Factorization by invariant embedding of elliptic problems: circular and star-shaped domains

Dissertação apresentada para obtenção do grau de Doutor em Matemática na especialidade de Equações Diferenciais, pela Universidade Nova de Lisboa,Faculdade de Ciências e TecnologiaThis thesis concerns the factorization of elliptic operators, namely the decomposition of a second order boundary value problem, de¯ned in an open bounded regular domain, in an uncoupled system of two ¯rst order initial value problems. The method presented here is inspired on the theory of Optimal Control. It is a return, in a new spatial approach, to the technique of the invariant temporal embedding, de¯ned originally in the context of Dynamic Programming, used in Control Theory for the computation of the optimal feedback. This technique consists in embedding the initial problem in a family of similar problems depending on a parameter, which are solved recursively. In our case, each problem is de¯ned over a sub-domain limited by a mobile boundary depending on the parameter. We introduce an operator relating the trace of the function de¯ned for each problem, and the trace of its normal derivative over the mobile boundary. Without loss of generality, we particularize the study to a Poisson's equation with, for example, a Dirichlet's boundary condition. We ¯rst consider a circular domain and we present for it two approaches: ¯rst, we apply an invariant embedding that starts on the boundary of the circle and go towards its center, followed by an invariant embedding in the opposite direction. Next, we generalize the method, applying it to the case of an arbitrary star shaped domain. In all cases, the family of curves which limits the subdomains de¯ned by the invariant embedding are homothetic to one another and homothetic to a point. This fact induces the appearing of a singularity.Fundação para a Ciência e a Tecnologia (FCT) - Praxis XXI, BD/ 21443/9

Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems

Dissertação para obtenção do Grau de Doutor em MatemáticaThe purpose of this thesis is the factorization of elliptic boundary value problems defined in cylindrical domains, in a system of decoupled first order initial value problems. We begin with the Poisson equation with mixed boundary conditions, and use the method of invariant embedding: we embed our initial problem in a family of similar problems, defined in sub-domains of the initial domain, with a moving boundary, and an additional condition in the moving boundary. This factorization is inspired by the technique of invariant temporal embedding used in Control Theory when computing the optimal feedback, for, in fact, as we show, our initial problem may be defined as an optimal control problem. The factorization thus obtained may be regarded as a generalized block Gauss LU factorization. From this procedure emerges an operator that can be either the Dirichlet-to-Neumann or the Neumann-to-Dirichlet operator, depending on which boundary data is given on the moving boundary. In any case this operator verifies a Riccati equation that is studied directly by using an Yosida regularization. Then we extend the former results to more general strongly elliptic operators. We also obtain a QR type factorization of the initial problem, where Q is an orthogonal operator and R is an upper triangular operator. This is related to a least mean squares formulation of the boundary value problem. In addition, we obtain the factorization of overdetermined boundary value problems, when we consider an additional Neumann boundary condition: if this data is not compatible with the initial data, then the problem has no solution. In order to solve it, we introduce a perturbation in the original problem and minimize the norm of this perturbation, under the hypothesis of existence of solution. We deduce the normal equations for the overdetermined problem and, as before, we apply the method of invariant embedding to factorize the normal equations in a system of decoupled first order initial value problems

Algebraic Closed Geodesics on a Triaxial Ellipsoid

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in ${\mathbb R}^3$. Such geodesics are either connected components of spatial elliptic curves or rational curves. Our approach is based on elements of the Weierstrass--Poncar\'e reduction theory for hyperelliptic tangential covers of elliptic curves and the addition law for elliptic functions. For the case of 3-fold and 4-fold coverings, explicit formulas for the cutting algebraic surfaces are provided and some properties of the corresponding geodesics are discussed.Comment: 15 figure

Screened poisson hyperfields for shape coding

We present a novel perspective on shape characterization using the screened Poisson equation. We discuss that the effect of the screening parameter is a change of measure of the underlying metric space. Screening also indicates a conditioned random walker biased by the choice of measure. A continuum of shape fields is created by varying the screening parameter or, equivalently, the bias of the random walker. In addition to creating a regional encoding of the diffusion with a different bias, we further break down the influence of boundary interactions by considering a number of independent random walks, each emanating from a certain boundary point, whose superposition yields the screened Poisson field. Probing the screened Poisson equation from these two complementary perspectives leads to a high-dimensional hyperfield: a rich characterization of the shape that encodes global, local, interior, and boundary interactions. To extract particular shape information as needed in a compact way from the hyperfield, we apply various decompositions either to unveil parts of a shape or parts of a boundary or to create consistent mappings. The latter technique involves lower-dimensional embeddings, which we call screened Poisson encoding maps (SPEM). The expressive power of the SPEM is demonstrated via illustrative experiments as well as a quantitative shape retrieval experiment over a public benchmark database on which the SPEM method shows a high-ranking performance among the existing state-of-the-art shape retrieval methods

Free Vibrations of Axisymmetric Shells: Parabolic and Elliptic cases

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{\'e} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{\'e} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{\'e} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=\gamma \varepsilon^{-\beta}$ with $\beta=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $\beta$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented