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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

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bject to the vertical force f . In this context there exist some positivity results: If the domain# is the unit disk B = # x # R 2 : |x| < 1 # (see [Bo]) or if# is close to the disk B in a suitable sense (see [GS1]), then it is known that 0 ## f # 0 implies u > 0, i.e., upwards pushing yields (globally) upwards bending. So, at least in these domains, nonconstant supersolutions of the clamped plate equation (1) are strictly positive. Here we call a function u # C 4 (## # C 1 ( # ) a supersolution of (1), if it solves (1) with some f # 0. It should be stressed that, in spite of the seemingly quadratic structure of (1), the so called Dirichlet boundary condit

We introduce a class of nonlinear transformations called "resizing rules" which associate to optimal shape design problems certain equivalent distributed control problems, while preserving the state of the system. This puts into evidence the duality principle that the class of system states that can be achieved, under a prescribed force, via modifications of the structure (shape) of the system can be as well obtained via the modifications of the force action, under a prescribed structure. We apply such transformations to the optimization of beams and plates and, in the simply supported or in the cantilevered cases, the obtained control problems are even convex. In all cases, we establish existence theorems for optimal pairs, by assuming only boundedness conditions. Moreover, in the simply supported case, we also prove the uniqueness of the global minimizer. A general algorithm that iterates between the original problem and the transformed one is introduced and studied. The applications...

The main result in this paper is that the solution operator for the bi-laplace problem with zero Dirichlet boundary conditions on a bounded smooth 2d-domain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part contains the singularity and the negative part inherits the full regularity of the boundary. Such a splitting allows one to find a priori estimates for fourth order problems similar to the one proved via the maximum principle in second order elliptic boundary value problems. The proof depends on a careful approximative fill-up of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bi-laplacian on each of these domains is strictly positive.

Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being written as a system of second order boundary value problems. On the other hand, the biharmonic Green’s function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. Previously it was shown that this property also remains under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3. 1.

nic operators, and similarly a function u satisfying \Deltaw \Gamma1 \Deltau = 0 will be called a w-biharmonic function. The latter means, by definition, that u is a C 2 function on\Omega n S such that w \Gamma1 \Deltau extends by continuity to a harmonic function on all of \Omega\Gamma The w-biharmonic Green function U(x; y) is the solution to the boundary value problem 8 ? ? ! ? ? : \Delta x 1 w(x) \Delta x U(x; y) = ffi(x \Gamma y) (the delta function), U(x; y) = @U @n x (x; y) = 0 for x<F70.93

In the present note we give a simpler proof of the recent result of Hedenmalm that the Green function for the weighted biharmonic operator ∆|z | 2α ∆, α>−1, on the unit disc D with the Dirichlet boundary conditions is positive. The main ingredient, which in the special case of the unweighted biharmonic operator ∆ 2 is due to Loewner and which is of an independent interest, is a lemma characterizing, for a positive C 2 weight function w, the second-order linear differential operators which take any function u satisfying ∆w−1∆u =0into a harmonic function. Another application of this lemma concerning positivity of the Poisson kernels for the biharmonic operator ∆2 is also given. Let Ω be a bounded domain in the complex plane with smooth boundary, w a function which is C2-continuous and positive on Ω except for a finite set S of isolated singularities in Ω, where it can have a zero or become + ∞ (or even not be defined at all); such functions will be termed weights. We will be interested in the weighted biharmonic operators ∆w−1∆onΩ, where ∆ stands for the Laplacian. For brevity, let us call the above w-biharmonic operators, and similarly a function u satisfying ∆w−1∆u =0 will be called a w-biharmonic function. The latter means, by definition, that u is a C 2 function on Ω \ S such that w −1 ∆u extends by continuity to a harmonic function on all of Ω. The w-biharmonic Green function U(x, y) is the solution to the boundary value problem 1