# Browsing by Subject "Boundary value problems"

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Item Linear boundary value problems for ordinary differential equations and their associated difference equations(1927) Whyburn, William M. (William Marvin), 1901-1972; Ettlinger, H. J. (Hyman Joseph), 1889-Show more Item Local elliptic boundary value problems for the dirac operator(2006) Scholl, Matthew Gregory; Freed, DanShow more Item Properties of solutions of an infinite system of ordinary linear differential equations of the first order with auxiliary boundary conditions(1929) Reid, William T. (William Thomas), 1907 October 4-1977; Ettlinger, H. J. (Hyman Joseph), 1889-1986Show more Item Regularity of free boundary in variational problems(2005) Teixeira, Eduardo Vasconcelos Oliveira; Caffarelli, Luis A.Show more We study the existence and geometric properties of an optimal configurations to a variational problem with free boundary. More specifically, we analyze the nonlinear optimization problem in heat conduction which can be described as follows: given a surface ∂D ⊂ R n and a positive function ϕ defined on it (temperature distribution of the body D), we want to find an optimal configuration Ω ⊃ ∂D (insulation), that minimizes the loss of heat in a stationary situation, where the amount of insulating material is prescribed. This situation also models problems in electrostatic, potential flow in fluid mechanics among others. The quantity to be minimized, the flow of heat, is given by a monotone operator on the flux uµ. Mathematically speaking, let D ⊂ R n be a given smooth bounded domain and ϕ: ∂D → R+ a positive continuous function. For each domain Ω surrounding D such that Vol.(Ω \ D) = 1, we consider the potential associated to the configuration Ω, i.e., the harmonic function on Ω\D taking boundary data u ∂D ≡ ϕ and u ∂Ω ≡ 0, and compute J(Ω) := Z ∂D Γ(x,uµ(x))dσ, vii where µ is the inward normal vector defined on ∂D and Γ is a continuous family of convex functions. Our goal is to study the existence and geometric properties of an optimal configuration related to the functional J. In other words, our purpose is to study the problem: minimize { J(u) := Z ∂D Γ(x,uµ(x))dσ : u: DC → R, u = ϕ on ∂D, ∆u = 0 in {u > 0} and Vol.(supp u) = 1 } Among other regularity properties of an optimal configuration, we prove analyticity of the free boundary up to a small singular set. We also establish uniqueness and symmetry results when ∂D has a given symmetry. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary.Show more Item Uniqueness and existence results on viscosity solutions of some free boundary problems(2002-08) Kim, Christina; Souganidis, PanagiotisShow more Item Weakly non-local arbitrarily-shaped absorbing boundary conditions for acoustics and elastodynamics theory and numerical experiments(2004) Lee, Sanghoon; Kallivokas, Loukas F.Show more In this dissertation we discuss the performance of a family of local and weakly non-local in space and time absorbing boundary conditions, prescribed on trun cation boundaries of elliptical and ellipsoidal shape for the solution of two- and three-dimensional scalar wave equations, respectively, in both the time- and frequency-domains. The elliptical and ellipsoidal artiﬁcial boundaries are de rived as particular cases of general arbitrarily-shaped convex boundaries for which the absorbing conditions are developed. From the mathematical per spective, the development of the conditions is based on earlier work by Kalli vokas et al [72–77]; herein an incremental modiﬁcation is made to allow for the spatial variability of the conditions’ absorption characteristics. From the appli cations perspective, the obtained numerical results appear herein for the ﬁrst time. It is further shown that the conditions, via an operator-splitting scheme, lend themselves to easy incorporation in a variational form that, in turn, leads to a standard Galerkin ﬁnite element approach. The resulting wave absorbing ﬁnite elements are shown to preserve the sparsity and symmetry of standard ﬁnite element schemes in both the time- and frequency-domains. Herein, we also extend the applicability of elliptically-shaped truncation boundaries to semi-inﬁnite acoustic media. Numerical experiments for transient and time harmonic cases attest to the computational savings realized when elongated scatterers are surrounded by elliptically- or ellipsoidally-shaped boundaries, as opposed to the more commonly used circular or spherical truncation geome tries in either the full- or half-space cases (near-surface scatterers). Lastly, we treat the two-dimensional elastodynamics case based on a Helmholtz decomposition of the displacement vector ﬁeld. The decomposi tion allows for scalar wave equations to be written for the scalar and vector potential components. Thus, absorbing conditions similar to the ones writ ten for acoustics can be used for the elastodynamics case. The stability of the elastodynamics conditions for time-domain applications remains an open question.Show more