arXiv subjects

Ariel Barton

Publications and source records attributed to Ariel Barton.

21 records · Page 2Linked to original sources

arXiv2013

The Dirichlet problem for higher order equations in composition form

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=\Div B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian Δ^2 from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L^2, and with optimal estimates in terms of nontangential maximal functions and square functions.

math.AP↗

arXiv2012

A new class of harmonic measure distribution functions

Let D be a planar domain containing 0. Let h_D(r) be the harmonic measure at 0 in D of the part of the boundary of D within distance r of 0. The resulting function h_D is called the harmonic measure distribution function of D. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X_n such that h_{X_n} converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f = h_D, where D is a subsequential limit of bounded simply connected domains that approximate the domains X_n. Further, the limit domain is unique in a class of suitably symmetric domains. Thus f = h_D for a unique symmetric bounded simply connected domain D.

math.CV↗

arXiv2009

Elliptic Partial Differential Equations with Complex Coefficients

In this paper we investigate elliptic partial differential equations on Lipschitz domains in the plane whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates. We show that for Dirichlet boundary data in L^p for p large enough, solutions exist and are controlled by the L^p-norm of the boundary data. Similarly, for Neumann boundary data in L^q, or for Dirichlet boundary data whose tangential derivative is in L^q (regularity boundary data), for q small enough, we show that solutions exist and are controlled by the L^q-norm of the boundary data. We prove similar results for Neumann or regularity boundary data in the Hardy space H^1, and for bounded or BMO Dirichlet boundary data. Finally, we show some converses: if the solutions are controlled in some sense, then Dirichlet, Neumann, or regularity boundary data must exist.

math.AP↗