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On positivity for the biharmonic operator under Steklov boundary conditions. (English) Zbl 1155.35019
Let us consider $$\Omega$$ being a bounded smooth domain in $$\mathbb R^n,$$ $$n \geq 2,$$ and the following elliptic problem, called Steklov problem, having parameter in the boundary condition:
$\Delta^2 u= f\quad\text{in }\Omega, \qquad u=f\text{ and }\Delta u= \alpha u_\nu\quad \text{on }\partial \Omega$
where $$\alpha \in C(\partial \Omega),$$ $$f \in L^2(\Omega)$$ and $$\nu$$ is the outside normal.
The authors are interested in conditions on $$\alpha$$ which guarantee that the above Steklov problem is positivity preserving, meaning $$f \geq 0$$ implies that $$u \geq 0.$$ It is proved that this property is associated to the parameter considered in the boundary condition. Gazzola and Sweers point out that for Navier and Dirichlet boundary conditions it is known that all $$\alpha$$ for which the positivity preserving property represent an interval. This result is similar to that is true obtained for the Steklov problem.

##### MSC:
 35J40 Boundary value problems for higher-order elliptic equations 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
##### Keywords:
Biharmonic operators; Steklov boundary value problem
Full Text:
##### References:
 [1] Adams R.A. (1975) Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic, London · Zbl 0314.46030 [2] Agmon S., Douglis A., Nirenberg L. (1959) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12: 623–727 · Zbl 0093.10401 [3] Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 , 620–709 (1976). (Errata in: SIAM Rev. 19, 1977, vii) (1977) · Zbl 0345.47044 [4] Berchio E., Gazzola F., Mitidieri E. (2006) Positivity preserving property for a class of biharmonic elliptic problems. J. Differ. Equ. 229: 1–23 · Zbl 1142.35016 [5] Berchio E., Gazzola F., Weth T. (2007) Critical growth biharmonic elliptic problems under Steklov-type boundary conditions. Adv. Differ. Equ. 12: 381–406 · Zbl 1155.35018 [6] Boggio T. (1905) Sulle funzioni di Green d’ordine m. Rend. Circ. Mat. Palermo 20: 97–135 · JFM 36.0827.01 [7] Coffman C.V., Duffin R.J. (1980) On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle. Adv. Appl. Math. 1: 373–389 · Zbl 0452.35002 [8] Dall’Acqua A., Sweers G. (2004) Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems. J. Differ. Equ. 205, 466–487 · Zbl 1154.35338 [9] Dall’Acqua A., Sweers G. (2005) The clamped plate equation for the limaçon. Ann. Mat. Pura Appl. 184: 361–374 · Zbl 1223.35145 [10] Destuynder P., Salaun M. (1996) Mathematical Analysis of Thin Plate Models. Springer, Berlin · Zbl 0860.73001 [11] Ferrero A., Gazzola F., Weth T. (2005) On a fourth order Steklov eigenvalue problem. Analysis 25, 315–332 · Zbl 1112.49035 [12] Garabedian P.R. (1951) A partial differential equation arising in conformal mapping. Pac. J. Math. 1, 253–258 · Zbl 0045.05102 [13] Gilbarg D., Trudinger N.S. (1983) Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Heidelberg · Zbl 0562.35001 [14] Grunau H.-Ch., Sweers G. (1996) Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions. Math. Nachr. 179, 89–102 · Zbl 0863.35016 [15] Grunau, H.-Ch., Sweers, G.: The maximum principle and positive principal eigenfunctions for polyharmonic equations. Reaction Diffusion Systems(Eds. Caristi G. and Mitidieri E.) Papers from the meeting held at the Università di Trieste, October 2–7, 1995. Lect. Notes Pure Appl. Math., vol. 194, Marcel Dekker, New York, pp. 163–182, 1998 [16] Grunau H.-Ch., Sweers G. (2002) Sharp estimates for iterated Green functions. Proc. R. Soc. Edinb. Sect. A 132, 91–120 · Zbl 1115.35009 [17] Grunau H.-Ch., Sweers G. (1999) Sign change for the Green function and for the first eigenfunction of equations of clamped-plate type. Arch. Ration. Mech. Anal. 150: 179–190 · Zbl 0973.74044 [18] Kuttler J.R. (1972) Remarks on a Stekloff eigenvalue problem. SIAM J. Numer. Anal. 9, 1–5 · Zbl 0233.35071 [19] Kuttler J.R. (1979) Dirichlet eigenvalues. SIAM J. Numer. Anal. 16, 332–338 · Zbl 0405.65055 [20] Kuttler J.R., Sigillito V.G. (1968) Inequalities for membrane and Stekloff eigenvalues. J. Math. Anal. Appl. 23: 148–160 · Zbl 0167.45701 [21] Lakes R.S. (1987) Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 [22] Love A.E.H. (1927) A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, London [23] Mitidieri E., Sweers G. (1995) Weakly coupled elliptic systems and positivity. Math. Nachr. 173, 259–286 · Zbl 0834.35041 [24] Nazarov S., Sweers G. (2007) A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners. J. Differ. Equ. 233: 151–180 · Zbl 1108.35043 [25] Payne L.E. (1970) Some isoperimetric inequalities for harmonic functions. SIAM J. Math. Anal. 1, 354–359 · Zbl 0199.16902 [26] Sassone E. (2007) Positivity for polyharmonic problems on domains close to a disk. Ann. Math. Pura Appl. 186: 419–432 · Zbl 1121.35045 [27] Shapiro H.S., Tegmark M. (1994) An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign. SIAM Rev. 36, 99–101 · Zbl 0794.35044 [28] Sweers G. (1994) Positivity for a strongly coupled elliptic system by Green function estimates. J. Geom. Anal. 4: 121–142 · Zbl 0792.35048 [29] Stekloff, W. Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. École Norm. Sup. Sér. 3 19, 191–259 and 455–490 (1902) · JFM 33.0800.01 [30] Villaggio P. (1997) Mathematical Models for Elastic Structures. Cambridge University Press, London · Zbl 0978.74002 [31] Zhao Z. (1986) Green function for Schrödinger operator and conditioned Feynman– Kacgauge. J. Math. Anal. Appl. 116, 309–334 · Zbl 0608.35012 [32] Zhao, Z. Green functions and conditioned gauge theorem for a 2-dimensional domain. Seminar on Stochastic Processes, (Eds. Cinlar E. et al.) Birkhäuser, Basel, 283–294, 1988
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