Variational Analysis of Non-Coercive Optimization Problems in Anisotropic Sobolev Spaces

This article is concerned with the application of the Newton-imbedding iteration procedure to nonlinear boundary value problems in Sobolev spaces. A simple model problem for the second-order semilinear elliptic equations is considered to illustrate the main idea. The essence of this method hinges on the a priori estimates of solutions of the associated linear problem in appropriate Sobolev spaces. It is to our surprise that H 1(Ω)-solution is not smooth enough to guarantee the convergence of the sequence generated by the procedure. Existence and uniqueness of solution to the original nonlinear problem are established constructively. An application of this approach to the Lamé system with nonlinear body force and its generalization to contain a nonlinear surface traction in elasticity is also discussed. †Dedicated to Professor Guochun Wen on the occasion of his 70th birthday.

Previous article Next article Optimal Control Problems in Sobolev Spaces with WeightsClaudia SimionescuClaudia Simionescuhttps://doi.org/10.1137/0314011PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe consider an optimal control problem in certain Sobolev spaces with weights used systematically by F. Trèves in [1], [2]. The notations and definitions are the same as in [2] and [3].[1] François Trèves, Domination et problèmes aux limites de type mixte, C. R. Acad. Sci. Paris, 245 (1957), 2454–2457 MR0098912 (20:5362) 0079.33202 Google Scholar[2] François Trèves, Relations de domination entre opérateurs différentiels, Acta Math., 101 (1959), 1–139 MR0125322 (23:A2625) CrossrefISIGoogle Scholar[3] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Avant propos de P. Lelong, Dunod, Paris, 1968xiii+426 MR0244606 (39:5920) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Bibliography6 October 2017 Cross Ref Linearization by means of optimal control for a model in plasma physicsInternational Journal of Engineering Science, Vol. 18, No. 2 Cross Ref Control for an hereditary systemMathematics and Computers in Simulation, Vol. 21, No. 2 Cross Ref Notes On a Mathematical Model in Plasma Physics Cross Ref Optimal control problems in sobolev spaces with weights. Numerical approaches applications to plasma optimal control and time delay problems21 May 2005 Cross Ref Numerical methods for a generalized optimal control problem Cross Ref Volume 14, Issue 1| 1976SIAM Journal on Control and Optimization History Submitted:27 June 1974Published online:03 August 2006 InformationCopyright © 1976 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0314011Article page range:pp. 137-143ISSN (print):0363-0129ISSN (online):1095-7138Publisher:Society for Industrial and Applied Mathematics

We consider the Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity and potential in the body based on the measurement of the boundary voltages on the m m electrodes for a given electrode current. The variational formulation is introduced as a PDE constrained coefficient optimal control problem in Sobolev spaces with dominating mixed smoothness. Electrical conductivity and boundary voltages are control parameters, and the cost functional is the L 2 L_2 -norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. EIT optimal control problem is fully discretized using the method of finite differences. The existence of the optimal control and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.

We prove the parametric version of Forstnerič splitting lemma for certain families of weakly pseudoconvex domains allowing regularity and stability of Kohn's solution to ∂ ¯ -problem in Sobolev spaces with exponent 2.

In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips.

We consider the most general class of multipoint boundary-value problems for systems of linear ordinary differential equations of the first order whose solutions belong to a given Sobolev space $$ {W}_p^n $$n ϵ ℕ, 1 ≤ p ≤ ∞: Sufficient constructive conditions under which the solutions of these problems are continuous with respect to the parameter e for e = 0 in the space $$ {W}_p^n $$ are established.

The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the solvability of such problems is investigated, their Fredholm properties are established, and their indexes and the dimensions of their kernels and co-kernels are found. In addition, necessary and sufficient conditions of continuity in the parameter of the solutions of the introduced classes of boundary-value problems in Sobolev spaces of an arbitrary order are obtained.

‎‎In this paper‎, ‎we prove the existence of infinitely many solutions of a system of boundary value problems involving flux boundary conditions in anisotropic variable exponent Sobolev spaces‎, ‎by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces‎.

The purpose of the textbook is to present the theory of generalized functions, its methods and its application to solving problems of mathematical physics in various spaces. The book covers the main spaces of generalized functions, including the space of generalized functions of slow growth and Sobolev spaces. Much attention is paid to methods associated with the use of the Fourier transform in these spaces, including methods of pseudodifferential operators, which are used to study elliptic problems in Sobolev spaces. Examples are given of the application of the theory of generalized functions to solving a number of problems of mathematical physics in spaces of functions of slow growth and Sobolev spaces. The book is intended for students of the Faculty of Computational Mathematics and Cybernetics of Moscow State University named after M.V. Lomonosov. It can be used by students and graduate students of mathematical specialties from other universities.

In this work we discuss the generalized treatment of the deformable registration problem in Sobolev spaces. We extend previous approaches in two points: 1) by employing a general energy model which includes a regularization term, and 2) by changing the notion of distance in the Sobolev space by problem-dependent Riemannian metrics. The actual choice of the metric is such that it has a preconditioning effect on the problem, it is applicable to arbitrary similarity measures, and features a simple implementation. The experiments demonstrate an improvement in convergence and runtime by several orders of magnitude in comparison to semi-implicit gradient flows in L2. This translates to increased accuracy in practical scenarios. Furthermore, the proposed generalization establishes a theoretical link between gradient flow in Sobolev spaces and elastic registration methods.

We introduce function spaces useful for the analysis of elliptic equations and for the error estimate of the finite element method. In particular, we review Sobolev spaces and the important properties of functions in these spaces, including the extension, embedding, and trace theorems. We also discuss the well-posedness and regularity estimates for elliptic boundary value problems in Sobolev spaces. In turn, we present key steps to derive the finite element error analysis. The trace and regularity results usually depend on the smoothness of the domain, and the nonsmooth points on the boundary can lead to singularities in the solution. For the conciseness of the presentation, some results are summarized without proofs. Readers will be referred to specific references for more details. This chapter is suitable for readers who need a review of basic results in Sobolev spaces and who are starting to work on finite element error analysis for elliptic equations.

Studying the problem of unsteady waves on the surface of an infinitely deep heavy incompressible ideal fluid, we derive equations for the height of the free surface as well as the vertical and horizontal components of velocity on the free surface. We prove that the initial-boundary value water waves problem is short-time solvable in Sobolev spaces.

We study the solvability of the Riemann-Hilbert and Poincare problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W21 (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincare problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.

A fundamental tool for the application of the direct method of the calculus of variations to minimum problems for functionals defined on Sobolev spaces is the following well-known theorem (see e.g. Adams (1975)).

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces.