scholarly journals Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Shengda Zeng ◽  
Yunru Bai ◽  
Leszek Gasiński
Keyword(s):  
Obstacle Problem ◽  
Maximal Monotone ◽  
Main Tool ◽  
Partial Differential Operator ◽  
Obstacle Problems ◽  
Multivalued Operator ◽  
Convection Term ◽  
Nonlinear Elliptic ◽  
Multivalued Functions ◽  
The Right

AbstractIn this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.

OBSTACLE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS WITH NONHOMOGENEOUS NONLINEAR BOUNDARY CONDITIONS

2008 ◽  
Vol 18 (11) ◽  
pp. 1869-1893 ◽  
Author(s):  
FUENSANTA ANDREU ◽  
NOUREDDINE IGBIDA ◽  
JOSÉ M. MAZÓN ◽  
JULIÁN TOLEDO
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Maximal Monotone ◽  
Nonlinear Boundary Conditions ◽  
Obstacle Problems ◽  
Laplacian Operator ◽  
Usual Sense ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic

In this paper we study the questions of existence and uniqueness of solutions for equations of type - div a(x,Du) + γ(u) ∋ ϕ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x,Du) · η + β(u) ∋ ψ. The nonlinear elliptic operator div a(x,Du) modeled on the p-Laplacian operator Δp(u) = div (|Du|p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) ∩ β(0), [Formula: see text] and the data ϕ ∈ L1(Ω) and ψ ∈ L1(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solution has to be introduced.

scholarly journals Convergence analysis for double phase obstacle problems with multivalued convection term

2020 ◽  
Vol 10 (1) ◽  
pp. 659-672
Author(s):  
Shengda Zeng ◽  
Yunru Bai ◽  
Leszek Gasiński ◽  
Patrick Winkert
Keyword(s):  
Convergence Analysis ◽  
Obstacle Problem ◽  
Solution Set ◽  
Obstacle Problems ◽  
Convection Term ◽  
Solution Sets ◽  
Upper Limit ◽  
Double Phase

Abstract In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢n the solution sets of approximating problems, we prove the following convergence relation $$\begin{array}{} \displaystyle \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{array}$$ where w-lim supn→∞ 𝓢n and s-lim supn→∞ 𝓢n denote the weak and the strong Kuratowski upper limit of 𝓢n, respectively.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

Global weighted Orlicz estimates for irregular obstacle problems with general growth over bounded nonsmooth domains

2018 ◽  
Vol 20 (08) ◽  
pp. 1750083
Author(s):  
Yumi Cho
Keyword(s):  
Variational Inequality ◽  
Sobolev Spaces ◽  
Obstacle Problem ◽  
Obstacle Problems ◽  
Generalized Variational Inequality ◽  
Nonsmooth Domain ◽  
General Growth ◽  
Nonsmooth Domains

We study a generalized variational inequality with an irregular obstacle in the frame of Orlicz–Sobolev spaces. Over a bounded nonsmooth domain having a sufficiently flat boundary in the Reifenberg sense, a global weighted Orlicz estimate is established for the gradient of the solution to the obstacle problem assumed BMO smallness of a coefficient.

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

2015 ◽  
Vol 8 (2) ◽  
pp. 199-219 ◽  
Author(s):  
Chunxiao Wu ◽  
Justin W.L. Wan
Keyword(s):  
Obstacle Problem ◽  
Multigrid Method ◽  
Multigrid Methods ◽  
Coarse Grid ◽  
Obstacle Problems ◽  
Hjb Equations ◽  
Fine Grid ◽  
Multigrid Algorithm ◽  
Prolongation Operator ◽  
Approximate Scheme

AbstractIn this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.

scholarly journals Problematic issues of identification of a person as the main tool of combating money laundering in the circulation of electronic money

2020 ◽  
pp. 84-90
Author(s):  
M.V. Starynskyi ◽  
Zh.V. Khutorіanets
Keyword(s):  
Money Laundering ◽  
Main Tool ◽  
Basic Mechanism ◽  
Electronic Money ◽  
Bank Institution ◽  
Banking Institutions ◽  
Illegal Activities ◽  
Identification Of Persons ◽  
The Right ◽  
Combat Money Laundering

The article investigates some problematic issues concerning identification of persons that perform financial operations as the main way to combat money laundering in the cycle of e-cash. It draws readers’ attention to the speedy development of the Internet technologies that results in the increasing popularity of new payment systems including e-cash. It is emphasized that the advantages provided to some people by the use of e-cash in their economic operations can be used by other persons for certain criminal purposes. In this aspect e-cash has found its extensive application in the grey sector of the world economy and as such requires a more intensive control over its use. The authors of the investigation prove that identification of the persons using payment tools is currently one of the most effective ways to prevent e-cash from being used in illegal activities. International organizations whose purpose is to combat money laundering always recommend different states to improve their mechanisms of the identification of the persons involved in financial operations. The analysis of person’s identification applied to the customers of financial operations proves that the basic mechanism of reliable identification is the list of documents that identify a person being fixed in the national laws. The range of these documents and their quality vary in different countries, thus proving the peculiar features of these states. The issue of the identification of financial operation customers is in Ukraine in the phase of its intense formation. The laws establish only the main documents to identify a person. At the same time, in our opinion, the legislation currently in effect must foresee complete lists of the documents that a bank institution has the right and possibility to ask for with view of person’s identification. Such lists must take into consideration not only average situations but also cases of natural disasters, socio-political changes in the society, presence of the person on the territory of another country, etc. The approval of such lists should allow banking institutions to duly observe the provisions of our national legislation designed to prevent money laundering.

scholarly journals IMPROVEMENT OF TECHNOLOGICAL PROCESS OF DRILLING THIN SHEET STEELS

2020 ◽  
pp. 48-54
Author(s):  
Anatoliy Ostrovsky
Keyword(s):  
Sheet Metal ◽  
Technological Process ◽  
Agricultural Sector ◽  
Sheet Material ◽  
Thin Sheet ◽  
Small Diameter ◽  
Main Tool ◽  
Drilling Process ◽  
Different Diameters ◽  
The Right

This article considers the technological process of drilling holes in sheet metal. The existing typical technological process, with detailed analysis, provides grounds for substantiating the idea of an innovative solution. There is a connection between the improvement of economic indicators of this technological operation with the change of geometric parameters of the tool. The comparison of existing technological processes, operations, and transitions of production of round openings with the use of spiral drills is given. The traditional technique involves a clear sequence of the technological process with the use of tools for marking, kerning, and sequential use of screw drills of different diameters. Therefore, after the previous operation of marking the center of the two holes, perform the operation of kerning. The core (recess in the metal) prevents the deviation of the small diameter screw drill from the marking lines. After the above steps, the traditional technological process involves drilling a small diameter hole in order to direct the main tool in the right direction. To make the main hole perform special sharpening of the working part of the twist drill so that the diameter of the Central part of the tool is equal to the diameter of the first guide hole to avoid further displacement of the main hole and the next operation is to drill the hole to the desired diameter. As a result, the article raises the topic, given the widespread use of sheet metal, including in the agricultural sector, the feasibility of using an innovative idea to improve the technological process associated with the processing of sheet material, namely thin metal sheets. It is worth noting that for example, we consider a material that is characterized by its versatility, namely steel Ст 3. Despite the numerous sources devoted to the study of improving the drilling process, there are a number of issues, including the extension of this direction surfaces with a covering and without it, various thicknesses within the thin-walled hire. The most significant result of the improved technological process of drilling will be used in the system of efficient maintenance of agricultural machinery.

scholarly journals Regularity for the fully nonlinear parabolic thin obstacle problem

2021 ◽  
pp. 2150011
Author(s):  
Georgiana Chatzigeorgiou
Keyword(s):  
Parabolic Equation ◽  
Viscosity Solution ◽  
Nonlinear Parabolic Equation ◽  
Obstacle Problem ◽  
Elliptic Case ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Nonlinear Parabolic ◽  
Parabolic Case ◽  
Thin Obstacle

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

scholarly journals Qualitative Methods for the Inverse Obstacle Problem: A Comparison on Experimental Data

2019 ◽  
Vol 5 (4) ◽  
pp. 47 ◽  
Author(s):  
Martina T. Bevacqua ◽  
Roberta Palmeri
Keyword(s):  
Experimental Data ◽  
Qualitative Methods ◽  
Obstacle Problem ◽  
Sampling Method ◽  
Obstacle Problems ◽  
Morphological Properties ◽  
Joint Sparsity ◽  
Qualitative Approaches ◽  
Linear Sampling Method ◽  
Wide Range

Qualitative methods are widely used for the solution of inverse obstacle problems. They allow one to retrieve the morphological properties of the unknown targets from the scattered field by avoiding dealing with the problem in its full non-linearity and considering a simplified mathematical model with a lower computational burden. Very many qualitative approaches have been proposed in the literature. In this paper, a comparison is performed in terms of performance amongst three different qualitative methods, i.e., the linear sampling method, the orthogonality sampling method, and a recently introduced method based on joint sparsity and equivalence principles. In particular, the analysis is focused on the inversion of experimental data and considers a wide range of (distinct) working frequencies and different kinds of scattering experiments.

Sign in / Sign up

Export Citation Format

Share Document