Emeritus Professor, Department
of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana, Illinois 61801
Office: 309
Altgeld Hall
Office Telephone: 333-3699
e-mail: l-helms@math.uiuc.edu
General Information
Ph. D., Purdue University, 1956
Mathematical Interests
My interests lie in three interrelated topics: heat equations associated
with second-order elliptic operators, Markov or diffusion processes, and
potential theory. In the early 1950s, W. Feller characterized one-dimensional
diffusions by representing their infinitesimal generators intrinsically
and determined all possible boundary conditions which determine the domain
of the generator. In 1959, Ventcel characterized the infinitesimal generators
of general diffusion processes on bounded domains in higher dimensions
as a second-order elliptic operator subject to boundary conditions involving
diffusion, absorption, reflection, and viscosity at the boundary. The problem
of showing that a second-order elliptic operator subject to such boundary
conditions generates a Markov or diffusion process is in its infancy. The
best results obtained so far involve a nondegenerate second-order elliptic
operator subject to oblique derivative boundary conditions.
Selected
Publications and Comments
Books
Tables of contents for the second and third books of the following list can be viewed.