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Recent documents in Department of Mathematical Sciencesen-usThu, 03 Sep 2015 02:13:29 PDT3600Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in higher dimensions
http://repository.cmu.edu/math/697
http://repository.cmu.edu/math/697Thu, 17 May 2012 15:18:03 PDT
Abstract: "In this work we study the generation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in several space variables, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We first examine the way the characteristics cross by identifying the set of critical points of the characteristic manifold with the caustic set of the related lagrangian mapping. We construct the viscosity solution by selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics. We finally discuss how the shocks propagate and undergo catastrophe in the case of two space variables."
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Georgios T. KossiorisOptimal order and minimal complexity of one-step methods for initial value problems
http://repository.cmu.edu/math/696
http://repository.cmu.edu/math/696Thu, 17 May 2012 15:17:56 PDTArthur G. WerschulzComputational complexity of one-step methods for systems of differential equations
http://repository.cmu.edu/math/695
http://repository.cmu.edu/math/695Thu, 17 May 2012 15:17:49 PDTArthur G. WerschulzComputational complexity of one-step methods for a scalar autonomous differential equation
http://repository.cmu.edu/math/694
http://repository.cmu.edu/math/694Thu, 17 May 2012 15:17:42 PDTArthur G. WerschulzRegularity of stochastic delay equations under pth order degeneracy
http://repository.cmu.edu/math/693
http://repository.cmu.edu/math/693Thu, 17 May 2012 15:17:34 PDTDenis R. Bell et al.Formation of singularities for viscosity solutions of Hamilton-Jacobi equations in one space variable
http://repository.cmu.edu/math/692
http://repository.cmu.edu/math/692Thu, 17 May 2012 15:17:26 PDT
Abstract: "In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacobi equations in one space variable, under no assumption on convexity or concavity of the hamiltonian. We study the problem in the class of viscosity solutions, which are the correct class of weak solutions. We obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing explicitly the proper rarefaction waves."
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Georgios T. KossiorisOn a class of invariant functionals
http://repository.cmu.edu/math/691
http://repository.cmu.edu/math/691Thu, 17 May 2012 15:17:18 PDT
Abstract: "A characterization of a class of functionals invariant under isochoric changes of domain is obtained. This class contains strictly the null Lagrangians."
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Irene Fonseca et al.Variational problems for liquid crystals with variable degree of orientation
http://repository.cmu.edu/math/690
http://repository.cmu.edu/math/690Thu, 17 May 2012 15:17:10 PDT
Abstract: "The analysis of a nematic liquid crystal, filling a bounded cylindrical container, whose free energy is a (simplified) version of Ericksen's model with variable degree of orientation, leads to a variational problem of the form F[s,[phi]] = [integral]┬╣ΓéÇ[k (s╩╣)┬▓ + s┬▓ (([phi]╩╣)┬▓ + cos┬▓[phi]/r┬▓)]rdr subject to s(1) = sΓéÇ, [phi](1) = 0, with k a positive constant. It will be shown that a surprisingly explicit solution is obtainable. Moreover an interesting bifurcation takes place at k = 1."
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Victor J. MizelRemarks on variational problems for defective crystals
http://repository.cmu.edu/math/689
http://repository.cmu.edu/math/689Thu, 17 May 2012 15:17:03 PDT
Abstract: "In the context of a continuum theory of crystals with defects, one can define a particular list of tensors which remain unchnaged [sic] when the crystal is deformed elastically. In our model, the defect notion relies on the assumption that deformations that leave these elements invariant do not change the defects. This class of deformations strictly includes the elastic deformations; nonelastic defect-preserving deformations are called neutral and generally involve some kind of rearrangement, or slip, of the crystal lattice. Here we deal with slightly defective crystals, i.e. where defects are so few that a lattice is distinguishable at the microscopic scale. We factor neutral deformations into components which are exclusively elastic at the macroscopic level or exclusively slip at the microscopic level. Using direct methods of the calculus of variations we determine equalibrium [sic] configurations for defective crystals. As in Chipot & Kinderlehrer, we study the behavior of minimizing sequences and their state functions, and in order to take into account possible oscillations of the minimizing sequences we assume that solutions may be measure-valued."
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Irene Fonseca et al.A one dimensional nearest neighbor model of coarsening
http://repository.cmu.edu/math/688
http://repository.cmu.edu/math/688Thu, 17 May 2012 15:16:56 PDT
Abstract: "A one dimensional model of coarsening is developed in which the domain boundaries are points on a line, either finite (with periodic boundary conditions) or (doubly) infinite, and a domain is an interval between any two adjacent points. The postulated equation of motion for the length l of a given interval depends only on the two nearest neighbor interval lengths, yields a zero average rate of change of interval lengths and makes the state of equal interval lengths unstable. It is proved that coarsening occurs by the disappearance of intervals. A special power law form of the equation of motion, treated by an approximation which ignores correlations of the lengths of neighboring intervals, shows a self-similar behavior with an asymptotic distribution of reduced interval lengths at long times that is time-independent. Comparison of the approximate results with computer simulations is made."
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W. W. MullinsInterpretation of the Lavrentiev phenomenon by relaxation
http://repository.cmu.edu/math/687
http://repository.cmu.edu/math/687Thu, 17 May 2012 15:16:47 PDT
Abstract: "We consider functionals of the calculus of variations of the form F(u) = [integral]┬╣ΓéÇ f(x,u,u)╠ü dx defined for u [element of] W[superscript 1,infinity](0,1), and we show that the relaxed functional Γü╗F with respect to weak W[superscript 1,1](0,1) convergence can be written as Γü╗F(u) = [integral]┬╣ΓéÇ f(x,u,u)╠üdx + L(u), where the additional term L(u), called the Lavrentiev term, is explicitly identified in terms of F."
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Giuseppe Buttazzo et al.Equilibrium configurations of defective crystals
http://repository.cmu.edu/math/686
http://repository.cmu.edu/math/686Thu, 17 May 2012 15:16:39 PDT
Abstract: "A special class of defect preserving deformations in solid crystals, the neutral deformations, is studied. Using the theory of compensated compactness it is shown that neutral deformations are closed with respect to weak convergence and a variational analysis of such states is undertaken. In particular, by means of the div-curl lemma minimizing sequences and their respective Young's probability measures are characterized."
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Irene Fonseca et al.Relaxation of multiple integrals in the space BV(Ω;Rp)
http://repository.cmu.edu/math/685
http://repository.cmu.edu/math/685Thu, 17 May 2012 15:16:32 PDT
Abstract: "A characterization of the surface energy density for the relaxation in BV([omega];R[superscript p]) of the functional u -> [integral subscript omega]f(x, u(x), [delta]u(x))dx is obtained. A 'slicing' technique is used allowing to patch together sequences without increasing their total energy."
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Irene Fonseca et al.Models of pattern formation
http://repository.cmu.edu/math/684
http://repository.cmu.edu/math/684Thu, 17 May 2012 15:16:25 PDT
Abstract: "Patterned structures are represented by means of a potential equal to the sum of a non-convex functional with the perimeter functional. This is also a model of stable and metastable states in two-phase systems with surface tension. A generalization based on an extension of Fleming-Rishel's coarea formula allows to deal with very irregular configurations, with boundary of fractional dimension."
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A VisintinA variational problem arising from a model in thermodynamics
http://repository.cmu.edu/math/683
http://repository.cmu.edu/math/683Thu, 17 May 2012 15:16:18 PDT
Abstract: "We study a model for the thermodynamics of equilibrium of materials for which the free energy density depends not only on the concentration u but also on its first and second gradients."
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M MarcusPhase transitions and generalized motion by mean curvature
http://repository.cmu.edu/math/682
http://repository.cmu.edu/math/682Thu, 17 May 2012 15:16:11 PDT
Abstract: "We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a model for phase transition in polycrystalline material. We rigourously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities."
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L. C. Evans et al.A regularized equation for anisotropic motion-by-curvature
http://repository.cmu.edu/math/681
http://repository.cmu.edu/math/681Thu, 17 May 2012 15:16:03 PDT
Abstract: "For realistic interfacial energies the equations of anisotropic motion-by-curvature exhibit backward-parabolic behavior over portions of their domain, thereby inducing phenomena such as the formation of facets and wrinkles. In this paper we derive a physically consistent regularized equation that may be used to analyze such phenomena."
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Antonio Di Carlo et al.Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions
http://repository.cmu.edu/math/680
http://repository.cmu.edu/math/680Thu, 17 May 2012 15:15:55 PDT
Abstract: "We study the equations of viscoelasticity in a multidimensional setting for the 'no-traction' boundary data. For the sake of modeling phase transitions we do not assume ellipticity of the stored energy function W. We construct dynamics in W[superscript 1,2]([omega], r[superscript n]) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show certain kind of decay in time of solutions for arbitrary initial conditions."
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Piotr RybkaA minimization problem involving variation of the domain
http://repository.cmu.edu/math/679
http://repository.cmu.edu/math/679Thu, 17 May 2012 15:15:48 PDT
Abstract: "Existence, regularity properties and uniqueness of solutions for a minimization problem involving variation of the domain are obtained. This model arises in the study of defective solid crystals and it is shown that direct methods of the Calculus of Variations cannot be applied."
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Dacorogna et al.Some remarks on the Stefan problem with surface structure
http://repository.cmu.edu/math/678
http://repository.cmu.edu/math/678Thu, 17 May 2012 15:15:39 PDT
Abstract: "This paper discusses a generalized Stefan problem which allows for supercooling and superheating and for capillarity in the interface between phases. Simple solutions are obtained indicating the chief differences between this problem and the classical Stefan problem. A weak formulation of the general problem is given."
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Morton E. Gurtin et al.