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Yury Grabovsky, a member of the Mathematics Department since 1999, performs the majority of his research in the field of calculus of variations and its applications to continuum mechanics. His work in the calculus of variations recently led to the solution of an important open problem originating in the early 20th century.
Calculus of variations was born in 1696 when Johann Bernoulli posed the question of which curve between two points provides the quickest descent, assuming a constant force, like gravity, and no friction. The answer was discovered to be the arc of a cycloid, which is the curve traced by a point on a rolling wheel. In general, classical variational problems try to find a curve in space that is a local minimum (represented visually by the lowest point on a graph) of a functional (a function on functions) — a quantity that depends on the curve as a whole, such as descent time in Bernoulli's problem.
From the time of their birth in the late 17th century, variational problems were solved by finding critical points of functionals like the cycloid minimizer in Bernoulli's problem. Then, if several solutions were found, the "correct" one was selected based on physical intuition. This method that didn't deal rigorously with saddle points; minima in some dimensions, but not others, where the graph of a function looks like a saddle or mountain pass. It was not until Karl Weierstrass in the early 20th century that a set of necessary and sufficient conditions was found that guarantees that a critical point is a local minimum rather than a saddle point.
Using calculus of variations tools developed over the last century, Grabovsky, along with former graduate student Tadele Mengesha, solved a modern, vastly more complicated, long-standing version of the Weierstrass sufficiency problem in which the unknown is a vector field — a function assigning a vector to every point in n-dimensional space. The ability to tell which critical points are local minima has implications for the study of shape-memory materials and non-linear elasticity.
Grabovsky's next step with regard to the calculus of variations is investigating how to treat vector fields that have singularities like phase boundaries, the regions where physical fields change abruptly, as observed in shape-memory alloys. Singularities give rise to a slew of new necessary conditions, but there is currently very little understanding of their true nature and interrelations.
The other branch of Grabovsky's research deals with the problem of exact relations for composite materials. Composite materials are never entirely homogeneous, though they may behave that way on a macroscopic level. Grabovsky and his colleagues are attempting to describe special conditions under which some properties of a composite are independent of the composite's microstructure.
Grabovsky's interest in materials has led to the sort of interdisciplinary work that can inject fresh viewpoints into disciplines. He has published articles in journals such as Mechanics of Material as well as Advances in Applied Mathematics.
Continuously funded by the National Science Foundation for over a decade, Grabovsky awards include a Sloan Doctoral Dissertation Fellowship, the New York Academy of Sciences Minoru & Ethel Tsutsui Distinguished Graduate Research Award, the Courant Institute Magnus Award, and the Monroe H. Martin Prize.
(posted April 2009)