Revista Matemática Iberoamericana

Analysis of the free boundary for the $p$-parabolic variational problem $(p\ge 2)$

Henrik Shahgholian

Full-text: Open access

Abstract

Variational inequalities (free boundaries), governed by the $p$-parabolic equation ($p\geq 2$), are the objects of investigation in this paper. Using intrinsic scaling we establish the behavior of solutions near the free boundary. A consequence of this is that the time levels of the free boundary are porous (in $N$-dimension) and therefore its Hausdorff dimension is less than $N$. In particular the $N$-Lebesgue measure of the free boundary is zero for each $t$-level.

Article information

Source
Rev. Mat. Iberoamericana, Volume 19, Number 3 (2003), 797-812.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1077293806

Mathematical Reviews number (MathSciNet)
MR2053564

Zentralblatt MATH identifier
1060.35065

Subjects
Primary: 35K55: Nonlinear parabolic equations 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40] 35K65: Degenerate parabolic equations 35R35: Free boundary problems

Keywords
variational problem inhomogeneous $p$-parabolic equation free boundary porosity

Citation

Shahgholian, Henrik. Analysis of the free boundary for the $p$-parabolic variational problem $(p\ge 2)$. Rev. Mat. Iberoamericana 19 (2003), no. 3, 797--812. https://projecteuclid.org/euclid.rmi/1077293806


Export citation

References

  • Apushkinskaya, D. E., Shahgholian, H. and Uraltseva, N. N.: Boundary estimates for solutions to the parabolic free boundary problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000). Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 39-55, 313.
  • Barenblatt, G. I.: On self-similar motions of a compressible fluid in a porous medium. (Russian)Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 679-698.
  • Caffarelli, L., Karp, L. and Shahgholian, H.: Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 152 (2000), 269-292.
  • Caffarelli, L., Petrosyan, A. and Shahgholian, H.: Regularity of a free boundary in parabolic potential theory, manuscript.
  • DiBenedetto, E.: Degenerate parabolic equations. Universitext. Springer-Verlag, New York, 1993.
  • Friedman, A.: Variational principles and free-boundary problems (second edition). Robert E. Krieger Publishing Co., Inc., Malabar, Florida, 1988.
  • Heinonen, J., Kilpeläinen, T. and Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.
  • Karp, L. and Shahgholian, H.: On the optimal growth of functions with bounded Laplacian. Electron. J. Differential Equations 2000, no. 03, 9 pp.
  • Karp, L., Kilpeläinen, T., Petrosyan, A. and Shahgholian, H.: On the porosity of free boundaries in degenerate variational inequalities. J. Differential Equations 164 (2000), no. 1, 110-117.
  • Kinderlehrer, D. and Stampacchia, G.: An introduction to variational inequalities and their applications. Pure and Applied Mathematics 88. Academic Press, Inc., New York-London, 1980.
  • Lindqvist, P.: A criterion of Petrowsky's kind for a degenerate quasilinear parabolic equation. Rev. Mat. Iberoamericana 11 (1995) no. 3, 569-578.
  • Lee, K. and Shahgholian, H.: Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 54 (2001), 43-56.
  • Martio, O. and Vuorinen, M.: Whitney cubes, $p$-capacity, and Minkowski content. Exposition. Math. 5 (1987), no. 1, 17-40.
  • Trudinger, N. S.: Pointwise estimates and quasilinear parabolic equations. Comm. Pure. Appl. Math. 21 (1968), 205-226.