Advertisement

Nonexistence of an a priori bound at the center in terms of anL 1 bound on the boundary for solutions of uniformly elliptic equations on a sphere

  • 26 Accesses

  • 3 Citations

Summary

Positive solutions on the unit sphere of second order uniformly elliptic nonvariational equations are found which exhibit behavior sharply differing from that of positive harmonic functions, despite the fact that the coefficients may be uniformly arbitrarily close (at least for n=2) to those of the Laplacian. A solution is found whose boundary integral on expanding concentric subspheres tends to zero and another is found for which this boundary iutegral tends to infinity.

References

  1. [1]

    K. Miller,Barriers on cones for uniformly elliptic operators, (to appear).

  2. [2]

    C. Miranda,Equazioni alle Derivate Parziali di Tipo Ellitico, Springer-Verlag, Berlin, 1955, p. 128 and p. 162.

  3. [3]

    C. Pucci,Operatori ellittici estremanti, « Ann Mat. pura ed appl. », LXXII (1966), pp. 141–170.

Download references

Author information

Additional information

This work was supported by the Air Force Office of Scientific Research and the National Academy of Sciences through a Postdoctoral Research Fellowship for 1964–1965 at the University of Genova.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Miller, K. Nonexistence of an a priori bound at the center in terms of anL 1 bound on the boundary for solutions of uniformly elliptic equations on a sphere. Annali di Matematica 73, 11–16 (1966). https://doi.org/10.1007/BF02415078

Download citation

Keywords

  • Harmonic Function
  • Elliptic Equation
  • Unit Sphere
  • Positive Harmonic Function