The eigenvalue problem for the \(\infty \)-Bilaplacian

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Abstract

We consider the problem of finding and describing minimisers of the Rayleigh quotient
$$\begin{aligned} \Lambda _\infty \, :=\, \inf _{u\in \mathcal {W}^{2,\infty }(\Omega )\setminus \{0\} }\frac{\Vert \Delta u\Vert _{L^\infty (\Omega )}}{\Vert u\Vert _{L^\infty (\Omega )}}, \end{aligned}$$
where \(\Omega \subseteq \mathbb {R}^n\) is a bounded \(C^{1,1}\) domain and \(\mathcal {W}^{2,\infty }(\Omega )\) is a class of weakly twice differentiable functions satisfying either \(u=0\) on \(\partial \Omega \) or \(u=|\mathrm {D}u|=0\) on \(\partial \Omega \). Our first main result, obtained through approximation by \(L^p\)-problems as \(p\rightarrow \infty \), is the existence of a minimiser \(u_\infty \in \mathcal {W}^{2,\infty }(\Omega )\) satisfying
$$\begin{aligned} \left\{ \begin{array}{ll} \Delta u_\infty \, \in \, \Lambda _\infty \mathrm {Sgn}(f_\infty ) &{} \text { a.e. in }\Omega , \\ \Delta f_\infty \, =\, \mu _\infty &{} \text { in }\mathcal {D}'(\Omega ), \end{array} \right. \end{aligned}$$
for some \(f_\infty \in L^1(\Omega )\cap BV_{\text {loc}}(\Omega )\) and a measure \(\mu _\infty \in \mathcal {M}(\Omega )\), for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue \(\Lambda _\infty \) on the domain, establishing the validity of a Faber–Krahn type inequality: among all \(C^{1,1}\) domains with fixed measure, the ball is a strict minimiser of \(\Omega \mapsto \Lambda _\infty (\Omega )\). This result is shown to hold true for either choice of boundary conditions and in every dimension.

Keywords

\(\infty \)-Laplacian \(\infty \)-Bilaplacian \(\infty \)-Eigenvalue problem Calculus of variations in \(L^\infty \) Differential inclusions Multi-valued functions Faber–Krahn inequality Symmetrisations Bathtub principle 

Mathematics Subject Classification

35G30 35G20 35P15 35P30 49R05 35D99 35D40 35J91 
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References

  1. 1.
    Aronsson, G.: Minimization problems for the functional \(sup_x \cal{F}(x, f(x), f^{\prime }(x))\). Ark. Mat. 6, 33–53 (1965)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aronsson, G.: Minimisation problems for the functional \(sup_x \cal{F}(x, f(x), f^{\prime }(x))\) II. Ark. Mat. 6, 409–431 (1966)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ashbaugh, M., Benguria, R.: On Rayleigh’s conjecture for the clamped plate and its generalisation to three dimensions. Duke Math. J. 78, 1–17 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ashbaugh, M., Laugesen, R.: Fundamental tones and buckling loads of clamped plates. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(2), 383–402 (1996)MathSciNetMATHGoogle Scholar
  6. 6.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Universitext (2011) reprintGoogle Scholar
  7. 7.
    Coffman, ChV: On the structure of solutions to \({\Delta }^2 u = \lambda u\) which satisfy the clamped plate conditions on a right angle. SIAM J. Math. Anal. 13(5), 746–757 (1982)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Coffman, ChV, Duffin, R.J.: On the fundamental eigenfunctions of a clamped punctured disk. Adv. Appl. Math. 13, 142–151 (1992)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Crandall, M.G.: A visit with the \(\infty \)-Laplacian, in Calculus of Variations and Non-Linear Partial Differential Equations, Springer Lecture Notes in Mathematics 1927. CIME, Cetraro, Italy (2005)Google Scholar
  10. 10.
    Drábek, P., Otani, M.: Global bifurcation result for the \(p\)-biharmonic operator. Electron. J. Differ. Equ. 48, 1–19 (2001)MathSciNetMATHGoogle Scholar
  11. 11.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. AMS, Providence (1998)Google Scholar
  12. 12.
    Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Cambridge (2015). (revised edition)MATHGoogle Scholar
  13. 13.
    Fonseca, I., Leoni, G.: Modern methods in the Calculus of Variations: \(L^p\) Spaces, Springer Monographs in Mathematics. Springer, Berlin (2007)MATHGoogle Scholar
  14. 14.
    Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics. Springer, Berlin (2010)MATHGoogle Scholar
  15. 15.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, reprint of the 1998 edition. SpringerGoogle Scholar
  16. 16.
    Hardt, R., Simon, L.: Nodal sets for solutions of elliptic equations. J. Differ. Geom. 30, 505–522 (1989)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Juutinen, P., Lindqvist, P., Manfredi, J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2), 89–105 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Katzourakis, N.: An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in \(L^\infty \), Springer Briefs in Mathematics. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-12829-0 MATHGoogle Scholar
  19. 19.
    Katzourakis, N.: Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems. J. Differ. Equ. 263(1), 641–686 (2017).  https://doi.org/10.1016/j.jde.2017.02.048 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Katzourakis, N.: Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in \(L^\infty \). Calc. Var. Partial Differ. Equ. 56(1), 1–25 (2017).  https://doi.org/10.1007/s00526-016-1099-z MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Katzourakis, N., Moser, R.: Existence, Uniqueness and Structure of Second Order Absolute Minimisers. ArXiv preprint, http://arxiv.org/pdf/1701.03348.pdf
  22. 22.
    Katzourakis, N., Pryer, T.: On the numerical approximation of \(\infty \)-Harmonic mappings. NoDEA Nonlinear Differ. Equ. Appl. 23(6), 1–23 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Katzourakis, N., Pryer, T.: 2nd Order \(L^\infty \) Variational Problems and the \(\infty \)-Polylaplacian. ArXiv preprint, http://arxiv.org/pdf/1605.07880.pdf
  24. 24.
    Kesavan, S.: Symmetrisations and Applications. World Scientific, Singapore (2006)CrossRefGoogle Scholar
  25. 25.
    Kesavan, S.: Some remarks on a result of Talenti. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15, 453–465 (1988)MathSciNetMATHGoogle Scholar
  26. 26.
    Lieb, E., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  27. 27.
    Moser, R., Schwetlick, H.: Minimisers of a weighted maximum of the Gauss curvature. Ann. Glob. Anal. Geom. 41(2), 199–207 (2012)CrossRefMATHGoogle Scholar
  28. 28.
    Nadirashvili, N.: Rayleigh’s conjecture on the principal frequency of the clamped plate. Arch. Ration. Mech. Anal. 129, 1–10 (1995)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Parini, E., Ruf, B., Tarsi, C.: The eigenvalue problem for the 1-biharmonic operator. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13(2), 307–332 (2014)MathSciNetMATHGoogle Scholar
  30. 30.
    Parini, E., Ruf, B., Tarsi, C.: Higher-order functional inequalities related to the clamped 1-biharmonic operator. Ann. Mat. Pura Appl. (4) 194(6), 1835–1858 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sakellaris, Z.: Minimisation of scalar curvature in conformal geometry. Ann. Global Anal. Geom. 51(1), 73–89 (2017)Google Scholar
  32. 32.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3, 697–718 (1976)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of ReadingReadingUK
  2. 2.CNRS Centrale Marseille I2MAix Marseille UnivMarseilleFrance

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