The obstacle problem for nonlinear elliptic equations with variable growth and L1-data

The aim of this paper is twofold: to prove, for L ¹-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L ¹.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

Verified computation of solutions for obstacle problems with guaranteed L∞ error bound

In this paper, we consider a numerical enclosure method with guaranteed L^∞ error bound for the solutions of obstacle problems. Using the finite-element approximations and the explicit a priori error estimates for obstacle problems, we present an effective verification procedure that automatically generates on a computer a set which includes the exact solution. A particular emphasis is that our method needs no assumption of the existence of the solution of the original obstacle problems, but it follows as the result of computation itself. A numerical example for an obstacle problem is presented.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Error estimates for MAC-like approximations to the linear Navier-Stokes equations

Summary In this paper a priori error estimates are derived for the discretization error which results when the linear Navier-Stokes equations are solved by a method which closely resembles the MAC-method of Harlow and Welch. General boundary conditions are permitted and the estimates are in terms of the discreteL ² norm. A solvability result is given which also applies to a generalization of the method to the nonlinear case. This generalization is used in the last section to produce a numerical solution to the problem of flow around an obstacle.This work supported in part by Westinghouse Nuclear Energy Systems. Research Report #76-13     

Elliptic gradient constraint problem

We study the existence and regularity of gradient constraint problem. It arises in elastoplasticity and finance. First, we consider linear double obstacle problem which comes from viscosity solution to Hamilton–Jacobi equation and find the solution has C^1,α regularity by estimating Campanato-type integral oscillation. Then, by perturbation method and fixed point theorem in C^1,α space, we prove the existence of C^1,α solution.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Optimal Control of the Obstacle for an Elliptic Variational Inequality

An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H ¹ ₀ (Ω) so that the state is close to the desired profile while the H ¹ (Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal pairs are established. Accepted 11 September 1996     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

The obstacle problem for nonlinear elliptic equations with variable growth and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-data

The aim of this paper is twofold: to prove, for L ¹-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L ¹. Current address: Manel Sanchón, Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08007 Barcelona, Spain; e-mail: msanchon@maia.ub.es Authors’ addresses: J. F. Rodrigues, CMUC, Department of Mathematics, University of Coimbra, and FCUL/Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal; M. Sanchón and J. M. Urbano, CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal     

On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L² convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H⁻¹-type norm, but not L²-coercive. Nevertheless, the L² convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L² norm, or by properly choosing the finite element spaces, providing implicit control of the L² norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L² convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).     

Existence of Multiple Positive Solutions of Inhomogeneous Semi–Linear Elliptic Problems Involving Critical Exponents ∗

We consider the obstacle problem for the degenerate Monge-Ampére equation. We prove the existence of the greatest viscosity sub-solution u(x) below a given obstacle φ(x), and its C ^{1, 1}-regularity which is optimal. Then the solution satisfies the concave uniformly elliptic equation if it doesn't touch the obstacle. We use the author's previous work to show the C ^{1, α}-regularity of the free boundary, ?{u(x) = φ(x)}. Finally, we discuss the stability of this free boundary.     

Optimal L ∞(L 2)-Error Estimates for the DG Method Applied to Nonlinear Convection–Diffusion Problems with Nonlinear Diffusion

This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection–diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L ^∞(L ²)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular.     

Lp – Lq time decay estimate to the solution of the Cauchy problem for the system of equations describing nonlocal model of the thermoviscoelastic body

The aim of this paper is to present a new system of equations describing nonlocal model of thermoviscoelastic theory. We used the Papkin and Gurtin approach based on the constitutive relations for stress tensor σ(x), internal energy e(x) and heat flux q(x), with integral terms. Using the modified Cagniard-de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three-dimensional space. Basing on this matrix we represent in the explicit formula the solution of the Cauchy problem to this system of equations. Next, applying the method of Sobolev spaces, we proved the L^p–L^q time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

Finite volume method for the variational inequalities of first and second kinds

We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the H¹‐norm. For the simplified friction problem, we establish an abstract H¹‐error estimate, which implies the convergence if the exact solution u∈H¹(Ω) and the optimal error estimate if u∈H^{1 + α}(Ω),0 < α≤2. Copyright © 2015 John Wiley & Sons, Ltd.     

L^p- L^q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.