Obstacle Problem Research Articles

An obstacle problem for the p-elastic energy

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler–Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

An Optimal ADMM for Unilateral Obstacle Problems

We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study.

Tumor Growth with a Necrotic Core as an Obstacle Problem in Pressure

Tumor Growth with a Necrotic Core as an Obstacle Problem in Pressure

The parabolic thin obstacle problem for the weighted biLaplacian

The parabolic thin obstacle problem for the weighted biLaplacian

Markov Decision Processes with Observation Costs: Framework and Computation with a Penalty Scheme

We consider Markov decision processes where the state of the chain is only given at chosen observation times and of a cost. Optimal strategies involve the optimization of observation times as well as the subsequent action values. We consider the finite horizon and discounted infinite horizon problems as well as an extension with parameter uncertainty. By including the time elapsed from observations as part of the augmented Markov system, the value function satisfies a system of quasivariational inequalities (QVIs). Such a class of QVIs can be seen as an extension to the interconnected obstacle problem. We prove a comparison principle for this class of QVIs, which implies the uniqueness of solutions to our proposed problem. Penalty methods are then utilized to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications that illustrate our framework. Funding: J. Tam is supported by the Engineering and Physical Sciences Research Council [Grant 2269738].

Screening of Germplasm and Construction of Evaluation System for Autotoxicity Tolerance during Seed Germination in Cucumber

Due to the widespread use of intensive cropping patterns, the problem of continuous cropping obstacle, which is dominated by autotoxicity, has been becoming more and more prominent. Although many methods have been proposed to overcome the continuous cropping obstacle of cucumber, no study has reported the screening and evaluation of cucumber germplasm resistant to autotoxicity. In this study, 28 physiological indices related to the cucumber bud stage under cinnamic acid (CA) treatment were determined. In total, 45 cucumber cultivars were classified into three groups using principal component analysis and cluster analysis, and a model for evaluating cucumber resistance to autotoxicity was developed. The evaluation model was validated using autotoxicity-tolerant and non-autotoxicity-tolerant cultivars. The results showed that the growth of non-autotoxicity-tolerant cultivars was significantly inhibited compared to autotoxicity-tolerant cultivars. This indicated that the evaluation model of cucumber autotoxicity tolerance is reliable. The results of this study provide a valuable reference for the application of cucumber autotoxicity-tolerant germplasm resources and the development of autotoxicity-tolerant genes.

Finite element method for fractional order parabolic obstacle problem with nonlinear source term

This paper considers the finite element method for a fractional order parabolic obstacle problem in time with a nonlinear source term. An error estimation of the spatially semi-discrete problem is discussed. A fully discrete problem is proposed by introducing the finite difference approximation for the time variable. The L2-unconditional stability and convergence of the approximate solution have been proven.

On a class of obstacle problem via Young measure in generalized Sobolev space

On a class of obstacle problem via Young measure in generalized Sobolev space

Regularity results for solutions to elliptic obstacle problems in limit cases

We prove the Lewy–Stampacchia’s inequality for elliptic variational inequalities with obstacle involving Leray–Lions type operator whose simpler model case is given by the following u∈W01,N(Ω)↦-ΔNu-divB(x)|u|N-2u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u \\in W^{1,N}_0(\\Omega )\\mapsto -\\Delta _N u-\ ext {div}\\left( B (x) |u|^{N-2}u \\right) \\end{aligned}$$\\end{document}where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is a smooth bounded domain of RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document} with N⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\geqslant 2$$\\end{document}, ΔNu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _N u$$\\end{document} denotes the classical N–Laplacian operator and the coefficient B:Ω→RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B:\\Omega \\rightarrow \\mathbb {R}^N$$\\end{document} belongs to a suitable Lorentz–Zygmund space. For this kind of obstacle problems, we also provide regularity results and amongst them we give sufficient conditions to get boundedness of solutions.

A C0 Nonconforming Virtual Element Method for the Kirchhoff Plate Obstacle Problem

This paper investigates a novel C0 nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal degrees of freedom to the traditional lowest-order C0 nonconforming VEM, which originally lacked such degrees. This addition not only facilitates error estimation but also enhances its intuitiveness. Importantly, our novel C0 nonconforming VEM naturally satisfies the constraints of the obstacle problem. We then establish an a priori error estimate for our novel C0 nonconforming VEM, with the result indicating that the lowest order of our method achieves optimal convergence. Finally, we present a numerical example to validate the theoretical result.

Singular elliptic measure data problems with irregular obstacles

We investigate elliptic irregular obstacle problems with p-growth involving measure data. Emphasis is on the strongly singular case 1<p≤2−1/n, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.

An Extended Courant Element on a Polytope with Application in Approximating an Obstacle Problem

An Extended Courant Element on a Polytope with Application in Approximating an Obstacle Problem

Dynamic asset allocation and consumption ratcheting with costs

This paper delves into studying the optimal decisions regarding consumption and investment for an agent operating within a finite time frame, where any reduction in consumption is intolerable. In contrast to Jeon et al. (2018), we assume that the agent incurs a proportional utility cost with each increase in consumption. To tackle this problem, we employ the duality method to define the dual problem and investigate the resulting upper obstacle problem using PDE techniques. Our study reveals the existence of a critical time attributed to the utility cost, beyond which not only does the agent cease to further increase their consumption, but they also abstain from investing in the risky asset. Furthermore, the existence of the critical time makes free boundary analysis even more challenging, as this is because the presence of the critical time can also lead to the disruption of the monotonicity of the free boundary.

Free boundary regularity in the fully nonlinear parabolic thin obstacle problem

Abstract We study the regularity of the free boundary in the fully nonlinear parabolic thin obstacle problem. Under the assumption of time semiconvexity, our main result establishes that the free boundary is a C 1 C^{1} graph in x near any regular free boundary point.

Optimal regularity for the fully nonlinear thin obstacle problem

In this work, we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent \alpha_{F} such that u is C^{1,\alpha_F} on either side of the obstacle. In order to do that, we prove the uniqueness of blow-ups at regular points, as well as an expansion for the solution there. Finally, we also prove that if the operator is rotationally invariant, then \alpha_{F} \ge \frac{1}{2} and the solution is always C^{1,1/2} .

Obstacle Problems with Double Boundary Condition for Least Gradient Functions in Metric Measure Spaces

AbstractIn the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain $$\Omega $$ Ω of functions bounded between two obstacle functions inside $$\Omega $$ Ω , and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when $$\Omega $$ Ω is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of $$ \varepsilon $$ ε -weak solutions as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.

Pointwise adaptive non-conforming finite element method for the obstacle problem

Pointwise adaptive non-conforming finite element method for the obstacle problem

On the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients

This work is devoted to the study of the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems.

Accuracy of an implicit scheme for the finite element method with a penalty for a nonlocal parabolic obstacle problem

In order to solve a parabolic variational inequality with a nonlocal spatial operator and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained.

Numerical Approximation of the Solution of an Obstacle Problem Modelling the Displacement of Elliptic Membrane Shells via the Penalty Method

Numerical Approximation of the Solution of an Obstacle Problem Modelling the Displacement of Elliptic Membrane Shells via the Penalty Method