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August  2014, 7(4): 761-766. doi: 10.3934/dcdss.2014.7.761

A clamped plate with a uniform weight may change sign

1. 

Fakultät für Mathematik, Otto-von-Guericke Universität, Postfach 4120, 39016 Magdeburg, Germany

2. 

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Received  July 2013 Revised  September 2013 Published  February 2014

It is known that the Dirichlet bilaplace boundary value problem, which is used as a model for a clamped plate, is not sign preserving on general domains. It is also known that the corresponding first eigenfunction may change sign. In this note we will show that even a constant right hand side may result in a sign-changing solution.
Citation: Hans-Christoph Grunau, Guido Sweers. A clamped plate with a uniform weight may change sign. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 761-766. doi: 10.3934/dcdss.2014.7.761
References:
[1]

L. Bauer and E. Reiss, Block five diagonal metrics and the fast numerical computation of the biharmonic equation,, Math. Comp., 26 (1972), 311.  doi: 10.1090/S0025-5718-1972-0312751-9.  Google Scholar

[2]

T. Boggio, Sull'equilibrio delle piastre elastiche incastrate,, Rend. Acc. Lincei, 10 (1901), 197.   Google Scholar

[3]

T. Boggio, Sulle funzioni di Green d'ordine $m$,, Rend. Circ. Mat. Palermo, 20 (1905), 97.   Google Scholar

[4]

Ch. V. Coffman, 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 (1982), 746.  doi: 10.1137/0513051.  Google Scholar

[5]

Ch. V. Coffman and R. J. Duffin, On the fundamental eigenfunctions of a clamped punctured disk,, Adv. in Appl. Math., 13 (1992), 142.  doi: 10.1016/0196-8858(92)90006-I.  Google Scholar

[6]

Ch. V. Coffman, R. J. Duffin and D. H. Shaffer, The fundamental mode of vibration of a clamped annular plate is not of one sign,, in Constructive approaches to mathematical models (Proc. Conf. in honor of R. J. Duffin, (1978), 267.   Google Scholar

[7]

A. Dall'Acqua and G. Sweers, On domains for which the clamped plate system is positivity preserving,, in Partial Differential Equations and Inverse Problems, 362 (2004), 133.  doi: 10.1090/conm/362/06609.  Google Scholar

[8]

A. Dall'Acqua and G. Sweers, The clamped-plate equation for the limaçon,, Ann. Mat. Pura Appl., (4) 184 (2005), 361.  doi: 10.1007/s10231-004-0121-9.  Google Scholar

[9]

A. Dall'Acqua and G. Sweers, Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems,, J. Differential Equations, 205 (2004), 466.  doi: 10.1016/j.jde.2004.06.004.  Google Scholar

[10]

R. J. Duffin, On a question of Hadamard concerning super-biharmonic functions,, J. Math. Phys., 27 (1949), 253.   Google Scholar

[11]

M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization,, Proc. Natl. Acad. Sci. USA, 109 (2012), 14761.  doi: 10.1073/pnas.1120432109.  Google Scholar

[12]

P. R. Garabedian, A partial differential equation arising in conformal mapping,, Pacific J. Math., 1 (1951), 485.  doi: 10.2140/pjm.1951.1.485.  Google Scholar

[13]

F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains,, Lecture Notes in Mathematics, (1991).  doi: 10.1007/978-3-642-12245-3.  Google Scholar

[14]

H.-Ch. Grunau and F. Robert, Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions,, Arch. Ration. Mech. Anal., 195 (2010), 865.  doi: 10.1007/s00205-009-0230-0.  Google Scholar

[15]

H.-Ch. Grunau and G. Sweers, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions,, Math. Nachr., 179 (1996), 89.  doi: 10.1002/mana.19961790106.  Google Scholar

[16]

H.-Ch. Grunau and G. Sweers, Sign change for the Green function and the first eigenfunction of equations of clamped-plate type,, Arch. Ration. Mech. Anal., 150 (1999), 179.  doi: 10.1007/s002050050185.  Google Scholar

[17]

H.-Ch. Grunau and G. Sweers, In any dimension a "clamped plate" with a uniform weight may change sign,, Nonlinear Anal. A: T. M. A., 97 (2014), 119.  doi: 10.1016/j.na.2013.11.017.  Google Scholar

[18]

J. Hadamard, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées,, in Oeuvres de Jacques Hadamard, 33 (1968), 515.   Google Scholar

[19]

J. Hadamard, Sur certains cas intéressants du problème biharmonique,, in Oeuvres de Jacques Hadamard, (1968), 1297.   Google Scholar

[20]

V. A. Kozlov, V. A. Kondrat'ev and V. G. Maz'ya, On sign variation and the absence of "strong'' zeros of solutions of elliptic equations,, Math. USSR Izvestiya, 34 (1990), 337.   Google Scholar

[21]

Ch. Loewner, On generation of solutions of the biharmonic equation in the plane by conformal mappings,, Pacific J. Math., 3 (1953), 417.  doi: 10.2140/pjm.1953.3.417.  Google Scholar

[22]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rational Mech. Anal., 98 (1987), 167.  doi: 10.1007/BF00251232.  Google Scholar

[23]

G. Szegö, On membranes and plates,, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 210.  doi: 10.1073/pnas.36.3.210.  Google Scholar

[24]

G. Sweers, When is the first eigenfunction for the clamped plate equation of fixed sign?, in Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, (2000), 285.   Google Scholar

show all references

References:
[1]

L. Bauer and E. Reiss, Block five diagonal metrics and the fast numerical computation of the biharmonic equation,, Math. Comp., 26 (1972), 311.  doi: 10.1090/S0025-5718-1972-0312751-9.  Google Scholar

[2]

T. Boggio, Sull'equilibrio delle piastre elastiche incastrate,, Rend. Acc. Lincei, 10 (1901), 197.   Google Scholar

[3]

T. Boggio, Sulle funzioni di Green d'ordine $m$,, Rend. Circ. Mat. Palermo, 20 (1905), 97.   Google Scholar

[4]

Ch. V. Coffman, 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 (1982), 746.  doi: 10.1137/0513051.  Google Scholar

[5]

Ch. V. Coffman and R. J. Duffin, On the fundamental eigenfunctions of a clamped punctured disk,, Adv. in Appl. Math., 13 (1992), 142.  doi: 10.1016/0196-8858(92)90006-I.  Google Scholar

[6]

Ch. V. Coffman, R. J. Duffin and D. H. Shaffer, The fundamental mode of vibration of a clamped annular plate is not of one sign,, in Constructive approaches to mathematical models (Proc. Conf. in honor of R. J. Duffin, (1978), 267.   Google Scholar

[7]

A. Dall'Acqua and G. Sweers, On domains for which the clamped plate system is positivity preserving,, in Partial Differential Equations and Inverse Problems, 362 (2004), 133.  doi: 10.1090/conm/362/06609.  Google Scholar

[8]

A. Dall'Acqua and G. Sweers, The clamped-plate equation for the limaçon,, Ann. Mat. Pura Appl., (4) 184 (2005), 361.  doi: 10.1007/s10231-004-0121-9.  Google Scholar

[9]

A. Dall'Acqua and G. Sweers, Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems,, J. Differential Equations, 205 (2004), 466.  doi: 10.1016/j.jde.2004.06.004.  Google Scholar

[10]

R. J. Duffin, On a question of Hadamard concerning super-biharmonic functions,, J. Math. Phys., 27 (1949), 253.   Google Scholar

[11]

M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization,, Proc. Natl. Acad. Sci. USA, 109 (2012), 14761.  doi: 10.1073/pnas.1120432109.  Google Scholar

[12]

P. R. Garabedian, A partial differential equation arising in conformal mapping,, Pacific J. Math., 1 (1951), 485.  doi: 10.2140/pjm.1951.1.485.  Google Scholar

[13]

F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains,, Lecture Notes in Mathematics, (1991).  doi: 10.1007/978-3-642-12245-3.  Google Scholar

[14]

H.-Ch. Grunau and F. Robert, Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions,, Arch. Ration. Mech. Anal., 195 (2010), 865.  doi: 10.1007/s00205-009-0230-0.  Google Scholar

[15]

H.-Ch. Grunau and G. Sweers, Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions,, Math. Nachr., 179 (1996), 89.  doi: 10.1002/mana.19961790106.  Google Scholar

[16]

H.-Ch. Grunau and G. Sweers, Sign change for the Green function and the first eigenfunction of equations of clamped-plate type,, Arch. Ration. Mech. Anal., 150 (1999), 179.  doi: 10.1007/s002050050185.  Google Scholar

[17]

H.-Ch. Grunau and G. Sweers, In any dimension a "clamped plate" with a uniform weight may change sign,, Nonlinear Anal. A: T. M. A., 97 (2014), 119.  doi: 10.1016/j.na.2013.11.017.  Google Scholar

[18]

J. Hadamard, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées,, in Oeuvres de Jacques Hadamard, 33 (1968), 515.   Google Scholar

[19]

J. Hadamard, Sur certains cas intéressants du problème biharmonique,, in Oeuvres de Jacques Hadamard, (1968), 1297.   Google Scholar

[20]

V. A. Kozlov, V. A. Kondrat'ev and V. G. Maz'ya, On sign variation and the absence of "strong'' zeros of solutions of elliptic equations,, Math. USSR Izvestiya, 34 (1990), 337.   Google Scholar

[21]

Ch. Loewner, On generation of solutions of the biharmonic equation in the plane by conformal mappings,, Pacific J. Math., 3 (1953), 417.  doi: 10.2140/pjm.1953.3.417.  Google Scholar

[22]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,, Arch. Rational Mech. Anal., 98 (1987), 167.  doi: 10.1007/BF00251232.  Google Scholar

[23]

G. Szegö, On membranes and plates,, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 210.  doi: 10.1073/pnas.36.3.210.  Google Scholar

[24]

G. Sweers, When is the first eigenfunction for the clamped plate equation of fixed sign?, in Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, (2000), 285.   Google Scholar

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