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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), 311-326. doi: 10.1090/S0025-5718-1972-0312751-9.  Google Scholar

[2]

T. Boggio, Sull'equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei, 10 (1901), 197-205. Google Scholar

[3]

T. Boggio, Sulle funzioni di Green d'ordine $m$, Rend. Circ. Mat. Palermo, 20 (1905), 97-135. 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), 746-757. 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), 142-151. 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, Pittsburgh, Pa., 1978), Academic Press, 1979, 267-277.  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, (eds. C. Conca, R. Manasevich, G. Uhlmann and M. Vogelius), AMS, Contemp. Math., 362 (2004), 133-144. 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), 361-374. 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), 466-487. 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), 253-258.  Google Scholar

[11]

M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization, Proc. Natl. Acad. Sci. USA, 109 (2012), 14761-14766. doi: 10.1073/pnas.1120432109.  Google Scholar

[12]

P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math., 1 (1951), 485-524. 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, Springer-Verlag, Berlin, 2010. 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), 865-898. 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), 89-102. 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), 179-190. 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), 119-124. 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, Tome II, CNRS Paris, (1968), 515-641; reprint of Mémoires présentés par divers savants a l'Académie des Sciences, 33 (1908), 1-128. Google Scholar

[19]

J. Hadamard, Sur certains cas intéressants du problème biharmonique, in Oeuvres de Jacques Hadamard, Tome III, CNRS Paris, (1968), 1297-1299; reprint of Atti IV Congr. Intern. Mat. Rome, (1908), 12-14. 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), 337-353.  Google Scholar

[21]

Ch. Loewner, On generation of solutions of the biharmonic equation in the plane by conformal mappings, Pacific J. Math., 3 (1953), 417-436. 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), 167-177. doi: 10.1007/BF00251232.  Google Scholar

[23]

G. Szegö, On membranes and plates, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 210-216. 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), Electron. J. Diff. Eqns. Conf., 6, Southwest Texas State Univ., San Marcos, TX, 2001, 285-296.  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), 311-326. doi: 10.1090/S0025-5718-1972-0312751-9.  Google Scholar

[2]

T. Boggio, Sull'equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei, 10 (1901), 197-205. Google Scholar

[3]

T. Boggio, Sulle funzioni di Green d'ordine $m$, Rend. Circ. Mat. Palermo, 20 (1905), 97-135. 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), 746-757. 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), 142-151. 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, Pittsburgh, Pa., 1978), Academic Press, 1979, 267-277.  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, (eds. C. Conca, R. Manasevich, G. Uhlmann and M. Vogelius), AMS, Contemp. Math., 362 (2004), 133-144. 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), 361-374. 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), 466-487. 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), 253-258.  Google Scholar

[11]

M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization, Proc. Natl. Acad. Sci. USA, 109 (2012), 14761-14766. doi: 10.1073/pnas.1120432109.  Google Scholar

[12]

P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math., 1 (1951), 485-524. 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, Springer-Verlag, Berlin, 2010. 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), 865-898. 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), 89-102. 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), 179-190. 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), 119-124. 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, Tome II, CNRS Paris, (1968), 515-641; reprint of Mémoires présentés par divers savants a l'Académie des Sciences, 33 (1908), 1-128. Google Scholar

[19]

J. Hadamard, Sur certains cas intéressants du problème biharmonique, in Oeuvres de Jacques Hadamard, Tome III, CNRS Paris, (1968), 1297-1299; reprint of Atti IV Congr. Intern. Mat. Rome, (1908), 12-14. 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), 337-353.  Google Scholar

[21]

Ch. Loewner, On generation of solutions of the biharmonic equation in the plane by conformal mappings, Pacific J. Math., 3 (1953), 417-436. 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), 167-177. doi: 10.1007/BF00251232.  Google Scholar

[23]

G. Szegö, On membranes and plates, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 210-216. 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), Electron. J. Diff. Eqns. Conf., 6, Southwest Texas State Univ., San Marcos, TX, 2001, 285-296.  Google Scholar

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