## Grants and Contracts Details

### Description

Intellectual Merit. The intellectual focus of the present proposal is in the area of intersection
between harmonic analysis and partial dierential equations, with an emphasis on the study of
elliptic boundary value problems for second and higher order operators and for systems in non-
smooth domains (domains with either isolated singularities or Lipschitz domains). While elliptic
boundary value problems for the Laplacian have been well-understood for over 20 years, the theory
for second order elliptic systems and higher order elliptic equations is not fully developed. The
three main research directions of the proposal are brie
y described below.
1. Boundary value problems for higher order elliptic operators. The layer potential
method has proved to be very successful in the treatment of second order elliptic boundary value
problems, but currently this approach is insuciently developed to deal with the intricacies of
the theory of higher order operators. Building on recent progress in the employment of potential
methods for the Neumann problem for the biharmonic equation on non-smooth domains, this
project will undertake a systematic treatment of the spectral properties of the layer potential
operators naturally associated with general higher order elliptic boundary value problems in smooth
and non-smooth domains in two dimensions.
2. The spectral radius conjecture on Besov spaces. Solving boundary value problems
via the method of layer potentials typically reduces the original problem to that of inverting an
operator of the form I + K on appropriate (boundary) function spaces, where I is the identity
operator. This research theme examines conditions under which the spectrum of K on acting on
Besov spaces is strictly less than 1. Here, the operator K is the boundary version of the double
layer potential operator associated to the Laplacian or the Lame system. For arbitrary Lipschitz
domains, this issue is occasionally referred to as the spectral radius conjecture.
3. Well-posedness of the mixed problem. The Dirichlet and Neumann boundary value
problems are special cases of the more general situation when mixed boundary conditions are
considered. This research theme investigates open problems related to the well-posedness of the
mixed problem for the Laplacian and the Lame system of elastostatics. Specically, the investigation
will focus on the following two topics: a) sharp well-posedness of the mixed problem in Lipschitz
domains; b) the mixed problem for the Lame system of elastostatics.
Broader Impacts. The research themes of this proposal are motivated by problems that naturally
arise in mathematical physics and engineering. In this regard, the non-smooth setting in which these
problems are posed is fundamental since most realistic physical models involve irregular domains.
For example, (A) has applications to engineering in the context of modeling shallow shells, beam
bending, and clamped plates. Problems falling under (C) model the behavior of several physical
quantities such as the temperature in a metallurgical melting process, the thermoelastic potential of
an elastic solid punched or stamped by a heated object, or the seepage through a porous material.
In parallel with pursuing the research directions outlined above, the proposer intends to con-
tinue her commitment to activities aimed at increasing the participation of women and other
under-represented groups in mathematics. Two examples of these outreach activities are the Sonia
Kovalevsky High School Mathematics Day and the AWM presence at the USA Science and Engi-
neering Festival Expo. In the classroom, the PI will continue to develop a proof-writing course in
Number Theory for sophomores and juniors majoring in mathematics to help ease the transition
to upper division mathematics courses. The proposer will also seek to popularize and promote
mathematics by publishing expository articles related to mathematics. Recently, her articles have
appeared in the Notices of the AMS, the AWM Newsletter, the magazine Science, and SIAM News.
1

Status | Finished |
---|---|

Effective start/end date | 6/1/12 → 6/30/14 |

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