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Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions
 MATHEMATISCHE NACHRICHTEN
, 1996
"... ..."
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Nonexistence Of Local Minima Of Supersolutions For The Circular Clamped Plate
, 2001
"... 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 ..."
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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
Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions, submitted
"... Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equ ..."
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Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n, the negative part of the corresponding Green’s function is “small ” when compared with its singular positive part, provided n ≥ 3. Moreover, the biharmonic Green’s function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3. Keywords: Biharmonic Green’s functions, positivity, almost positivity, blowup procedure.
A Duality Approach in the Optimization of Beams and Plates
, 1997
"... 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 ..."
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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...
Separating positivity and regularity for fourth order Dirichlet problems in 2ddomains
"... The main result in this paper is that the solution operator for the bilaplace problem with zero Dirichlet boundary conditions on a bounded smooth 2ddomain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part conta ..."
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The main result in this paper is that the solution operator for the bilaplace problem with zero Dirichlet boundary conditions on a bounded smooth 2ddomain 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 fillup of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bilaplacian on each of these domains is strictly positive.
STABILITY OF THE POSITIVITY OF BIHARMONIC GREEN’S FUNCTIONS UNDER PERTURBATIONS OF THE DOMAIN
"... 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 ..."
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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.
POSITIVITY ISSUES OF BIHARMONIC GREEN’S FUNCTIONS UNDER DIRICHLET BOUNDARY CONDITIONS
"... Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equ ..."
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Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n, the negative part of the corresponding Green’s function is “small ” when compared with its singular positive part, provided n ≥ 3. Moreover, the biharmonic Green’s function in balls B ⊂ R n under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3. 1.