Differential Equations and Complex Analysis Research Group

Ronald Gariepy

Research Description

There is a satisfactory theory for a wide range of variational problems and partial differential equations that guarantees the existence of solutions if the term "solution" is interpreted in a suitably general sense. My research primarily concerns regularity theory which addresses the problem of determining the extent to which these "generalized solutions" are solutions in the usual or classical sense and describing the detailed, i.e., pointwise, behavior of such solutions. Many of the ideas used in this kind of study are described in the book [4].

The types of problems that I consider often arise as mathematical models for physical phemomena, e.g., the equilibrium behavior of a nonlinearly elastic material or the flow of water through partially saturated soil.

    Selected Publications

  1. (with L. C. Evans) Blow-up, compactness, and partial regularity in the calculus of variations. Indiana Univ. Math. J. 36 (1987), 361-371.
  2. A Caccioppoli inequality and partial regularity, Proc. Roy. Soc. Edinburgh Sect. A. 112A (1989), 249-255.
  3. (with L. C. Evans) Partial regularity for constrained minimizers of convex or quasiconvex funcitonals, Rend. Sem. Mat. univ. Politec. Torino, Fascicolo Speciale (1989), Nonlinear PDE's, 75-79.
  4. (with L. C. Evans) Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1991, ISBN 0-8493-7157-0.
  5. (with Ph. Bénilan) Strong solutions in L1 of degenerate parabolic equations, Journal of Differential Equations119 (1995), 473-502.
  6. (with Ph. Bénilan, L. Boccardo, Th. Gallouët, M. Pierre, J. Vasquez) An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa XXII, 2 (1995), 241-273.
  7. (with L. C. Evans and M. Feldman) Fast/slow diffusion and collapsing sandpiles, Journal of Differential Equations,  137 (1997) 166--209.
  8. (with M. Shillor and X. Xu) Existence of generalized weak solutions to a model for In Situ Vitrification, European Journal for Applied Mathematics, 9 (1998) 543--559.
  9. (with  L. C. Evans) On the partial regularity of energy-minimizing area-preserving maps, Calculus of Variations and Partial Differential Equations, 9  (1999) 357--372.
  10. (with M G. Crandall and L.C. Evans) Optimal Lipchitz extensions and the infinity Laplacian, Calcuus of Variations and Partial Differential Equations, 13  (2001), no. 2, 123--139.
  11. (with Ph. Bénilan and L. C. Evans) On some singular limits of homogeneous semigroups, Journal of Evolution Equations, 3  (2003), 169-201.