2014, 7(4): 839-855. doi: 10.3934/dcdss.2014.7.839

Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime

1. 

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

Received  August 2013 Revised  November 2013 Published  February 2014

We address the question, for which $\lambda \in \mathbb{R}$ is the boundary value problem \begin{equation*} \left\{ \begin{array}{cc} \Delta ^{2}u+\lambda u=f & \text{in }\Omega , \\ u=\Delta u=0 & \text{on }\partial \Omega , \end{array} \right. \end{equation*} positivity preserving, that is, $f\geq 0$ implies $u\geq 0$. Moreover, we consider what happens, when $\lambda $ passes the maximal value for which positivity is preserved.
Citation: Guido Sweers. Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 839-855. doi: 10.3934/dcdss.2014.7.839
References:
[1]

R. Bañuelos and T. Carroll, Extremal problems for conditioned Brownian motion and the hyperbolic metric,, Ann. Inst. Fourier (Grenoble), 50 (2000), 1507. doi: 10.5802/aif.1798.

[2]

R. F. Bass, J. Horák and P. J. McKenna, On the lift-off constant for elastically supported plates,, Proc. Amer. Math. Soc., 132 (2004), 2951. doi: 10.1090/S0002-9939-04-07428-3.

[3]

G. Caristi and E. Mitidieri, Further results on maximum principles for noncooperative elliptic systems,, Nonlinear Anal., 17 (1991), 547. doi: 10.1016/0362-546X(91)90063-7.

[4]

K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation,, Grundlehren der Mathematischen Wissenschaften, (1995). doi: 10.1007/978-3-642-57856-4.

[5]

M. Cranston, E. Fabes and Z. X. Zhao, Conditional gauge and potential theory for the Schrödinger operator,, Trans. Amer. Math. Soc., 307 (1988), 171. doi: 10.2307/2000757.

[6]

B. Dittmar, Local and global maxima for the expectation of the lifetime of a Brownian motion on the disk,, J. Anal. Math., 104 (2008), 59. doi: 10.1007/s11854-008-0016-6.

[7]

J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften, (1984). doi: 10.1007/978-1-4612-5208-5.

[8]

A. Dall'Acqua, H.-Ch. Grunau and G. Sweers, On a conditioned Brownian motion and a maximum principle on the disk,, J. Anal. Math., 93 (2004), 309. doi: 10.1007/BF02789311.

[9]

A. Dall'Acqua, On the lifetime of a conditional Brownian motion in the ball,, J. Math. Anal. Appl., 335 (2007), 389. doi: 10.1016/j.jmaa.2007.01.081.

[10]

M. Erven and G. Sweers, On the lifetime of a conditioned Brownian motion on a fish bowl,, Arch. Math. (Basel), 90 (2008), 87. doi: 10.1007/s00013-007-2387-9.

[11]

M. Erven and G. Sweers, On the lifetime of conditioned Brownian motion in domains connected through small gaps,, submitted., ().

[12]

M. Erven and G. Sweers, On the location of the maximal lifetime for a conditional Brownian motion,, in preparation., ().

[13]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Tracts in Mathematics, (2000). doi: 10.1017/CBO9780511569203.

[14]

Ph. Griffin, T. McConnell and G. Verchota, Conditioned Brownian motion in simply connected planar domains,, Ann. Inst. H. Poincaré Probab. Statist, 29 (1993), 229.

[15]

H.-Ch. Grunau and G. Sweers, Positivity for equations involving polyharmonic elliptic operators with Dirichlet boundary conditions,, Math. Ann., 307 (1997), 589. doi: 10.1007/s002080050052.

[16]

H.-Ch. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations,, in Reaction Diffusion systems, (1997), 163.

[17]

H.-Ch. Grunau and G. Sweers, Sharp estimates for iterated Green functions,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 91. doi: 10.1017/S0308210500001542.

[18]

B. Kawohl and G. Sweers, On ‘anti'-eigenvalues for elliptic systems and a question of McKenna and Walter,, Indiana Univ. Math. J., 51 (2002), 1023. doi: 10.1512/iumj.2002.51.2275.

[19]

B. Kawohl and G. Sweers, Among all two-dimensional convex domains the disk is not optimal for the lifetime of a conditioned Brownian motion,, J. Anal. Math., 86 (2002), 335. doi: 10.1007/BF02786655.

[20]

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

[21]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity,, Math. Nachr., 173 (1995), 259. doi: 10.1002/mana.19951730115.

[22]

S. A. Nazarov and G. Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners,, J. Differential Equations, 233 (2007), 151. doi: 10.1016/j.jde.2006.09.018.

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Corrected reprint of the 1967 original, (1967). doi: 10.1007/978-1-4612-5282-5.

[24]

P. Pucci and J. Serrin, The Maximum Principle,, Progress in Nonlinear Differential Equations and their Applications, (2007).

[25]

J. Schröder, Zusammenhängende Mengen inverspositiver Differentialoperatoren vierter Ordnung,, Math. Z., 96 (1967), 89. doi: 10.1007/BF01111581.

[26]

J. Schröder, Operator Inequalities,, Mathematics in Science and Engineering, (1980).

[27]

G. Sweers, A strong maximum principle for a noncooperative elliptic system,, SIAM J. Math. Anal., 20 (1989), 367. doi: 10.1137/0520023.

[28]

G. Sweers, Strong positivity in $C$ $( \bar{\Omega}) $ for elliptic systems,, Math. Z., 209 (1992), 251. doi: 10.1007/BF02570833.

[29]

G. Sweers, Positivity for a strongly coupled elliptic system by Green function estimates,, J. Geom. Anal., 4 (1994), 121. doi: 10.1007/BF02921596.

[30]

M. Ulm, The interval of resolvent-positivity for the biharmonic operator,, Proc. A.M.S., 127 (1999), 481. doi: 10.1090/S0002-9939-99-04556-6.

[31]

M. van den Berg, A. Dall'Acqua and G. Sweers, Estimates for the expected lifetime of conditioned Brownian motion,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1091. doi: 10.1017/S0308210506000448.

[32]

Z. X. Zhao, Uniform boundedness of conditional gauge and Schrödinger equations,, Comm. Math. Phys., 93 (1984), 19. doi: 10.1007/BF01218637.

[33]

Z. X. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge,, J. Math. Anal. Appl., 116 (1986), 309. doi: 10.1016/S0022-247X(86)80001-4.

show all references

References:
[1]

R. Bañuelos and T. Carroll, Extremal problems for conditioned Brownian motion and the hyperbolic metric,, Ann. Inst. Fourier (Grenoble), 50 (2000), 1507. doi: 10.5802/aif.1798.

[2]

R. F. Bass, J. Horák and P. J. McKenna, On the lift-off constant for elastically supported plates,, Proc. Amer. Math. Soc., 132 (2004), 2951. doi: 10.1090/S0002-9939-04-07428-3.

[3]

G. Caristi and E. Mitidieri, Further results on maximum principles for noncooperative elliptic systems,, Nonlinear Anal., 17 (1991), 547. doi: 10.1016/0362-546X(91)90063-7.

[4]

K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation,, Grundlehren der Mathematischen Wissenschaften, (1995). doi: 10.1007/978-3-642-57856-4.

[5]

M. Cranston, E. Fabes and Z. X. Zhao, Conditional gauge and potential theory for the Schrödinger operator,, Trans. Amer. Math. Soc., 307 (1988), 171. doi: 10.2307/2000757.

[6]

B. Dittmar, Local and global maxima for the expectation of the lifetime of a Brownian motion on the disk,, J. Anal. Math., 104 (2008), 59. doi: 10.1007/s11854-008-0016-6.

[7]

J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften, (1984). doi: 10.1007/978-1-4612-5208-5.

[8]

A. Dall'Acqua, H.-Ch. Grunau and G. Sweers, On a conditioned Brownian motion and a maximum principle on the disk,, J. Anal. Math., 93 (2004), 309. doi: 10.1007/BF02789311.

[9]

A. Dall'Acqua, On the lifetime of a conditional Brownian motion in the ball,, J. Math. Anal. Appl., 335 (2007), 389. doi: 10.1016/j.jmaa.2007.01.081.

[10]

M. Erven and G. Sweers, On the lifetime of a conditioned Brownian motion on a fish bowl,, Arch. Math. (Basel), 90 (2008), 87. doi: 10.1007/s00013-007-2387-9.

[11]

M. Erven and G. Sweers, On the lifetime of conditioned Brownian motion in domains connected through small gaps,, submitted., ().

[12]

M. Erven and G. Sweers, On the location of the maximal lifetime for a conditional Brownian motion,, in preparation., ().

[13]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Tracts in Mathematics, (2000). doi: 10.1017/CBO9780511569203.

[14]

Ph. Griffin, T. McConnell and G. Verchota, Conditioned Brownian motion in simply connected planar domains,, Ann. Inst. H. Poincaré Probab. Statist, 29 (1993), 229.

[15]

H.-Ch. Grunau and G. Sweers, Positivity for equations involving polyharmonic elliptic operators with Dirichlet boundary conditions,, Math. Ann., 307 (1997), 589. doi: 10.1007/s002080050052.

[16]

H.-Ch. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations,, in Reaction Diffusion systems, (1997), 163.

[17]

H.-Ch. Grunau and G. Sweers, Sharp estimates for iterated Green functions,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 91. doi: 10.1017/S0308210500001542.

[18]

B. Kawohl and G. Sweers, On ‘anti'-eigenvalues for elliptic systems and a question of McKenna and Walter,, Indiana Univ. Math. J., 51 (2002), 1023. doi: 10.1512/iumj.2002.51.2275.

[19]

B. Kawohl and G. Sweers, Among all two-dimensional convex domains the disk is not optimal for the lifetime of a conditioned Brownian motion,, J. Anal. Math., 86 (2002), 335. doi: 10.1007/BF02786655.

[20]

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

[21]

E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity,, Math. Nachr., 173 (1995), 259. doi: 10.1002/mana.19951730115.

[22]

S. A. Nazarov and G. Sweers, A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners,, J. Differential Equations, 233 (2007), 151. doi: 10.1016/j.jde.2006.09.018.

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Corrected reprint of the 1967 original, (1967). doi: 10.1007/978-1-4612-5282-5.

[24]

P. Pucci and J. Serrin, The Maximum Principle,, Progress in Nonlinear Differential Equations and their Applications, (2007).

[25]

J. Schröder, Zusammenhängende Mengen inverspositiver Differentialoperatoren vierter Ordnung,, Math. Z., 96 (1967), 89. doi: 10.1007/BF01111581.

[26]

J. Schröder, Operator Inequalities,, Mathematics in Science and Engineering, (1980).

[27]

G. Sweers, A strong maximum principle for a noncooperative elliptic system,, SIAM J. Math. Anal., 20 (1989), 367. doi: 10.1137/0520023.

[28]

G. Sweers, Strong positivity in $C$ $( \bar{\Omega}) $ for elliptic systems,, Math. Z., 209 (1992), 251. doi: 10.1007/BF02570833.

[29]

G. Sweers, Positivity for a strongly coupled elliptic system by Green function estimates,, J. Geom. Anal., 4 (1994), 121. doi: 10.1007/BF02921596.

[30]

M. Ulm, The interval of resolvent-positivity for the biharmonic operator,, Proc. A.M.S., 127 (1999), 481. doi: 10.1090/S0002-9939-99-04556-6.

[31]

M. van den Berg, A. Dall'Acqua and G. Sweers, Estimates for the expected lifetime of conditioned Brownian motion,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1091. doi: 10.1017/S0308210506000448.

[32]

Z. X. Zhao, Uniform boundedness of conditional gauge and Schrödinger equations,, Comm. Math. Phys., 93 (1984), 19. doi: 10.1007/BF01218637.

[33]

Z. X. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge,, J. Math. Anal. Appl., 116 (1986), 309. doi: 10.1016/S0022-247X(86)80001-4.

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