal solvability for the Dirichlet and Neumann problem in dimension two

We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p, p>2/3, where This in turn implies that solutions to the Dirichlet problem with data in the Holder class C^{1/2}(\partial D) are themselves in C^{1/2}(D). Both of these results are sharp. In fact, we prove a more general statement regarding the H^p solvability for divergence form elliptic equations with bounded measurable coefficients. We also prove similar solvability result for the regularity and Dirichlet problem for the biharmonic equation on Lipschitz domains

Counterexamples and uniqueness for Lp(∂Ω) oblique derivative problems

AbstractHarmonic functions defined in Lipschitz domains of the plane that have gradient nontangentially in L2 and have nonnegative oblique derivative almost everywhere on the boundary with respect to a continuous transverse vector field are shown to be constant. Explicit examples that have almost everywhere vanishing oblique derivative are constructed when L2 is replaced by Lp, p<2. Explicit examples with vanishing oblique derivative are constructed when p⩽2 and the continuous vector field is replaced by large perturbations of the normal vector field. Optimal bounds on the perturbation, depending on p⩽2 and the Lipschitz constant, are given which imply that only the constant solution has nonnegative oblique derivative almost everywhere. Examples are constructed in higher dimensions and the Fredholm properties of certain nonvariational layer potentials discussed

The mixed problem for harmonic functions in polyhedra

R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses