Coercive Problems Research Articles

Sequences of nodal solutions for critical double phase problems with variable exponents

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} and in the Musielak–Orlicz Sobolev space.

Sign changing solutions for critical double phase problems with variable exponent

In this paper, we deal with a double phase problem with variable exponent and a right-hand side consisting of a Carathéodory perturbation defined only locally and of a critical term. We stress that the presence of the critical term inhibits the possibility to apply results of the critical point theory to the corresponding energy functional. Instead, we use suitable cut-off functions and truncation techniques in order to work with a coercive functional. Then, using variational tools and an appropriate auxiliary coercive problem, we can produce a sequence of sign changing solutions to our main problem converging to 0 in L^{\infty} and in the Musielakk–Orlicz Sobolev space.

Multiplicity of solutions for nonlinear coercive problems

We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

Efficient Stability-Preserving Numerical Methods for Nonlinear Coercive Problems in Vector Space

Efficient Stability-Preserving Numerical Methods for Nonlinear Coercive Problems in Vector Space

기혼 남녀의 정서적 성숙 및 자기지각이 부부간 성적 자기결정권에 미치는 영향과 TV 불륜 드라마에 대한 비판의식의 매개효과 검증

This study is an empirical study to identify married men and women's perception of sexual self-determination and related factors, with the recognition that having a clear awareness of the right to sexual self-determination between husband and wife is an effective approach to preventing coercive sexual problems in extramarital relationships. This study aims to reveal the direct effect of emotional maturity and self-awareness on married men and women's perceptions of sexual self-determination between couples, as well as the mediating effect of criticism of TV dramas with the theme of infidelity. The research data are obtained by conducting an online survey of married men and women(614), and the data are analyzed based on the structural equation model. As a result, first, married people have a fairly good level of awareness of sexual self-determination, which shows higher levels of women than men. Second, it is found that married men and women's self-awareness does not directly affect their sexual self-determination rights, but can significantly affect their critical consciousness of adultery drama as mediators. Third, it is confirmed that married people's right to sexual self-determination is greatly affected by their emotional maturity. The positive effect of their emotional maturity on their sexual self-determination rights is further strengthened through the critical consciousness of married people against affair drama. It is verified that critical consciousness on infidelity drama can have a strong impact directly and indirectly on married couples' sexual self-determination rights. It shows that the use of media such as adultery dramas is effective in strengthening the right of sexual self-determination between married men and women. Emotional maturity and self-awareness directly or indirectly affect the right of sexual self-determination between couples. In addition, it has verified that it is a psychological characteristic that can significantly affect the consciousness of criticism of adultery dramas, which can have a significant impact on the right to sexual self-determination between them. The practical and policy implications of the research results are discussed to find ways to improve married men and women's awareness of the right to sexual self-determination between husband and wife.

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component – associated with the solution on interior boundaries – and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss–Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the a priori error analysis for linear coercive problems.

Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions

Abstract In the present paper, we study the existence as well as the non-existence of some positive solutions for the equation −Δ p(x) u = λ k(x) uq ± h(x) ur under Robin boundary condition in a regular open bounded domain Ω of ℝ N , N ≥ 2. Δ p(x) is the p(x)-Laplacian operator where p ∈ C 1(Ω) and p > 1. Our proofs are based on the sub solution-super solution method and also on variational arguments.

A Scalable and Robust Vertex-Star Relaxation for High-Order FEM

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate.

On the Accuracy of the Model Predictive Control Method

The paper investigates the accuracy of the model predictive control (MPC) method for finding on-line approximate optimal feedback control for Bolza-type problems on a fixed finite horizon. The predictions for the dynamics, the state measurements, and the solution of the auxiliary open-loop control problems that appear at every step of the MPC method may be inaccurate. The main result provides an error estimate of the MPC-generated solution compared with the optimal open-loop solution of the “ideal” problem, where all predictions and measurements are exact. The technique of proving the estimate involves an extension of the notion of strong metric subregularity of set-valued maps and utilization of a specific new metric in the control space, which makes the proof nonstandard. The result is specialized for two problem classes: coercive problems and affine problems.

Normalized solutions with positive energies for a coercive problem and application to the cubic–quintic nonlinear Schrödinger equation

In any dimension [Formula: see text], for given mass [Formula: see text] and when the [Formula: see text] energy functional [Formula: see text] is coercive on the mass constraint [Formula: see text] we are interested in searching for constrained critical points at positive energy levels. Under general conditions on [Formula: see text] and for suitable ranges of the mass, we manage to construct such critical points which appear as a local minimizer or correspond to a mountain pass or a symmetric mountain pass level. In particular, our results shed some light on the cubic–quintic nonlinear Schrödinger equation in [Formula: see text].

Regularity results for a noncoercive nonlinear Dirichlet problem

In this paper, we prove some regularity results for a noncoercive nonlinear Dirichlet problem in an unbounded domain. The main result is obtained in the hypothesis that suitable coefficients of the operator associated to the problem belong to a class of functional spaces strictly containing Lebesgue ones. This can be done by approximation via coercive nonlinear Dirichlet problems. Under additional assumptions, we also get the boundedness of a solution.

On obstacle problems for non coercive linear operators

In this paper we prove the existence and uniqueness of the solution to the one and the two obstacles problems associated with a linear elliptic operator, which is non coercive due to the presence of a convection term. We show that the operator is weakly T-monotone and, as a consequence, we establish the Lewy–Stampacchia dual estimates and we study the comparison and the continuous dependence of the solutions as the obstacles vary. As an application, we prove also the existence of solutions for a class of non coercive implicit obstacle problems.

J‐ADMM for a multi‐contact problem in electro‐elastostatics

AbstractIn this work we apply a regularized alternating direction method of multiplier of Jacobi type (J‐ADMM) to a frictional multi‐contact (both unilateral and bilateral) problem between an electro‐elastic material and a rigid non conductive foundations. The frictional contact is modelled by the Coulomb friction law. The resulted problem is non symmetric and non coercive. By dissociating the electric potential from the mechanical field, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply a J‐ADMM for the numerical approximation to the resulting perturbed elastic problem. The resolution of obtained sub‐problems is based on convex dualities. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.

Superconvergent HDG methods for Maxwell’s equations via the [formula omitted]-decomposition

The concept of the M-decomposition was introduced by Cockburn et al. (2017) to provide criteria to guarantee optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG) method for coercive elliptic problems. In that paper they systematically constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods to approximate the solutions of elliptic PDEs on unstructured meshes. In this paper, we use the M-decomposition to construct HDG methods for the Maxwell’s equations on unstructured meshes in two dimension. In particular, we show the any choice of spaces having an M-decomposition, together with sufficiently rich auxiliary spaces, has an optimal error estimate and superconvergence even though the problem is not in general coercive. Motivated by the elliptic case, we obtain a superconvergent rate for the curl and flux of the solution, and this is confirmed by our numerical experiments.

Optimal parameter for the stabilised five-field extended Hu–Washizu formulation

We present a mixed finite element method for the elasticity problem. We expand the standard Hu–Washizu formulation to include a pressure unknown and its Lagrange multiplier. By doing so, we derive a five-field formulation. We apply a biorthogonal system that leads to an efficient numerical formulation. We address the coercivity problem by adding a stabilisation term with a parameter. We also present an analysis of the optimal choices of parameter approximation.

On Some Boundary Value Problems for Multidimensional Higher Order Equations of Mixed Type

We describe the statements of boundary value problems for multidimensional higher order equations of mixed type and propose a method for constructing coercive boundary value problems. Also, we prove the generalized and regular solvability of these boundary value problems in Sobolev spaces.

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.

On some weakly coercive quasilinear problems with forcing

We consider the forced problem −Δpu − V(x)∣u∣p−2u = f(x), where Δp is the p-Laplacian (1 < p < ∞) in a domain Ω ⊂ ℝN, V ≥ 0 and QV(u)≔ ∫Ω ∣∇u∣pdx − ∫ΩV∣u∣pdx satisfies the condition (A) below. We show that this problem has a solution for all f in a suitable space of distributions. Then we apply this result to some classes of functions V which in particular include the Hardy potential (1.5) and the potential V(x)= λ1,p(Ω), where λ1,p(Ω) is the Poincare constant on an infinite strip.

Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number

Although superconvergence properties of the polynomial preserving recovery (PPR) for finite element methods (FEM) for coercive elliptic problems are well-understood, the superconvergence behavior of PPR for the Helmholtz equation with high wave number still remains unclear. We study the superconvergence property of the linear FEM with PPR for the two dimensional Helmholtz equation on triangulations satisfying the O(h1+α) approximate parallelogram property for some constant α>0. We prove the supercloseness on the H1-seminorm between the finite element solution and the linear interpolant of the exact solution under k(kh)2≤C0, where k is the wave number and h is the mesh size. Then, we obtain the superconvergence result based on the PPR technique which says that the PPR improves only the interpolation error but keeps the pollution error unchanged. Finally, we provide numerical tests to verify the superconvergence property and to demonstrate that the PPR combining with the continuous interior penalty technique is much effective for improving both the interpolation error and the pollution error.

Asymptotically vanishing nodal solutions for critical double phase problems

We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the original Dirichlet equation has a whole sequence of nodal (sign-changing) solutions which converge to zero in the Musielak–Orlice–Sobolev space and in L ∞ .