Spruck is well known in the field of elliptic
partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration with
Luis Caffarelli,
Joseph J. Kohn, and
Louis Nirenberg. These papers were among the first to develop a general theory of second-order elliptic differential equations which are fully nonlinear, with a regularity theory that extends to the boundary. Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of
geometric analysis since many geometric partial differential equations are amenable to its methods. With
Basilis Gidas, Spruck studied positive solutions of subcritical second-order elliptic partial differential equations of
Yamabe type. With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a
positive mass-style theorem on the asymptotic behavior of isolated singularities. In 1974, Spruck and
David Hoffman extended a
mean curvature-based
Sobolev inequality of James H. Michael and
Leon Simon to the setting of submanifolds of
Riemannian manifolds. This has been useful for the study of many analytic problems in geometric settings, such as for
Gerhard Huisken's study of
mean curvature flow in Riemannian manifolds and for
Richard Schoen and
Shing-Tung Yau's study of the Jang equation in their resolution of the
positive energy theorem in
general relativity. In the late 80s,
Stanley Osher and
James Sethian developed the
level-set method as a computational tool in
numerical analysis. In collaboration with
Lawrence Evans, Spruck pioneered the rigorous study of the level-set flow, as adapted to the
mean curvature flow. The level-set approach to mean curvature flow is important in the technical ease with topological change that can occur along the flow. The same approach was independently developed by Yun Gang Chen,
Yoshikazu Giga, and Shun'ichi Goto. The works of Evans–Spruck and Chen–Giga–Goto found significant application in
Gerhard Huisken and
Tom Ilmanen's solution of the
Riemannian Penrose inequality of
general relativity and
differential geometry, where they adopted the level-set approach to the
inverse mean curvature flow. In 1994 Spruck was an invited speaker at the
International Congress of Mathematicians in Zurich. ==Major publications==