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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime
Pages: 839 - 855, Issue 4, August 2014

doi:10.3934/dcdss.2014.7.839      Abstract        References        Full text (4362.5K)           Related Articles

Guido Sweers - Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany (email)

1 R. Bañuelos and T. Carroll, Extremal problems for conditioned Brownian motion and the hyperbolic metric, Ann. Inst. Fourier (Grenoble), 50 (2000), 1507-1532.       
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), 2951-2958.       
3 G. Caristi and E. Mitidieri, Further results on maximum principles for noncooperative elliptic systems, Nonlinear Anal., 17 (1991), 547-558.       
4 K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, Grundlehren der Mathematischen Wissenschaften, 312, Springer-Verlag, Berlin, 1995.       
5 M. Cranston, E. Fabes and Z. X. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc., 307 (1988), 171-194.       
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), 59-68.       
7 J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften, 262, Springer-Verlag, New York, 1984.       
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), 309-329.       
9 A. Dall'Acqua, On the lifetime of a conditional Brownian motion in the ball, J. Math. Anal. Appl., 335 (2007), 389-405.       
10 M. Erven and G. Sweers, On the lifetime of a conditioned Brownian motion on a fish bowl, Arch. Math. (Basel), 90 (2008), 87-96.       
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, 128, Cambridge University Press, Cambridge, 2000.       
14 Ph. Griffin, T. McConnell and G. Verchota, Conditioned Brownian motion in simply connected planar domains, Ann. Inst. H. Poincaré Probab. Statist, 29 (1993), 229-249.       
15 H.-Ch. Grunau and G. Sweers, Positivity for equations involving polyharmonic elliptic operators with Dirichlet boundary conditions, Math. Ann., 307 (1997), 589-626.       
16 H.-Ch. Grunau and G. Sweers, The maximum principle and positive principal eigenfunctions for polyharmonic equations, in Reaction Diffusion systems, Marcel Dekker Inc., New York, 1997, 163-182.       
17 H.-Ch. Grunau and G. Sweers, Sharp estimates for iterated Green functions, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 91-120.       
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), 1023-1040.       
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), 335-357.       
20 P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177.       
21 E. Mitidieri and G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr., 173 (1995), 259-286.       
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), 151-180.       
23 M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984.       
24 P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhäuser Verlag, Basel, 2007.       
25 J. Schröder, Zusammenhängende Mengen inverspositiver Differentialoperatoren vierter Ordnung, Math. Z., 96 (1967), 89-110.       
26 J. Schröder, Operator Inequalities, Mathematics in Science and Engineering, 147, Academic Press Inc., New York-London, 1980.       
27 G. Sweers, A strong maximum principle for a noncooperative elliptic system, SIAM J. Math. Anal., 20 (1989), 367-371.       
28 G. Sweers, Strong positivity in $C$ $( \bar{\Omega}) $ for elliptic systems, Math. Z., 209 (1992), 251-271.       
29 G. Sweers, Positivity for a strongly coupled elliptic system by Green function estimates, J. Geom. Anal., 4 (1994), 121-142.       
30 M. Ulm, The interval of resolvent-positivity for the biharmonic operator, Proc. A.M.S., 127 (1999), 481-489.       
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), 1091-1099.       
32 Z. X. Zhao, Uniform boundedness of conditional gauge and Schrödinger equations, Comm. Math. Phys., 93 (1984), 19-31.       
33 Z. X. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl., 116 (1986), 309-334.       

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