The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal. 144 (1998) 1–46] for a first-order perturbation model.This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies

$\[{\cal F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon} \int_{\partial \Omega} V(Tu),\]$

where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon\lambda_{\varepsilon}^{\frac{2}{3}} \sim 1$.

This paper is mainlyconcerned with a class of optimal control problems of systemsgoverned by the nonlinear dynamic systems on time scales.Introducing the reasonable weak solution of nonlinear dynamicsystems, the existence of the weak solution for the nonlineardynamic systems on time scales and its properties are presented.Discussing L1-strong-weak lower semicontinuity of integralfunctional, we give sufficient conditions for the existence ofoptimal controls. Using integration by parts formula and Hamiltonianfunction on time scales, the necessary conditions of optimality arederived respectively. Some examples on continuous optimal controlproblems, discrete optimal control problems, mathematicalprogramming and variational problems are also presented for demonstration.

We consider higher order functionals of the form

$F[u]=\int\limits_\Omega f(D^mu)\,{\rm d}x \qquad\text{for }u:\mathbb{R}^n\supset\Omega\to\mathbb{R}^N,$

where the integrand $f:{\textstyle \bigodot^m}(\R^{n},\R^{N})\to\mathbb{R}$, m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition.More precisely we assume that f fulfills the (p, q)-growth condition

\[\gamma|A|^p\le f(A)\le L(1+|A|^q)\qquad \mbox{for all }A \in {\textstyle \bigodot^m}(\R^{n},\R^{N}),\]

with γ, L > 0 and $1< p \le q<\min\big\{p+\frac1n,\frac{2n-1}{2n-2}p\big\}$. We study minimizers of thefunctional $F[\cdot]$ and prove a partial $C^{m,\alpha}_{\rm loc}$-regularity result.

We consider a time optimal control problem arisen from the optimalmanagement of a bioreactor devoted to the treatment ofeutrophicated water. We formulate this realistic problem as astate-control constrained time optimal control problem. Afteranalyzing the state system (a complex system of coupled partialdifferential equations with non-smooth coefficients foradvection-diffusion-reaction with Michaelis-Menten kinetics,modelling the eutrophication processes) we demonstrate theexistence of, at least, an optimal solution. Then we present adetailed derivation of a first order optimality condition(involving the corresponding adjoint systems) characterizing theseoptimal solutions. Finally, a numerical example is shown.

This paper studies the strong unique continuation property for theLamé system of elasticity with variable Lamé coefficientsλ, µ in three dimensions, ${\rm{div}}(\mu(\nabla u+\nablau^t))+ \nabla(\lambda{\rm{div}} u)+Vu=0$where λ and μ are Lipschitz continuous and V ∈ L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar-to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence results in the literature of solutions to algebraic Riccati equations do not apply to this class of problems. Here taking advantage of the fact that the semigroup of the state equation is exponentially stable and that the observation operator is a Hilbert-Schmidt operator, we are able to prove the existence and uniqueness of solution to the A.R.E. satisfied by the kernel of the operator which associates the 'optimal adjoint state' with the 'optimal state'. In part 2 [Buchot and Raymond, Appl. Math. Res. eXpress (2010) doi:10.1093/amrx/abp007], we study problems in which the feedback law is determined by the solution to the A.R.E. and another nonhomogeneous term satisfying an evolution equation involving nonhomogeneous perturbations of the state equation, and a nonhomogeneous term in the cost functional.

A zero-sum stochastic differential gameproblem on infinite horizon with continuous and impulse controls isstudied. We obtain the existence of the value of the game andcharacterize it as the unique viscosity solution of the associatedsystem of quasi-variational inequalities. We also obtain averification theorem which provides an optimal strategy of the game.

We consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

A variant of the Total OverlappingSchwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1Math.336(2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains.That same method turns to be an efficient toolto make numerical zoomsin regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture andthe reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper isto use this modified Schwarz procedure as a preconditioner to Krylovsubspaces methods so to accelerate the calculations. A detailed study concludes toa super-linear convergence ofGMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numericalexamples are also provided and commented that demonstrate the reliability of theTOS-preconditioner.

While alternans in a single cardiac cell appears through a simpleperiod-doubling bifurcation, in extended tissue the exact natureof the bifurcation is unclear. In particular, the phase ofalternans can exhibit wave-like spatial dependence, eitherstationary or travelling, which is known as discordantalternans. We study these phenomena in simple cardiac modelsthrough a modulation equation proposed by Echebarria-Karma. Asshown in our previous paper, the zero solution of their equationmay lose stability, as the pacing rate is increased, througheither a Hopf or steady-state bifurcation. Which bifurcationoccurs first depends on parameters in the equation, and for onecritical case both modes bifurcate together at a degenerate(codimension 2) bifurcation. For parameters close to thedegenerate case, we investigate the competition between modes,both numerically and analytically. We find that at sufficientlyrapid pacing (but assuming a 1:1 response is maintained), steadypatterns always emerge as the only stable solution. However, inthe parameter range where Hopf bifurcation occurs first, theevolution from periodic solution (just after the bifurcation) tothe eventual standing wave solution occurs through an interestingseries of secondary bifurcations.

We analyze the accuracy and well-posedness of generalized impedanceboundary value problems in the framework of scattering problemsfrom unbounded highly absorbing media. We restrict ourselves in this first workto the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficultiesin the analysis for the generalized impedance boundary conditions, sinceclassical compactness arguments are no longer possible. Our new analysisis based on the use of Rellich-type estimates and boundedness of L2solution operators. We also discuss some numerical experimentsconcerning these boundary conditions.