Boundary element methods for variational inequalities

In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in $\widetilde{H}^{1/2}(\Gamma )$ . In addition to error estimates in the energy norm we also provide, by applying the Aubin–Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including $L_2(\Gamma )$ . The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results.     

A projection subgradient method for solving optimization with variational inequality constraints

In this paper, we propose a projection subgradient method for solving some classical variational inequality problem over the set of solutions of mixed variational inequalities. Under the conditions that $T$ is a $\Theta $ -pseudomonotone mapping and $A$ is a $\rho $ -strongly pseudomonotone mapping, we prove the convergence of the algorithm constructed by projection subgradient method. Our algorithm can be applied for instance to some mathematical programs with complementarity constraints.     

Regularization and iterative methods for monotone inverse variational inequalities

We consider the monotone inverse variational inequality: find $x\in H$ such that $$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$ where $\Omega $ is a nonempty closed convex subset of a real Hilbert space $H$ and $f:H\rightarrow H$ is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.     

Minkowski Decomposition of Associahedra and Related Combinatorics

Realisations of associahedra with linear non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila et al. (Discret Comput Geom, 43:841–854, 2010). The coefficients $y_I$ of such a Minkowski decomposition can be computed by Möbius inversion if tight right-hand sides $z_I$ are known not just for the facet-defining inequalities of the associahedron but also for all inequalities of the permutahedron that are redundant for the associahedron. We show for certain families of these associahedra: (1) How to compute the tight value $z_I$ for any inequality that is redundant for an associahedron but facet-defining for the classical permutahedron. More precisely, each value $z_I$ is described in terms of tight values $z_J$ of facet-defining inequalities of the corresponding associahedron determined by combinatorial properties of $I$ . (2) The computation of the values $y_I$ of Ardila, Benedetti & Doker can be significantly simplified and depends on at most four values $z_{a(I)},\,z_{b(I)},\,z_{c(I)}$ and $z_{d(I)}$ . (3) The four indices $a(I),\,b(I),\,c(I)$ and $d(I)$ are determined by the geometry of the normal fan of the associahedron and are described combinatorially. (4) A combinatorial interpretation of the values $y_I$ using a labeled $n$ -gon. This result is inspired from similar interpretations for vertex coordinates originally described by Loday and well-known interpretations for the $z_I$ -values of facet-defining inequalities.     

Quasilinear parabolic variational inequalities with multi-valued lower-order terms

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.     

Existence and extremal solutions of parabolic variational–hemivariational inequalities

We consider quasilinear parabolic variational–hemivariational inequalities in a cylindrical domain $Q=\Omega \times (0,\tau )$ of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q j^o(x,t, u;v-u)\,dxdt\ge 0,\ \ \forall \ v\in K, \end{aligned}$$ where $K\subset X_0=L^p(0,\tau ;W_0^{1,p}(\Omega ))$ is some closed and convex subset, $A$ is a time-dependent quasilinear elliptic operator, and $s\mapsto j(\cdot ,\cdot ,s)$ is assumed to be locally Lipschitz with $(s,r)\mapsto j^o(x,t, s;r)$ denoting its generalized directional derivative at $s$ in the direction $r$ . The main goal of this paper is threefold: first, an existence and comparison principle is proved; second, the existence of extremal solutions within some sector of appropriately defined sub-supersolutions is shown; third, the equivalence of the above parabolic variational–hemivariational inequality with an associated multi-valued parabolic variational inequality of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q \eta \, (v-u)\,dxdt\ge 0,\ \ \forall \ v\in K \end{aligned}$$ with $\eta (x,t)\in \partial j(x,t, u(x,t))$ is established, where $s\mapsto \partial j(x,t, s)$ denotes Clarke’s generalized gradient of the locally Lipschitz function $s\mapsto j(\cdot ,\cdot ,s)$ .     

On the W^{2,1+\varepsilon }-estimates for the Monge-Ampère equation and related real analysis

We build upon the techniques introduced by De Philippis and Figalli regarding $W^{2,1+\varepsilon }$ bounds for the Monge-Ampère operator, to improve the recent $A_\infty $ estimates for $\Vert D^2 \varphi \Vert $ to $A_2$ ones. Also, we prove a $(1,2)-$ Poincaré inequality and weak $(q,p)-$ Poincaré inequalities associated to the Monge-Ampère quasi-metric structure. In turn, these Poincaré inequalities are used to prove Harnack’s inequality for non-negative solutions to the linearized Monge-Ampère under minimal geometric assumptions.     

The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}

Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).     

Convergence of inexact Newton methods for generalized equations

For solving the generalized equation $f(x)+F(x) \ni 0$ , where $f$ is a smooth function and $F$ is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by $$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$ where $Df$ is the derivative of $f$ and the sequence of mappings $R_k$ represents the inexactness. We show how regularity properties of the mappings $f+F$ and $R_k$ are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.     

Singular perturbation method for inhomogeneous nonlinear free boundary problems

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.     

Existence theorem for a class of generalized quasi-variational inequalities

In this paper we consider a class of generalized quasi-variational inequalities. The variational problem is studied in the convex set \(X\times Y\) , with \(Y\) bounded and \(X\) unbounded. In the latter settings, we investigate about the solvability of the problem. In particular, by using the perturbation theory, we give an existence result of the solution without requesting any coercivity hypothesis on the operator. Finally, we give an application to the obtained theoretical results in terms of an economic equilibrium problem.     

A generalized f-projection algorithm for inverse mixed variational inequalities

In this paper, a new inverse mixed variational inequality is introduced and studied in Hilbert spaces, which provides a model for the study of traffic network equilibrium control problems. An iterative algorithm involving the generalized $f$ -projection operator for solving inverse mixed variational inequalities is constructed and the convergence of sequences generated by the algorithm is given under some suitable conditions.     

Lattice-free sets,multi-branch split disjunctions,and mixed-integer programming

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734, 2008). By analyzing $n$ -dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with $n$ integer variables is a $t$ -branch split cut. We use this result to give a finite cutting-plane algorithm to solve mixed-integer programs. We also show that the minimum value $t$ , for which all facets of polyhedral mixed-integer sets with $n$ integer variables can be generated as $t$ -branch split cuts, grows exponentially with $n$ . In particular, when $n=3$ , we observe that not all facet-defining inequalities are 6-branch split cuts.     

Weighted Dunkl transform inequalities and application on radial Besov spaces

In Dunkl theory on $\mathbb R ^d$ which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the $d$ -dimensional Hardy inequalities whose constants are sharp and independent of $d$ . Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.     

On efficiency and mixed duality for a new class of nonconvex multiobjective variational control problems

In this paper, we extend the notions of \((\Phi ,\rho )\) -invexity and generalized \((\Phi ,\rho )\) -invexity to the continuous case and we use these concepts to establish sufficient optimality conditions for the considered class of nonconvex multiobjective variational control problems. Further, multiobjective variational control mixed dual problem is given for the considered multiobjective variational control problem and several mixed duality results are established under \((\Phi ,\rho )\) -invexity.     

Convergence of distributed optimal control problems governed by elliptic variational inequalities

First, let u _g be the unique solution of an elliptic variational inequality with source term g. We establish, in the general case, the error estimate between $u_{3}(\mu)=\mu u_{g_{1}}+ (1-\mu)u_{g_{2}}$ and $u_{4}(\mu)=u_{\mu g_{1}+ (1-\mu) g_{2}}$ for ????[0,1]. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy g for each positive heat transfer coefficient h given on a part of the boundary of the domain. For a given cost functional and using some monotony property between u ₃(??) and u ₄(??) given in Mignot (J.?Funct. Anal. 22:130?C185, 1976), we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter h goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot??s conical differentiability) which is a great advantage with respect to the proof given in Gariboldi and Tarzia (Appl. Math. Optim. 47:213?C230, 2003), for optimal control problems governed by elliptic variational equalities.     

New inequalities for q-ary constant-weight codes

Using double counting, we prove Delsarte inequalities for \(q\) -ary codes and their improvements. Applying the same technique to \(q\) -ary constant-weight codes, we obtain new inequalities for \(q\) -ary constant-weight codes.     

3-Variable Additive {\rho} -Functional Inequalities and Equations

In this paper, we introduce and investigate additive \({\rho}\) -functional inequalities associated with the following additive functional equations $$\begin{array}{lll} \,\,\,\,\,\,\, f(x+y+z) - f(x)-f(y)-f(z) \,\,\,\, = 0 \\ 2f \left(\frac{x+y}{2}+z \right) - f(x)-f(y)-2f(z) = 0 \\ \,\,2f \left(\frac{x+y+z}{2} \right) - f(x)-f(y)-f(z) = 0\end{array}$$ Furthermore, we prove the Hyers–Ulam stability of the additive \({\rho}\) -functional inequalities in complex Banach spaces and prove the Hyers–Ulam stability of additive \({\rho}\) -functional equations associated with the additive \({\rho}\) -functional inequalities in complex Banach spaces.     

Homogenization of Elasticity Problems with Boundary Conditions of Signorini type

In a perforated domain $\Omega ^\varepsilon = \Omega \cap \varepsilon \omega $ formed of a fixed domain Ω and an ε-compression of a 1-periodic domain ω, we consider problems of elasticity for variational inequalities with boundary conditions of Signorini type on a part of the surface $S_0^\varepsilon $ of perforation. We study the asymptotic behavior of solutions as ε → 0 depending on the structure of the set $S_0^\varepsilon $ . In the general case, the limit (homogenized) problem has the two distinguishing properties: (i) the limit set of admissible displacements is determined by nonlinear restrictions almost everywhere in the domain Ω, i.e., in the limit, the Signorini conditions on the surface $S_0^\varepsilon $ can turn into conditions posed at interior points of Ω (ii) the limit problem is stated for an homogenized Lagrangian which need not coincide with the quadratic form usually determining the homogenized elasticity tensor. Theorems concerning the homogenization of such problems were obtained by the two-scale convergence method. We describe how the limit set of admissible displacements and the homogenized Lagrangian depend on the geometry of the set $S_0^\varepsilon $ on which the Signorini conditions are posed.