p-Harmonic obstacle problems

SummaryWe develop an interior partial regularity theory for vector valued Sobolev functions which locally minimize degenerate variational integrals under the additional side condition that all comparison maps take their values in the closure of a smooth region of the target space. Our results apply to the case of penergy minimizing mappings X → Y between Riemannian manifolds including target manifolds Y with nonvoid boundary.

[1]  Jean-Michel Coron,et al.  Harmonic maps with defects , 1986 .

[2]  Karen K. Uhlenbeck.  Regularity for a class of non-linear elliptic systems , 1977 .

[3]  F. Duzaar.  Variational inequalities and harmonic mappings. , 1987 .

[4]  M. Fuchs,et al.  The regularity of minima of variational problems with graph obstacles , 1989 .

[5]  J. Hutchinson,et al.  C1, α Partial regularity of functions minimising quasiconvex integrals , 1985 .

[6]  J. Jost,et al.  Boundary regularity for minima of certain quadratic functionals , 1983 .

[7]  M. Giaquinta,et al.  Remarks on the regularity of the minimizers of certain degenerate functionals , 1986 .

[8]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[9]  M. Fuchs.  Variational inequalities for vector-valued functions with non-convex obstacles , 1985 .

[10]  M. Giaquinta.  Multiple integrals in the calculus of variations and non linear elliptic systems , 1981 .

[11]  S. Hildebrandt,et al.  Some regularity results for quasilinear elliptic systems of second order , 1975 .

[12]  M. Fuchs.  Some remarks on the boundary regularity for minima of variational problems with obstacles , 1985 .

[13]  Brian White,et al.  Infima of energy functionals in homotopy classes of mappings , 1986 .

[14]  Stefan Hildebrandt,et al.  On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order , 1977 .

[15]  Karen Uhlenbeck,et al.  A regularity theory for harmonic maps , 1982 .

[16]  L. Caffarelli,et al.  Existence and regularity for a minimum problem with free boundary. , 1981 .

[17]  Enrico Giusti,et al.  On the regularity of the minima of variational integrals , 1982 .

[18]  M. Fuchs.  A note on removable singularities for minima of certain vector-valued obstacle problems , 1987 .

[19]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[20]  Enrico Giusti,et al.  The singular set of the minima of certain quadratic functionals , 1984 .

[21]  M. Fuchs,et al.  Optimal regularity theorems for variational problems with obstacles , 1986 .

[22]  F. Lin,et al.  Mappings minimizing the Lp norm of the gradient , 1987 .

[23]  A note on affine planes with transitive collineation groups , 1977 .

[24]  Karen Uhlenbeck,et al.  Boundary regularity and the Dirichlet problem for harmonic maps , 1983 .

[25]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[26]  Klaus Gornik.  Ein Stetigkeitssatz für Variationsprobleme mit Ungleichungen als Nebenbedingung , 1976 .

[27]  Mariano Giaquinta,et al.  Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105 , 1984 .

[28]  Stefan Hildebrandt,et al.  An existence theorem for harmonic mappings of Riemannian manifolds , 1977 .

[29]  Fuchs Martin.  A regularity theorem for energy minimizing maps of riemannian manifolds , 1987 .

[30]  David Kinderlehrer,et al.  Existence and partial regularity of static liquid crystal configurations , 1986 .

[31]  Elliott H. Lieb,et al.  Co-Area, Liquid Crystals, and Minimal Surfaces , 1988 .