Noncoercive Problems Research Articles

A study of sign-changing Poisson-type equations in two configurations

Abstract. In this paper, We are interested in specific non-coercive issues concerning electromagnetic wave propagation in the presence of metals or particular metamaterials. We focus on some non-coercive problems which cannot be studied by a classical Lax-Milgram approach. We consider the sign-changing Poisson-type equations with the homogeneous Dirichlet boundary conditions in the circular configuration and in the three square configuration. We utilize a method called T-isomorphism, which allows the conversion of non-coercive problems into coercive ones, to examine specific Poisson-type equations with sign-changing parameters. We first use the classical method to transform the equations into the corresponding variational formulations, and then apply the T-isomorphism method to show the well-posedness of these variational formulations with some parameters. By applying appropriate isomorphisms, we can conclude the well-posedness of the related variational formulations with certain parameters, which constitute the main findings of this paper. After giving appropriate isomorphisms, we can deduce the well-posedness of the corresponding variational formulations with some parameters, which are our main results in this paper.

Non-coercive problems for elastic plates with thin junction

We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other.

Noncoercive problems for elastic bodies with thin elastic inclusions

The paper concerns boundary value problems for an elastic body with a thin elastic inclusion for a noncoercive case. The inclusion is assumed to be delaminated thus forming a crack between the inclusion and the surrounding elastic body. To provide a mutual nonpenetration between crack faces, we consider inequality type boundary conditions with unknown set of a contact. In the paper, a solution existence of the equilibrium problems is proved for the cases when one or two given points of the inclusion are fixed as well as and for the case without these restrictions.

Non-coercive problems for Kirchhoff–Love plates with thin rigid inclusion

Non-coercive problems for Kirchhoff–Love plates with thin rigid inclusion

A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences

Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.

Regularity results and asymptotic behavior for a noncoercive parabolic problem.

In this paper we study the regularity and the behavior in time of the solutions to a quasilinear class of noncoercive problems whose prototype is \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} u_{t} - \\mathrm{div}(a(x,t,u)\ abla u) = -\\mathrm{div}(u\\,E(x,t)) &{}\\quad \ ext{ in }\\, \\Omega \ imes (0,T), \\\\ u(x,t) = 0 &{}\\quad \ ext{ on }\\, \\partial \\Omega \ imes (0,T), \\\\ u(x,0) = u_{0}(x) &{}\\quad \ ext{ in }\\, \\Omega . \\end{array} \\right. \\end{aligned}$$\\end{document}ut-div(a(x,t,u)∇u)=-div(uE(x,t))inΩ×(0,T),u(x,t)=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ. In particular we show that under suitable conditions on the vector field E, even if the problem is noncoercive and although the initial datum u_0 is only an L^{1}(Omega ) function, there exist solutions that immediately improve their regularity and belong to every Lebesgue space. We also prove that solutions may become immediately bounded. Finally, we study the behavior in time of such regular solutions and we prove estimates that allow to describe their blow-up for t near zero.

On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems

We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required.

Quasiconvex optimization problems and asymptotic analysis in Banach spaces

ABSTRACT We use asymptotic analysis for dealing with quasiconvex optimization problems in reflexive Banach spaces. We study generalized asymptotic (recession) cones for nonconvex and nonclosed sets and its respective generalized asymptotic functions. We prove that the generalized asymptotic functions defined in previous works directly through closed formulae can also be generated from the generalized asymptotic cones. We establish three characterizations results for the nonemptiness and compactness of the solution set for noncoercive quasiconvex minimization problems using different asymptotic functions. Finally, we present a sufficient condition for the nonemptiness and boundedness of the solution set for quasiconvex pseudomonotone equilibrium problems.

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

ABSTRACTIn this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.

Second-order asymptotic analysis for noncoercive convex optimization

We use second-order asymptotic analysis to deal with the minimization problem of a noncoercive convex function in a reflexive Banach space. To that end, we first introduce the definition of a second-order asymptotic cone, and its respective function, based on previous results for the finite dimensional case. We provide necessary and sufficient conditions for the existence of solutions for noncoercive convex minimization problems. Examples for which our assumptions are easier to verify than other well-known results are also provided.

Stabilisation de problèmes non coercifs via une méthode numérique utilisant la mesure invariante

We study an advection–diffusion equation that is both non-coercive and advection-dominated. We present a possible numerical approach, to our best knowledge new, and based on the invariant measure associated with the original equation. We show that the approach allows for an unconditionally well-posed finite-element approximation. Two variants of the approach are studied. One of them is stable, and as accurate as a classical stabilization approach. This suggests a possible general strategy, applicable to a large class of non-coercive problems.

On the completeness of root functions of a holomorphic family of non-coercive mixed problem

We consider a non-coercive mixed boundary value problem in a bounded domain D of complex space for a second order parameter-dependent elliptic differential operator with complex-valued essentially bounded measured coefficients and complex parameter . The differential operator is assumed to be of divergent form in D, the boundary operator is of Robin type. The boundary of D is assumed to be a Lipschitz surface. Under reasonable assumptions the pair (A, B) induces a family of non-coercive mixed problems and a holomorphic family of Fredholm operators in suitable Hilbert spaces , of Sobolev type (here are the Sobolev-Slobodetskii spaces over D). If there is a Lipschitz function close enough to the (possibly discontinuous) argument of the complex-valued multiplier of the parameter in then we prove that the operators are continuously invertible for all with sufficiently large modulus on each angle on the complex plane where the operator is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family to be (doubly) complete in the spaces , and the Lebesgue space .

A Petrov–Galerkin reduced basis approximation of the Stokes equation in parameterized geometries

We present a Petrov–Galerkin reduced basis (RB) approximation for the parameterized Stokes equation. Our method, which relies on a reduced solution space and a parameter-dependent test space, is shown to be stable (in the sense of Babuška) and algebraically stable (a bound on the condition number of the online system can be established). Compared to other stable RB methods that can also be shown to be algebraically stable, our approach is among those with the smallest online time cost and it has general applicability to linear non-coercive problems without assuming a saddle-point structure.

On asymptotic exit-time control problems lacking coercivity

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem.

Unification of distance inequalities for linear variational problems

In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Cea’s or Babuska’s for coercive and non-coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang’s inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.

Existence and regularity results for solutions to non-coercive Dirichlet problems

We deal with non-coercive Dirichlet problems, whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = -{{\mathrm{div}}}\bigl (|u|^{\theta -1}\,u\,E(x)\bigr )+f &{} \text {in } {\Omega }\\ u=0 &{} \text {on } \partial {\Omega }, \end{array}\right. } \end{aligned}$$ where \({\Omega }\) is a bounded open set of \(\mathbb {R}^N\), \(E:{\Omega }\rightarrow \mathbb {R}^N\) and \(f:{\Omega }\rightarrow \mathbb {R}\) are in some Lebesgue space, and \(0<\theta <1\). We prove some existence and regularity results depending on the summability of the data \(E\) and \(f\).

RADIATION CONDITION FOR A NON-SMOOTH INTERFACE BETWEEN A DIELECTRIC AND A METAMATERIAL

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in H 1. This is due to the degeneration of the two dual singularities which then behave like r±iη = e±iη ln r with η ∈ ℝ*. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to H 1 one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.

Factorization method for the inverse Stokes problem

We propose an imaging technique for the detection of porous inclusions in a stationary flow based on the Factorization method. The stationary flow is described by the Stokes-Brinkmann equations, a standard model for a flow through a (partially) porous medium, involving the deformation tensor of the flow and the permeability tensor of the porous inclusion. On the boundary of the domain we prescribe Robin boundary conditions that provide the freedom to model, e.g., in- or outlets for the flow. The direct Stokes-Brinkmann problem to find a velocity field and a pressure for given boundary data is a mixed variational problem lacking coercivity due to the indefinite pressure part. It is well-known that indefinite problems are difficult to tackle theoretically using Factorization methods. Interestingly, the Factorization method can nevertheless be applied to this non-coercive problem, as long as one uses data consisting only of velocity measurements. We provide numerical experiments showing the feasibility of the proposed technique.

On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper correction term. Because the approximate problems are affine, an efficient offline/online computational scheme can be developed for the certified solution (error bounds and stability factors) of the parametric equations considered. We experiment with different correction terms suited for a posteriori error estimation of the reduced basis solution of elliptic coercive and noncoercive problems.

Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton–Jacobi equations with noncoercive Hamiltonians

We study large-time asymptotics for a class of noncoercive Hamilton–Jacobi equations with Dirichlet boundary condition in one space dimension. We prove that the average growth rate of a solution is constant only in a subset of the whole domain and give the asymptotic profile in the subset. We show that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. This work is an extension with more rigorous analysis of a recent paper by E. Yokoyama, Y. Giga and P. Rybka, in which a growing crystal model is established and the asymptotic behavior described above is first discovered.