## Taiwanese Journal of Mathematics

### MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION

#### Abstract

This paper presents the existence of minimizers and $\Gamma$-convergence for the energey functionals \begin{eqnarray*} E_\epsilon(u) = \int_\Omega \left\{ W(u(x))+\epsilon{|\nabla u(x)|^p}\right\} dx, \mbox{ for all }\epsilon>0,\quad p>1 \end{eqnarray*} with Neumann boundary condition and the constraint \begin{eqnarray*} \int_\Omega u(x) dx = m|\Omega|, \mbox{ where }0 \lt m \lt 1. \end{eqnarray*} The energy functionals discussed in this paper are associated with the Euler-Lagrange $p$-Laplacian equation. We employ the direct method in the calculus of variations to show the existence of minimizers. The $\Gamma$-convergence is achieved with the help of coarea formula and Young's inequality.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 6B (2009), 2021-2036.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405655

Digital Object Identifier
doi:10.11650/twjm/1500405655

Mathematical Reviews number (MathSciNet)
MR2583535

Zentralblatt MATH identifier
1192.49017

#### Citation

Chang, Mao-Sheng; Lee, Shu-Cheng; Yen, Chien-Chang. MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION. Taiwanese J. Math. 13 (2009), no. 6B, 2021--2036. doi:10.11650/twjm/1500405655. https://projecteuclid.org/euclid.twjm/1500405655

#### References

• A. Aftalion and F. Pacella, Morse index and uniqueness for positive solutions of radial $p$-Laplace equations, Trans. Amer. Math. Soc., 356 (2004), 4255-4272.
• S. Baldo, Minimal Interface Criterion for Phase Transitions in Mixtures of Cahn-Hiliard Fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(2) (1990), 67-90.
• P. A. Binding, P. Drábek and Y. X. Huang, On Neumann Boundary Value Problems for Some Quasilinear Elliptic Equations, Electronic Journal of Differential Equations, 1997(5) (1997), 1-11.
• A. Braides, $\Gamma$-convergence for beginners, Vol. 22 (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 2002.
• B. M. Brown and W. Reichel, Eigenvalues of the radially symmetric $p$-Laplacian in $\reals^n$, J. London Math. Soc., 69(2) (2004), 657-675.
• S. Conti, I. Fonseca and G. Leoni, A $\Gamma$-convergence result for the two-gradient theory of phase transitions, Comm. Pure Appl. Math., Vol. LV (2002), 0857-0936.
• B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, 1989.
• G. Dal Maso, An introduction to $\Gamma$-convergence, Birkhäuser Boston Inc., Boston, MA, 1993.
• L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998.
• L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
• I. Fonseca and L. Tartar, The Gradient Theory of Phase Transitions for Systems with Two Potential Wells, Proc. Roy. Soc. Edinburgh, A-111 (1989), 89-102.
• M. E. Gurtin, Some results and conjectures in the gradient theory of phase transitions, Institute for Mathematics and Its Applications, University of Minnesota preprint, n., 156 (1985).
• P. Hartman, Ordinary Differential Equations, John Wiley and Sons, 1964.
• W. Jin and R. V. Kohn, Singular perturbation and the energy of folds, J. Non-linear Sci., 10 (2000), 355-390.
• R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect., A. 111 (1989), 69-84.
• E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 1997.
• F. H. Lin and X. P. Yang, Geometric Measure Theory-An Introduction, Science Press, Cambridge, 2002.
• L. Modica, The Gradient Theory of Phase Transitions and the Minimal Interface Criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
• L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital., 14-B (1977), 285-299.
• N. C. Owen, Nonconvex Variational Problems with General Singular Perturbations, Tran. American Math. Soc., 310 (1988), 393-404.
• E. Sandier and S. Serfaty, Gamma-convergence of Gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., Vol. LVII (2004), 1627-1672.
• P. Sternberg, The Effect of a Singular Perturbation on Non-Convex Variational Problems, Arch. Rational Mech. Anal., 101 (1988), 209-260.