Parabolic Obstacle Problem Research Articles

Parabolic Boundary Harnack Inequalities with Right-Hand Side

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side f∈Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f \\in L^q$$\\end{document} for q>n+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q > n+2$$\\end{document}. In the case of the heat equation, we also show the optimal C1-ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1-\\varepsilon }$$\\end{document} regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} in the parabolic obstacle problem and in the parabolic Signorini problem.

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.

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.

Accuracy of an Implicit Scheme for the Finite Element Method with a Penalty for a Nonlocal Parabolic Obstacle Problem

Accuracy of an Implicit Scheme for the Finite Element Method with a Penalty for a Nonlocal Parabolic Obstacle Problem

On Lewy Stampacchia inequalities for a pseudomonotone parabolic obstacle problem with [formula omitted]-data

The aim of this paper is to prove the existence of entropy solution, and the associated Lewy-Stampacchia inequalities in a renormalized form, to some parabolic obstacle problems governed by a Leray-Lions pseudomonotone operator in the presence of L1-data and homogeneous Dirichlet boundary conditions.

Optimal regularity for supercritical parabolic obstacle problems

AbstractWe study the obstacle problem for parabolic operators of the type , where L is an elliptic integro‐differential operator of order 2s, such as , in the supercritical regime . The best result in this context was due to Caffarelli and Figalli, who established the regularity of solutions for the case , the same regularity as in the elliptic setting.Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C1, 1 in space and time, and that this is optimal. We also deduce the regularity of the free boundary. Moreover, at all free boundary points , we establish the following expansion: with , and .

Regularity theory for fully nonlinear parabolic obstacle problems

We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is C∞ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension n−1 which is also ε-flat in space, for any ε>0.

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems

<abstract><p>We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.</p></abstract>

$$C^{2,\alpha }$$ regularity of free boundaries in parabolic non-local obstacle problems

We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian \((-\Delta )^s\) (and more general integro-differential operators) in the regime \(s>\frac{1}{2}\). We prove that once the free boundary is \(C^1\) it is actually \(C^{2,\alpha }\). To do so, we establish a boundary Harnack inequality in \(C^1\) and \(C^{1,\alpha }\) (moving) domains, providing that the quotient of two solutions of the linear equation, that vanish on the boundary, is as smooth as the boundary. As a consequence of our results we also establish for the first time optimal regularity of such solutions to nonlocal parabolic equations in moving domains.

On the Hölder regularity for obstacle problems to porous medium type equations

We show that signed weak solutions to parabolic obstacle problems with porous medium-type structure are locally Hölder continuous, provided that the obstacle is Hölder continuous.

Systems of Fully Nonlinear Parabolic Obstacle Problems with Neumann Boundary Conditions

We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron’s method. Our motivation stems from so called optimal switching problems on bounded domains.

On the time fractional heat equation with obstacle

We study a Caputo time fractional degenerate diffusion equation which we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α∈(0,1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. We also study the problem from the numerical point of view, comparing some finite different approaches, and showing the results of some tests. These results extend what recently proved in [1] for the case α=1.

A General Error Estimate For Parabolic Variational Inequalities

Abstract The gradient discretisation method (GDM) is a generic framework designed recently, as a discretisation in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical approximations of parabolic obstacle problems. This gives the convergence rates of several well-known conforming and non-conforming numerical methods. Numerical experiments based on the hybrid finite volume method are provided to verify the theoretical results.

The Elastic Flow with Obstacles: Small Obstacle Results

We consider a parabolic obstacle problem for Euler’s elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably ‘small’. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.

A convergence analysis of semi-discrete and fully-discrete nonconforming FEM for the parabolic obstacle problem

We propose and analyse a nonconforming finite element method for numerical approximation of the solution of a parabolic variational inequality associated with general obstacle. In this article, we carry out the error analysis for both the semi-discrete and fully-discrete schemes. We use the backward Euler method for time discretization and the lowest order Crouzeix-Raviart nonconforming finite element method for space discretization. The main motivation for the space discretization with Crouzeix-Raviart nonconforming finite element method to the parabolic obstacle problem is that it gives a natural -stable interpolation which is commutative with the time derivative. By taking full advantage of this commutative property together with -stability of interpolation, we derive an error estimate of the order for semi-discrete scheme, and error estimate of order for fully-discrete scheme in a certain norm defined precisely in the article.

Orlicz estimates for general parabolic obstacle problems with p(t,x)-growth in Reifenberg domains

This article shows a global gradient estimate in the framework of Orlicz spaces for general parabolic obstacle problems with p(t,x)-Laplacian in a bounded rough domain. It is assumed that the variable exponent p(t,x) satisfies a strong log-Holder continuity, that the associated nonlinearity is measurable in the time variable and have small BMO semi-norms in the space variables, and that the boundary of the domain has Reifenberg flatness.

On the asymptotic behavior of the solutions to parabolic variational inequalities

Abstract We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem

In this article, we propose and analyze conforming and discontinuous Galerkin (DG) finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in Rd(d=2,3). For fully discrete conforming method, we use globally continuous and piecewise linear finite element space. Whereas for the fully-discrete DG scheme, we employ piecewise linear finite element space for spatial discretization. The time discretization has been done by using the implicit backward Euler method. We present the error analysis for the conforming and the DG fully discrete schemes and derive an error estimate of optimal order O(h+Δt) in a certain energy norm defined precisely in the article. The analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that ut∈L2(0,T;L2(Ω)). The obstacle constraints are incorporated at the Lagrange nodes of the triangular mesh and the analysis exploits the Lagrange interpolation. We present some numerical experiment to illustrate the performance of the proposed methods.

Parabolic obstacle problems. Quasi-convexity and regularity

In a wide class of the so called Obstacle Problems of parabolic type it is shown how to improve the optimal regularity of the solution and as a consequence how to obtain space-time regularity of the corresponding free boundary.