A generalized Stefan problem in several space variables

A multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered. Regularization techniques and monotonicity methods are exploited. Existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.     

A singular-perturbed two-phase Stefan problem

The asymptotic behavior of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit, the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.     

A nonlinear two-phase Stefan problem with melting point gradient: a constructive approach

We consider a one-dimensional two-phase Stefan problem, modeling a layer of solid material floating on liquid. The model includes internal heat sources, variable total mass (resulting e.g. from sedimentation or erosion), and a pressure-dependent melting point. The problem is reduced to a set of nonlinear integral equations, which provides the basis for an existence and uniqueness proof and a new numerical method. Numerical results are presented.     

The variational inequality approach to the

Considering the one-phase Stefan problem, we present an account of some recent mathematical results within the framework of variational inequalities. We discuss several situations corresponding to different boundary conditions and different geometries, like the exterior problem, the continuous casting model, and the degenerate case of the quasi-steady model. We develop a few continuous-dependence results explaining their relevance to the stability properties of the solution and of the free boundary, including the asymptotic behaviour for large time, the stability for homogenization, and the perturbation of the Dirichlet boundary conditions.     

A variational problem in Hilbert space

Let? be a Hilbert space over the field of complex numbers, with inner product (g,h). Letf be a fixed element in?, and letH be a compact, self-adjoint linear operator on?. We find the maximum value ofQ _f (u)=|(f,u)|² in the classU of elementsu in? for which (u,u)=1, (u, Hu)=0.     

Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints

In this paper, we study Levitin–Polyak type well-posedness for generalized vector variational inequality problems with abstract and functional constraints. Various criteria and characterizations for these types of well-posednesses are given. This research is partially supported by the National Science Foundation of China and Shanghai Pujiang Program.     

The semigroup approach to first order quasilinear equations in several space variables

The Cauchy problem for u _t + Σ _{i = 1} ⁿ (φ _i (u)) _xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σ _{i = 1} ⁿ (φ _i (u)) _xi = h for h∈L ¹(Rⁿ), which in turn is studied via the regularized equation $$ u + \sum\nolimits_{i = 1}^n {\left( {\phi _i \left( u \right)} \right)} _{xi} - \varepsilon \Delta u = h $$ .     

A heuristic algorithm for constrained multi-source Weber problem – The variational inequality approach

For solving the well-known multi-source Weber problem (MWP), each iteration of the heuristic alternate location–allocation algorithm consists of a location phase and an allocation phase. The task of the location phase is to solve finitely many single-source Weber problems (SWP), which are reduced by the heuristic of nearest center reclassification for the customers in the previous allocation phase. This paper considers the more general and practical case – the MWP with constraints (CMWP). In particular, a variational inequality approach is contributed to solving the involved constrained SWP (CSWP), and thus a new heuristic algorithm for CMWP is presented. The involved CSWP in the location phases are reformulated into some linear variational inequalities, whose special structures lead to a new projection–contraction (PC) method. Global convergence of the PC method is proved under mild assumptions. The new heuristic algorithm using the PC method in the location phases approaches to the heuristic solution of CMWP efficiently, which is verified by the preliminary numerical results reported in this paper.     

Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

We present two extensions of Korpelevich's extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevich's extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.     

A non-interior-point smoothing method for variational inequality problem

In this paper, we focus on the variational inequality problem. Based on the Fischer-Burmeister function with smoothing parameters, the variational inequality problem can be reformulated as a system of parameterized smooth equations, a non-interior-point smoothing method is presented for solving the problem. The proposed algorithm not only has no restriction on the initial point, but also has global convergence and local quadratic convergence, moreover, the local quadratic convergence is established without a strict complementarity condition. Preliminary numerical results show that the algorithm is promising.     

A numerical approach to optimization problems with variational inequality constraints

Optimization problems with variational inequality constraints are converted to constrained minimization of a local Lipschitz function. To this minimization a non-differentiable optimization method is used; the required subgradients of the objective are computed by means of a special adjoint equation. Besides tests with some academic examples, the approach is applied to the computation of the Stackelberg—Cournot—Nash equilibria and to the numerical solution of a class of quasi-variational inequalities.Corresponding author.     

A generalized mixed vector variational-like inequality problem

In this paper, we introduce relaxed η-α-P-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η-complete semicontinuity and η-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

On generalized implicit vector variational inequality problems

In this paper, we introduce and study the generalized implicit vector variational inequality problems with set valued mappings in topological vector spaces. We establish existence theorems for the solution set of these problems be nonempty compact and convex. Our results extend the results by Fang and Huang [ Existence results for generalized implicit vector variational inequalities with multivalued mappings, Indian J. Pure and Appl. Math. 36(2005), 629–640.]     

A variational approach to the Steiner network problem

Supposen points are given in the plane. Their coordinates form a 2n-vectorX. To study the question of finding the shortest Steiner networkS connecting these points, we allowX to vary over a configuration space. In particular, the Steiner ratio conjecture is well suited to this approach and short proofs of the casesn=4, 5 are discussed. The variational approach was used by us to solve other cases of the ratio conjecture (n=6, see [11] and for arbitraryn points lying on a circle). Recently, Du and Hwang have given a beautiful complete solution of the ratio conjecture, also using a configuration space approach but with convexity as the major idea. We have also solved Graham's problem to decide when the Steiner network is the same as the minimal spanning tree, for points on a circle and on any convex polygon, again using the variational method.     

An evolution variational inequality related to a diffusion-absorption problem

An implicit non-steady free boundary problem is transformed into a variational inequality, which is solved by means of a semi-discretization technique.     

A variational inequality approach to constrained control problems for parabolic equations

A distributed optimal control problem for parabolic systems with constraints in state is considered. The problem is transformed to control problem without constraints but for systems governed by parabolic variational inequalities. The new formulation presented enables the efficient use of a standard gradient method for numerically solving the problem in question. Comparison with a standard penalty method as well as numerical examples are given.     

On the domain dependence of solutions to the two-phase Stefan problem

We prove that solutions to the two-phase Stefan problem defined on a sequence of spatial domains _n ^N converge to a solution of the same problem on a domain where is the limit of _n in the sense of Mosco. The corresponding free boundaries converge in the sense of Lebesgue measure on ^N.