In this paper we study the limit as p → ∞ of minimizers of thefractional Ws,p-norms. In particular, weprove that the limit satisfies a non-local and non-linear equation. We also prove theexistence and uniqueness of solutions of the equation. Furthermore, we prove the existenceof solutions in general for the corresponding inhomogeneous equation. By making strong useof the barriers in this construction, we obtain some regularity results.

The aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$$\mathrm{See\; PDF}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathbb{L}^{p}$$\mathrm{See\; PDF}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$$\mathrm{See\; PDF}$ problems in continuous and lower semicontinuous setting.

Bounds for Hardy differences, that is, improvements and reverses of the well-known Hardy inequality, are obtained.

We present a stability analysis of steady-state solutions of a continuous-time predator–prey population dynamics model subject to Allee effects on the prey population which occur at low population density. Numerical simulations show that the system subject to an Allee effect takes a much longer time to reach its stable steady-state solution. This result differs from that obtained for the discrete-time version of the same model.

A class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze the relationship between atomistic and Cauchy–Born stability regions, and the convergence of atomistic stability regions as the cell size tends to infinity.

In this paper sufficient second order optimality conditions for optimal control problemssubject to stationary variational inequalities of obstacle type are derived. Sinceoptimality conditions for such problems always involve measures as Lagrange multipliers,which impede the use of efficient Newton type methods, a family of regularized problems isintroduced. Second order sufficient optimality conditions are derived for the regularizedproblems as well. It is further shown that these conditions are also sufficient forsuperlinear convergence of the semi-smooth Newton algorithm to be well-defined andsuperlinearly convergent when applied to the first order optimality system associated withthe regularized problems.

In several practically interesting applications of electromagnetic scattering theory like, e.g., scattering from small point-like objects such as buried artifacts or small inclusions in non-destructive testing, scattering from thin curve-like objects such as wires or tubes, or scattering from thin sheet-like objects such as cracks, the volume of the scatterers is small relative to the volume of the surrounding medium and with respect to the wave length of the applied electromagnetic fields.This smallness typically causes problems when solving direct scattering problems due to the need to discretize the objects and also when solving inverse scattering problems because small objects have very little effect on electromagnetic fields. In this paper we consider an asymptotic representation formula for scattered electromagnetic waves caused by low volume objects contained in some otherwise homogeneous three-dimensional bounded domain, assuming only that the scatterers are measurable and well-separated from the boundary of the domain.The formula yields a very general asymptotic model for electromagnetic scattering due to low volume objects that can either be used to simulate the corresponding electromagnetic fields or as the foundation of efficient reconstruction methods for inverse scattering problems with low volume scatterers.Our analysis extends results originally obtained in [Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal. 37 (2003) 159–173] for steady state voltage potentials to time-harmonic Maxwell's equations.

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete ellipticequations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice $\mathbb{Z}^d$with independent and identically distributed conductivities on the associated edges.Recent results by Otto and the author quantify the error made by approximatingthe homogenized coefficient by the averaged energy of a regularizedcorrector (with parameter T) on some box of finite size L. In this article, we replace the regularizedcorrector (which is the solution of a problem posed on $\mathbb{Z}^d$) by some practically computable proxy on some box of size R ≥ L, and quantify the associated additional error.In order to improve the convergence, one may also consider N independentrealizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients.A natural optimization problem consists in properly choosing T, R, L and N in order toreduce the error at given computational complexity.Our analysis is sharp and sheds some light on this question.In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errorswe derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.

In this paper we consider linear Hamiltonian differential systems without thecontrollability (or normality) assumption. We prove the Rayleigh principle for thesesystems with Dirichlet boundary conditions, which provides a variational characterizationof the finite eigenvalues of the associated self-adjoint eigenvalue problem. This resultgeneralizes the traditional Rayleigh principle to possibly abnormal linear Hamiltoniansystems. The main tools are the extended Picone formula, which is proven here for thisgeneral setting, results on piecewise constant kernels for conjoined bases of theHamiltonian system, and the oscillation theorem relating the number of proper focal pointsof conjoined bases with the number of finite eigenvalues. As applications we obtain theexpansion theorem in the space of admissible functions without controllability and aresult on coercivity of the corresponding quadratic functional.

We consider an evolution equation similar to that introduced by Vese in [Comm.Partial Diff. Eq. 24 (1999) 1573–1591] and whose solutionconverges in large time to the convex envelope of the initial datum. We give a stochasticcontrol representation for the solution from which we deduce, under quite generalassumptions that the convergence in the Lipschitz norm is in fact exponential in time.

We consider general surface energies, which areweighted integrals over a closed surface with a weight functiondepending on the position, the unit normal andthe mean curvature of the surface. Energiesof this form have applications in many areas, such as materials science,biology and image processing. Often one is interested in findinga surface that minimizes such an energy, which entails finding its firstvariation with respect to perturbations of the surface.We present a concise derivation of the first variation of thegeneral surface energy using tools from shape differential calculus.We first derive a scalar strong form and nexta vector weak form of the first variation. The latter reveals thevariational structure of the first variation, avoids dealingexplicitly with the tangential gradient of the unit normal,and thus can be easily discretized using parametric finite elements.Our results are valid for surfaces in any number of dimensionsand unify all previous results derived for specific examples ofsuch surface energies.

We prove estimates for the partial derivatives of the solution to a time-fractional diffusion equation posed over a bounded spatial domain. Such estimates are needed for the analysis of effective numerical methods, particularly since the solution is typically less regular than in the familiar case of classical diffusion.

We present a convergence analysis of a cell-based finite volume (FV) discretization scheme applied to a problem of control in the coefficients of a generalized Laplace equation modelling, for example, a steady state heat conduction. Such problems arise in applications dealing with geometric optimal design, in particular shape and topology optimization, and are most often solved numerically utilizing a finite element approach. Within the FV framework for control in the coefficients problems the main difficulty we face is the need to analyze the convergence of fluxes defined on the faces of cells, whereas the convergence of the coefficients happens only with respect to the “volumetric” Lebesgue measure. Additionally, depending on whether the stationarity conditions are stated for the discretized or the original continuous problem, two distinct concepts of stationarity at a discrete level arise.We provide characterizations of limit points, with respect to FV mesh size, of globally optimal solutions and two types of stationary points to the discretized problems. We illustrate the practical behaviour of our cell-based FV discretization algorithm on a numerical example.

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to the small parameter ε and obtain optimal bound for the modelling error outside the sheet of order $\varepsilon^{N+1}$ for the condition of order N. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.

In this paper, we discuss an hp-discontinuous Galerkin finiteelement method (hp-DGFEM) for the laser surface hardening ofsteel, which is a constrained optimal control problem governed by asystem of differential equations, consisting of an ordinarydifferential equation for austenite formation and a semi-linearparabolic differential equation for temperature evolution. The spacediscretization of the state variable is done using an hp-DGFEM,time and control discretizations are based on a discontinuousGalerkin method. A priori error estimates are developed atdifferent discretization levels. Numerical experimentspresented justify the theoretical order of convergence obtained.

The paper deals with a Dirichlet spectral problem for an elliptic operator withε-periodic coefficients in a 3D bounded domain of small thicknessδ. We study the asymptotic behavior of the spectrum asε and δ tend to zero. This asymptotic behavior dependscrucially on whether ε and δ are of the same order(δ ≈ ε), or ε is much less thanδ(δ = ετ, τ < 1),or ε is much greater thanδ(δ = ετ, τ > 1).We consider all three cases.

Second-order sufficient conditions of a bounded strong minimum are derived for optimalcontrol problems of ordinary differential equations with initial-final state constraintsof equality and inequality type and control constraints of inequality type. The conditionsare stated in terms of quadratic forms associated with certain tuples of Lagrangemultipliers. Under the assumption of linear independence of gradients of active controlconstraints they guarantee the bounded strong quadratic growth of the so-called “violationfunction”. Together with corresponding necessary conditions they constitute a no-gap pairof conditions.