Elliptic Obstacle Problems Research Articles

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.

Besov regularity for a class of elliptic obstacle problems with double-phase Orlicz growth

In this paper, we consider local Besov regularity of to the elliptic variational inequality with double-phase Orlicz growth:∫Ω〈A(x,Du),D(u−φ)〉dx≤∫Ω〈B(x,F),D(u−φ)〉dx∀φ∈Kψ(Ω), where Kψ(Ω) is an admissible set constrained by an obstacle function ψ(x). Here, each of nonlinearities A and B is supposed to be (Φ1,Φ2) double-phase with Orlicz type growth, respectively, under the assumption supt∈R+⁡Φ2(t)Φ1(t)+Φ11+α/n(t)<+∞ for a Hölder exponent α∈(0,1] of the coefficient a(x). We prove that a fractional differentiability of Du is reflected by extra differentiability assumption on the obstacle function and the external force, under a suitable regularity on the coefficient 0≤a(x)∈C0,α(Ω). This is an extension of recent work [44] from the elliptic problems of (p,q) growth to the setting of double phase Orlicz growth.

Calderón-Zygmund estimates for nonlinear elliptic obstacle problems with log-BMO matrix weights

Calderón-Zygmund estimates for nonlinear elliptic obstacle problems with log-BMO matrix weights

On solving elliptic obstacle problems by constant abs-linearization

We consider optimal control problems governed by an elliptic variational inequality of the first kind, namely the obstacle problem. The variational inequality is treated by penalization, which leads to optimization problems governed by a nonsmooth semi-linear elliptic PDE. The CALi algorithm is then applied for the efficient solution of these nonsmooth optimization problems. The special feature of the optimization algorithm CALi is the treatment of the nonsmooth Lipschitz-continuous operators abs, max and min, which allows to explicitly exploit the nonsmooth structure. Stationary points are located by appropriate decomposition of the optimization problem into so-called smooth constant abs-linearized problems. Each of these constant abs-linearized problems can be solved by classical means. The comprehensive algorithmic concept is presented, and its performance is discussed through examples.

Systems of fully nonlinear degenerate elliptic obstacle problems with Dirichlet boundary conditions

In this paper, we prove existence and uniqueness of viscosity solutions to the following system: For $$ i\in \left\{ 1,2,\dots ,m\right\} $$ $$\begin{aligned}{} & {} \min \biggl \{ F\bigl ( y,x,u_{i}(y,x),D u_{i}(y,x),D^2 u_{i}(y,x)\bigl ), u_{i}(y,x)-\max _{j\ne i}\bigl ( u_{j}(y,x)-c_{ij}(y,x)\bigl )\biggl \}\\{} & {} =0, \left( y,x \right) \in \Omega _{L}\\{} & {} u_{i}(0,x)=g_{i}(x), x\in \bar{\Omega },\ u_i(y,x)=f_i(y,x), (y,x)\in (0,L)\times \partial {\Omega } \end{aligned}$$ where $$ \Omega \subset \mathbb {R}^n $$ is a bounded domain, $$ \Omega _{L}:=(0,L)\times \Omega $$ and $$ F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}$$ is a general second-order partial differential operator which covers even the fully nonlinear case. (We will call a second-order partial differential operator $$F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}$$ fully nonlinear if and only if, it has the following form $$\begin{aligned} F \left( y,x,u,D_x u,D_{xx}^2 u\right) :=\sum _{|\alpha |=2}\alpha _{\alpha }\left( y,x,u,D_x u,D_{xx}^2 u \right) D^{\alpha }u(y,x)+\alpha _{0}\left( y,x,u,D_x u \right) \end{aligned}$$ with the restriction that at least one of the functional coefficients $$ \alpha _{\alpha },\ |\alpha |=2, $$ contains a partial derivative term of second order.) Moreover, F belongs to an appropriate subclass of degenerate elliptic operators. Regarding uniqueness, we establish a comparison principle for viscosity sub and supersolutions of the Dirichlet problem. This system appears among others in the theory of the so-called optimal switching problems on bounded domains.

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Optimal control of elliptic obstacle problems with mixed boundary conditions

We study optimal control of an elliptic obstacle problem with mixed boundary conditions whose weak formulation is a nonlinear variational inequality. We put the control on both the boundary and obstacles. Existence of optimal solutions is proved and the necessary conditions of optimality are derived by the Lagrange multiplier rule and approximation techniques.

Pointwisea posteriorierror analysis of a discontinuous Galerkin method for the elliptic obstacle problem

AbstractWe present a posteriori error analysis in the supremum norm for the symmetric interior penalty discontinuous Galerkin method for the elliptic obstacle problem. We construct discrete barrier functions based on appropriate corrections of the conforming part of the solution obtained via a constrained averaging operator. The corrector function accounts properly for the nonconformity of the approximation and it is estimated by direct use of the Green’s function of the unconstrained elliptic problem. The use of the continuous maximum principle guarantees the validity of the analysis without mesh restrictions but with shape regularity. The proposed residual-type estimators are shown to be reliable and efficient. Numerical results in two dimensions are included to verify the theory and validate the performance of the error estimator.

Higher differentiability for solutions of a general class of nonlinear elliptic obstacle problems with Orlicz growth

Higher differentiability for solutions of a general class of nonlinear elliptic obstacle problems with Orlicz growth

Two neural-network-based methods for solving elliptic obstacle problems

Two neural-network-based numerical schemes are proposed to solve the classical obstacle problems. The schemes are based on the universal approximation property of neural networks, and the cost functions are taken as the energy minimization of the obstacle problems. We rigorously prove the convergence of the two schemes and derive the convergence rates with the number of neurons N. In the simulations, two example problems (both 1-D and 2-D) are used to verify the convergence rate of the methods and the quality of the results.

On W^{2,p}-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions

This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and W^{2,p}-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.

Inverse problems for a class of elliptic obstacle problems involving multivalued convection term and weighted $(p,q)$-Laplacian

In this paper, we study the inverse problem of estimating discontinuous parameters and boundary data in a nonlinear elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is given as a sum of a weighted p-Laplacian and a weighted q-Laplacian (called the weighted -Laplacian), a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. Under general hypotheses on the data, we prove the existence of a nontrivial solution to the elliptic obstacle problem, which depends on the first eigenvalue of the Steklov eigenvalue problem for the p-Laplacian. We also establish the boundedness and the closedness of the solution set to the problem. For the parameter-to-solution map, we give a convergence result of the Kuratowski type and prove the solvability of the inverse problem.

W 1,γ(·)-estimate to non-uniformly elliptic obstacle problems with borderline growth

ABSTRACT In this paper, we are devoted to a global -estimate for elliptic obstacle problem with double phase growth for the borderline setting in nonsmooth domain. More precisely, we consider the variational inequality whose characteristics of ellipticity and growth switch between a type of polynomial and its logarithm, then we prove a variable exponent Calderón-Zygmund estimate with an implication that provided that the nonlinearity satisfies a small BMO condition while the boundary of the underlying domain is flat in the sense of Reifenberg.

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.

Pointwise a posteriori error analysis of quadratic finite element method for the elliptic obstacle problem

In this article, we study a posteriori error analysis of quadratic finite element method in the maximum norm for the elliptic obstacle problem. We discuss the reliability and the efficiency of the proposed a posteriori error estimator. In the analysis, regularized Green’s function plays a crucial role and together with that, in obtaining the sign of the discrete Lagrange multiplier we have exploited the property that midpoint quadrature rules are exact for quadratic polynomials. Numerical results are performed to illustrate the convergence behavior of a posteriori error estimator through various test examples.

Gradient potential estimates in elliptic obstacle problems with Orlicz growth

In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. Furthermore, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on \(C^{1,\alpha }\)-regularity of solutions.

Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

In 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.

The domain derivative for semilinear elliptic inverse obstacle problems

&lt;p style='text-indent:20px;'&gt;We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.&lt;/p&gt;

Besov regularity for the gradients of solutions to non-uniformly elliptic obstacle problems

In this paper, we use the finite difference argument to prove a higher fractional differentiability in the scale of Besov spaces for the gradients of solutions of the double phase elliptic problems with unilateral obstacle:∫Ω〈A(x,Du),D(u−v)〉dx≤∫Ω〈|F|p−2F+a(x)|F|q−2F,D(u−v)〉dx. A key feature of our study consists in assuming that the nonlinearity A(x,Du) has an anisotropic (p,q)-growth for any 2≤p<q and qp<1+αn with 0≤a(⋅)∈C0,α(Ω‾) for α∈(0,1].

Inverse problems for nonlinear quasi-variational hemivariational inequalities with application to obstacle problems of elliptic type

In this paper we study an inverse problem governed by a constrained nonlinear elliptic quasi-variational hemivariational inequality. The inequality involves two nondifferentiable functions which directly depend on solutions. We solve the direct problem and obtain the properties of weak closedness and uniform boundedness of solutions, which develops some existing results in literature. Then, using a regularized optimization method, we deal with an inverse problem to seek the identification of parameter functions appearing in the leading operator and constraint set. Finally, we apply the abstract result to a concrete inverse problem of identifying the parameter and obstacle functions in elliptic obstacle problems with the p-Laplace operator and mixed boundary conditions.