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ABSTRACT Problems in elasticity are described by relations between the displacement field u, the strain or deformation tensor ε, and the stress tensor σ. In 2D elasticity and if volume forces are absent, one may also consider an elastic... more
ABSTRACT Problems in elasticity are described by relations between the displacement field u, the strain or deformation tensor ε, and the stress tensor σ. In 2D elasticity and if volume forces are absent, one may also consider an elastic potential. We explain these settings for a domain Ω with finitely many holes. On such domains the problem for the displacement and the potential problem are not necessarily equivalent. We will also recall the relation between these quantities. We conclude the paper with the potential setting for some specific problems in semi-periodic domains with infinite many holes. Also the setting for non-smooth domains is addressed.
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The conjecture named after Boggio and Hadamard that a biharmonic Green function on convex domains is of fixed sign is known to be false. One might ask what happens for the triharmonic Green function on convex domains. On disks and balls... more
The conjecture named after Boggio and Hadamard that a biharmonic Green function on convex domains is of fixed sign is known to be false. One might ask what happens for the triharmonic Green function on convex domains. On disks and balls it is known to be positive. We will show that also this Green function is not positive on some eccentric ellipse.
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... Filippo Gazzola Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 Guido Sweers Mathematisches Institut Universit¨at zu K ... de 20133 Milano, Italy filippo.gazzola@ polimi.it Hans-Christoph Grunau Institut... more
... Filippo Gazzola Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 Guido Sweers Mathematisches Institut Universit¨at zu K ... de 20133 Milano, Italy filippo.gazzola@ polimi.it Hans-Christoph Grunau Institut f¨ur Analysis und Numerik Otto von Guericke ...
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Research Interests: Mathematics and Grid
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For most `nice' elliptic boundary value problems there is a general expectation that the ̄rst eigenfunction is unique and of ̄xed sign. And indeed, for second order elliptic di®erential equations with Dirichlet boundary conditions... more
For most `nice' elliptic boundary value problems there is a general expectation that the ̄rst eigenfunction is unique and of ̄xed sign. And indeed, for second order elliptic di®erential equations with Dirichlet boundary conditions such a result holds as a consequence of the maximum principle. It is well known that such a maximum principle does not have a direct generalization to higher order elliptic problems. Nevertheless, the hypothesis that the principal eigenfunction for the biharmonic Dirichlet problem is of ̄xed sign does appear in earlier papers, see for example [32] from 1950. Let us be more precise.
It is known that the rst eigenfunction of the clamped plate equation, 2'='in with'= @ @n'=0on@, is not necessarily ofxed sign. Inthis article, we survey therelations between domains andthe sign of that rst eigenfunction.
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ABSTRACT The goal of this chapter is to explain in some detail which models and equations are considered in this book and to provide some background information and comments on the interplay between the various problems. Our motivation... more
ABSTRACT The goal of this chapter is to explain in some detail which models and equations are considered in this book and to provide some background information and comments on the interplay between the various problems. Our motivation arises on the one hand from equations in continuum mechanics, biophysics or differential geometry and on the other hand from basic questions in the theory of partial differential equations.
Research Interests: Mathematics and Physics
As already mentioned in Section 1.2, in general one does not have positivity preserving for higher order Dirichlet problems. Nevertheless, in Chapter 6 we shall identify some families of domains where the biharmonic-or more generally the... more
As already mentioned in Section 1.2, in general one does not have positivity preserving for higher order Dirichlet problems. Nevertheless, in Chapter 6 we shall identify some families of domains where the biharmonic-or more generally the polyharmonic-Dirichlet problem enjoys a positivity preserving property. Moreover, there we shall prove “galmost positivity” for the biharmonic Dirichlet problem in any bounded smooth domain Ω ⊂Rn.
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In Section 1.2 we mentioned that although the Green functionG Δ2 Ω for the clamped plate boundary value problem $$ \left\{\begin{array}{l}\Delta^2 u = f\ in\ \Omega \\ u\ =\ |\nabla u|\ =\ 0\ on\ \partial\ \Omega \end{array}\right.\ $$ is... more
In Section 1.2 we mentioned that although the Green functionG Δ2 Ω for the clamped plate boundary value problem $$ \left\{\begin{array}{l}\Delta^2 u = f\ in\ \Omega \\ u\ =\ |\nabla u|\ =\ 0\ on\ \partial\ \Omega \end{array}\right.\ $$ is in general sign changing, it is very hard to display its negative part in numerical simulations or in real world experiments. Moreover, numerical work in nonlinear elliptic fourth order equations suggests that maximum or comparison principles are violated only to a “small extent”. Nevertheless, we do not yet have tools at hand to give this feeling a precise form and, in particular, a quantitative form which might prove to be useful also for nonlinear higher order equations.
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This last chapter serves to give a first existence result for a priori bounded classical solutions of the Dirichlet problem forWillmore surfaces and thereby to outline possible directions of further research. In order to see which kind of... more
This last chapter serves to give a first existence result for a priori bounded classical solutions of the Dirichlet problem forWillmore surfaces and thereby to outline possible directions of further research. In order to see which kind of phenomena and results concerning compact embedded solutions in R3 of boundary value problems for the corresponding equationmight be expected,we investigateDirichlet problems in a particularly symmetric situation.
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Summary. In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an el-liptic partial differential equation with rapidly varying coefficients. For a regular... more
Summary. In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an el-liptic partial differential equation with rapidly varying coefficients. For a regular discretization this mapping has to be invertible. We will show that such result holds for general elliptic operators (in two dimensions). The Carleman-Hartman-Wintner Theorem will be fundamental in our proof. We will also explain why such a general result cannot be expected to hold for the (three-dimensional) cube. Mathematics Subject Classification (1991): 35J25, 65M50, 76B05 1.
Abstract. Boggio proved in 1905 that the clamped plate equation is positivity preserving for a disk. It is known that on many other domains such a property fails. In this paper we will show that an affirmative result holds on still a... more
Abstract. Boggio proved in 1905 that the clamped plate equation is positivity preserving for a disk. It is known that on many other domains such a property fails. In this paper we will show that an affirmative result holds on still a large class of domains. We also survey the available methods in obtaining domains with such property. 1.
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We consider fully coupled cooperative systems on \begin{document}$ \mathbb{R}^n $\end{document} with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the... more
We consider fully coupled cooperative systems on \begin{document}$ \mathbb{R}^n $\end{document} with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on \begin{document}$ \mathbb{R}^2 $\end{document} with exponential nonlinearity, are nondegenerate.
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SummaryThe 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... more
SummaryThe 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.
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appears in a model for the vertical displacement of a two-dimensional grid that consists of two perpendicular sets of elastic fibers or rods. We are interested in the behaviour of such a grid that is clamped at the boundary and more... more
appears in a model for the vertical displacement of a two-dimensional grid that consists of two perpendicular sets of elastic fibers or rods. We are interested in the behaviour of such a grid that is clamped at the boundary and more specifically near a corner of the domain. Kondratiev supplied the appropriate setting in the sense of Sobolev type spaces tailored to find the optimal regularity. Inspired by the Laplacian and the Bilaplacian models one expect, except maybe for some special angles that the optimal regularity improves when angle decreases. For the homogeneous Dirichlet problem with this special non-isotropic fourth order operator such a result does not hold true. We will show the existence of an interval ( 1 2 pi, ω?), ω?/pi ≈ 0.528... (in degrees ω? ≈ 95.1...◦), in which the optimal regularity
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It is known that the first eigenfunction of the clamped plate equation, ∆φ = λφ in Ω with φ = ∂ ∂n φ = 0 on ∂Ω, is not necessarily of fixed sign. In this article, we survey the relations between domains Ω and the sign of that first... more
It is known that the first eigenfunction of the clamped plate equation, ∆φ = λφ in Ω with φ = ∂ ∂n φ = 0 on ∂Ω, is not necessarily of fixed sign. In this article, we survey the relations between domains Ω and the sign of that first eigenfunction.
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On any bounded domain the Dirichlet or Neumann–Laplace operator has a first eigenfunction which is the unique one of fixed sign. For the bilaplace operator, as for any fourth order operator, there is no direct maximum principle and hence... more
On any bounded domain the Dirichlet or Neumann–Laplace operator has a first eigenfunction which is the unique one of fixed sign. For the bilaplace operator, as for any fourth order operator, there is no direct maximum principle and hence no obvious argument for a first eigenfunction of one sign. We will address some examples where fourth order eigenvalue problems have an unexpected behaviour concerning positivity of an eigenfunction.
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A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a... more
A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction between the influence on the solution of the positive and the negative part of the right-hand side.