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We are interested in the existence of normalized solutions to the problem (- Δ) m u + μ | y | 2 m u + λ u = g (u) , x = (y , z) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the L 2 -ball. Moreover, we find also a solution to the related curl–curl problem ∇ × ∇ × U + λ U = f (U) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence and multiplicity of bounded state solutions for a critical (p(x), q(x))-curl system which arises in electromagnetism. The existence of nontrivial bounded state solutions is first obtained by applying the mountain pass lemma. Then the existence of ground state solutions is studied by restricting the analysis to the Nehari manifold. Furthermore, under some suitable assumptions, we prove that the mountain pass solution is actually a ground state solution. Finally, the existence of infinitely many solutions is investigated by using the genus theory combined with a truncated argument. The results obtained in this paper develop and complement several contributions concerning the p-curl operator and we focus on new existence results which are due to the presence of nonhomogeneous (p(x), q(x))-curl operator and critical nonlinearity. To the best of our knowledge, our results are new even in the semilinear case. [ABSTRACT FROM AUTHOR]

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, to guarantee well-posedness, are known [(see, e.g., Exercise 2.14 of Ern and Guermond (Theory and practice of finite elements, Applied mathematical, sciences, Springer, 2004)]. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately. [ABSTRACT FROM AUTHOR]

We perform a stochastic homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure. [ABSTRACT FROM AUTHOR]

We develop a framework shedding light on common features of coupled variational formulations arising in electromagnetic scattering and acoustics. We show that spurious resonances haunting coupled domain-boundary formulations based on direct boundary integral equations of the first kind originate from the formal structure of their Calderón identities. Using this observation, the kernel of the coupled problem is characterized explicitly, and we show that it completely vanishes under the exterior representation formula. [ABSTRACT FROM AUTHOR]

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A−1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems. [ABSTRACT FROM AUTHOR]

In this article, a fitted finite element method is proposed and analyzed for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media which has a discontinuous coefficients. Convergence properties of the semidiscrete and fully discrete schemes are investigated in the L 2 norm. Optimal a priori error estimates for both the schemes are proved. Numerical experiment is conducted to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

We propose a fitted finite element method for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media that have a heterogeneous conductivity. Well-posedness of the model interface problem is established. A continuous piecewise linear finite element space is employed for the spatially semidiscrete approximation and the temporal discretization is based on backward scheme. Optimal order error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ (H 1) norm. Finally, we give numerical examples to verify our theoretical results. The new results and finite element schemes can be applied in the fields of engineering, medicine, and biotechnology. [ABSTRACT FROM AUTHOR]

For the case of singularly perturbed elliptic transmission problems we demonstrate the use of asymptotic expansion techniques both for establishing regularity results for the solution and for deriving a priori error estimates for boundary element Galerkin discretisation. The dependence of the corresponding bounds on the singular perturbation parameter is studied in detail. This dependence clearly manifests itself in numerical experiments. [ABSTRACT FROM AUTHOR]

Existence of steady states in elastic media at small strains with diffusion of a solvent or fluid due to Fick's or Darcy's laws is proved by combining usage of variational methods inspired from static situations with Schauder's fixed-point arguments. In the plain variant, the problem consists in the force equilibrium coupled with the continuity equation, and the underlying operator is non-potential and non-pseudomonotone so that conventional methods are not applicable. In advanced variants, electrically-charged multi-component flows through an electrically charged elastic solid are treated, employing critical points of the saddle-point type. Eventually, anisothermal variants involving heat-transfer equation are treated, too. [ABSTRACT FROM AUTHOR]

This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary. [ABSTRACT FROM AUTHOR]

This work consists in the asymptotic analysis of the solution of Poisson equation in a bounded domain of RP(P= 2, 3) with a thin layer. We use a method based on hierarchical variational equations to derive an explicitly asymptotic expansion of the solution with respect to the thickness of the thin layer. We determine the first two terms of the expansion and prove the error estimate made by truncating the expansion after a finite number of terms. Next, using the first two terms of the asymptotic expansion, we show that we can model the effect of the thin layer by a problem with transmission conditions of order two. [ABSTRACT FROM PUBLISHER]

In this paper, we present modified variational iteration methods (MVIM) to solve the sine-Gordon equation which play a very important part in applied and engineering sciences. Two versions of MVIM are made by the classical variational iteration method coupled with other methods such as homotopy perturbation method and Adomian decomposition method (ADM). The reliability and efficiency of the proposed methods are discussed and illustrated by numerical examples. Moreover, it has the advantage of higher precision and is easier to implement when compared with the ADM. [ABSTRACT FROM AUTHOR]

In this paper, the variational iteration method is used to solve a system of third-order boundary value problem associated with obstacle, unilateral and contact problems. Numerical solution obtained by the method is of high accuracy. The numerical example compared with those considered by other authors shows that the method is more efficient. [ABSTRACT FROM AUTHOR]