223 research outputs found
Analysis of a class of penalty methods for computing singular minimizers
Get PDFAmongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computa- tional calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalization strategy. We establish convergence results under mild conditions on the stored energy function. Moreover, we present practical strategies for the solution of the discretized problems and for the choice of the penalty parameter
Nonconforming finite-element discretization of convex variational problems
Get PDFThe Lavrentiev gap phenomenon is a well-known effect in the calculus of variations, related to singularities of minimizers. In its presence, conforming finite-element methods are incapable of reaching the energy minimum. By contrast, it is shown in this work that for convex variational problems the nonconforming Crouzeix–Raviart finite-element discretization always converges to the correct minimizer and that the discrete energy converges to the correct limit
Lavrentiev Phenomenon in Microstructure Theory
No full textA variational problem arising as a model in martensitic phase transformation
including surface energy is studied. It explains the complex,
multi-dimensional pattern of twin branching which is often observed in a
martensitic phase near the austenite interface.
We prove that a Lavrentiev phenomenon can occur
if the domain is a rectangle. We show that this phenomenon
disappears under arbitrarily small shears
of the domain. We also prove that other perturbations of the problem lead to
an extinction of the Lavrentiev phenomenon
MESH-INDEPENDENCE OF SEMISMOOTH NEWTON METHODS FOR LAVRENTIEV-REGULARIZED STATE CONSTRAINED NONLINEAR OPTIMAL CONTROL PROBLEMS
No full textA class of nonlinear elliptic optimal control problems with mixed control-state constraints arising, e.g., in Lavrentiev-type regularized state constrained optimal control is considered. Based on its first order necessary optimality conditions, a semismooth Newton method is proposed and its fast local convergence in function space as well as a mesh-independence principle for appropriate discretizations are proved. The paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing Lavrentiev-parameter. The latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the Lavrentiev-parameter
Instrumentation for study of nanomaterials in NPI REZ (New laboratory for material study in Nuclear Physics Institute in REZ)
Get PDFNano-sized materials become irreplaceable component of a number of devices for every aspect of human life. The development of new materials and deepening of the current knowledge require a set of specialized techniques-deposition methods for preparation/modification of the materials and analytical tools for proper understanding of their properties. A thoroughly equipped research centers become the requirement for the advance and development not only in nano-sized field. The Center of Accelerators and Nuclear Analytical Methods (CANAM) in the Nuclear Physics Institute (NPI) comprises a unique set of techniques for the synthesis or modification of nanostructured materials and systems, and their characterization using ion beam, neutron beam and microscopy imaging techniques. The methods are used for investigation of a broad range of nano-sized materials and structures based on metal oxides, nitrides, carbides, carbon-based materials (polymers, fullerenes, graphenes, etc.) and nano-laminate composites (MAX phases). These materials can be prepared at NPI using ion beam sputtering, physical vapor deposition and molecular beam epitaxy. Based on the deposition method and parameters, the samples can be tuned to possess specific properties, e.g., composition, thickness (nm-μm), surface roughness, optical and electrical properties, etc. Various nuclear analytical methods are applied for the sample characterization. RBS, RBS-channeling, PIXE, PIGE, micro-beam analyses and Transmission Spectroscopy are accomplished at the Tandetron 4130MC accelerator, and additionally the Neutron Depth Profiling (NDP) and Prompt Gamma Neutron Activation (PGNA) analyses are performed at an external neutron beam from the LVR-15 research reactor. The multimode AFM facility provides further surface related information, magnetic/electrical properties with nano-metric precision, nano-indentation, etc
Composition operators on hardy spaces on Lavrentiev domains
No full textFor any simply connected domain., we prove that a Littlewood type inequality is necessary for boundedness of composition operators on H-p(Omega), 1 <= p < infinity, whenever the symbols are finitely-valent. Moreover, the corresponding "little-oh" condition is also necessary for the compactness. Nevertheless, it is shown that such an inequality is not sufficient for characterizing bounded composition operators even induced by univalent symbols. Furthermore, such inequality is no longer necessary if we drop the extra assumption on the symbol of being finitely-valent. In particular, this solves a question posed by Shapiro and Smith (2003). Finally, we show a striking link between the geometry of the underlying domain. and the symbol inducing the composition operator in H-p(Omega), and in this sense, we relate both facts characterizing bounded and compact composition operators whenever. is a Lavrentiev domain.Plan Nacional I+DGobierno de AragónDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu
Lavrentiev phenomenon, relaxation and some regularity results for anisotropic functionals
No full textWe study local minimizers of anisotropic variational integrals of the form J[u]=\int_{\Omega}f(\cdot,\nabla u)dx with integrand f satisfying a (p,\bar{q})-growth condition w.r.t. \nabla u and with D_{P}f(x,P) satisfying a Lipschitz condition w.r.t. x\in\Omega. If the Lavrentiev gap functional \mathcal{L} relative to J vanishes for all balls B_{R}\Subset\Omega and if \bar{q}<p(1+1/), then (partial) C^{1,\alpha}-regularity holds. Moreover, the bound on the exponents can be replaced by \bar{q}<p+1 provided we study locally bounded minimizers.
We also investigate the relaxation of global minimization problems and discuss the regularity of the corresponding solutions. The importance of the condition \mathcal{L}\equiv0 was recently discovered by Esposito, Leonetti and Mingione in [ELM], where besides other results the higher integrability of the gradient is proved even under weaker assumptions than used here
Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
Get PDFWe consider the classical functional of the Calculus of Variations of the form
I(u)=∫ΩF(x,u(x),∇u(x))dx,
where Ω is a bounded open subset of Rn and F : Ω × R × Rn → R is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function φ on ∂Ω. We formulate some conditions under which a given function in φ + W1,p0(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with φ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth
Beyond the point defect limit: Simulation methods for solid solutions and highly disordered systems
Get PDFWe discuss how two techniques, based on (1) lattice statics/lattice dynamics simulations and (2) Monte Carlo methods may be used to calculate the thermodynamic properties of solid solutions and highly disordered systems. The lattice statics/lattice dynamics calculations involve a full free-energy structural optimization of each of a number of configurations, followed by thermodynamic averaging. The Monte Carlo simulations include the explicit interchange of cations and use the semigrand canonical ensemble for chemical potential differences. Both methods are readily applied to high pressures and elevated temperatures without the need for any new parameterization; at agreement between the two techniques is better at high pressures where anharmonic terms are smaller. Vibrational contributions to thermodynamic quantities of mixing are examined. A range of examples, including binary oxides, garnets and carbonates, are used to illustrate the methods.Fil: Allan, N. L.. University of Bristol; Reino UnidoFil: Barrera, Gustavo Daniel. Universidad Nacional de la Patagonia; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Centro Nacional Patagónico; ArgentinaFil: Lavrentiev, M. Yu. Culham Science Centre; Reino UnidoFil: Freeman, C. L. University of Bristol; Reino UnidoFil: Todorov, I. T.. CLRC Daresbury Laboratory; Reino UnidoFil: Purton, J. A.. CLRC Daresbury Laboratory; Reino Unid
THE LAVRENTIEV GAP PHENOMENON FOR HARMONIC MAPS INTO SPHERES HOLDS ON A DENSE SET OF ZERO DEGREE BOUNDARY DATA
No full textAbstract. We prove that for each positive integer N the set of smooth, zero degree maps ψ: S2 → S2 which have the following three properties: (i) there is a unique minimizing harmonic map u: B3 → S2 which agrees with ψ on the boundary of the unit ball; (ii) this map u has at least N singular points in B3; (iii) the Lavrentiev gap phenomenon holds for ψ, i. e., the infimum of the Dirichlet en-ergies E(w) of all smooth extensions w: B3 → S2 of ψ is strictly larger than the Dirichlet energy B3 |∇u|2 of the (irregular) minimizer u, is dense in the set of all smooth zero degree maps φ: S2 → S2 endowed with the H1/2– topology. 1
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