We consider a nonlinear elliptic equation of the formdiv [a(∇u)] + F[u] = 0on a domain Ω, subject to a Dirichlet boundary conditiontru = φ. We do not assume that the higher order terma satisfies growth conditions from above. We prove the existence ofcontinuous solutions either when Ω is convex and φ satisfies a one-sidedbounded slope condition, or when a is radial:\hbox{$a(\xi)=\fr{l(|\xi|)}{|\xi|} \xi$}$\mathit{a}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{=}\frac{\mathit{l}\mathrm{\left(}\mathrm{\right|}\mathit{\xi }\mathrm{\left|}\mathrm{\right)}}{\mathrm{\left|}\mathit{\xi }\mathrm{\right|}}\mathit{\xi }$ for some increasingl:ℝ+ → ℝ+.

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.

Optimal nonanticipating controls are shown to exist in nonautonomous piecewisedeterministic control problems with hard terminal restrictions. The assumptions needed arecompletely analogous to those needed to obtain optimal controls in deterministic controlproblems. The proof is based on well-known results on existence of deterministic optimalcontrols.

We study here the impulse control minimax problem. We allow the cost functionals anddynamics to be unbounded and hence the value functions can possibly be unbounded. We provethat the value function of the problem is continuous. Moreover, the value function ischaracterized as the unique viscosity solution of an Isaacs quasi-variational inequality.This problem is in relation with an application in mathematical finance.

This paper is concerned with the stochastic linear quadratic optimal control problems (LQproblems, for short) for which the coefficients are allowed to be random and the costfunctionals are allowed to have negative weights on the square of control variables. Wepropose a new method, the equivalent cost functional method, to deal with the LQ problems.Comparing to the classical methods, the new method is simple, flexible and non-abstract.The new method can also be applied to deal with nonlinear optimization problems.

In this paper, we are concerned with the existence of multi-bump solutions for anonlinear Schrödinger equations with electromagnetic fields. We prove under some suitableconditions that for any positive integer m, there existsε(m) > 0 such that, for0 < ε < ε(m),the problem has an m-bump complex-valued solution. As a result, whenε → 0, the equation has more and more multi-bumpcomplex-valued solutions.

We study the partial differential equation

        max{Lu − f, H(Du)} = 0

where u is the unknownfunction, L is a second-order elliptic operator, f is agiven smooth function and H is a convex function. This is a modelequation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. Weestablish the existence of a unique viscosity solution of the Dirichlet problem that has aHölder continuous gradient. We also show that if H is uniformlyconvex, the gradient of this solution is Lipschitz continuous.

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

In the context of a variational model for the epitaxial growth of strained elastic films,we study the effects of the presence of anisotropic surface energies in the determinationof equilibrium configurations. We show that the threshold effect that describes thestability of flat morphologies in the isotropic case remains valid for weak anisotropies,but is no longer present in the case of highly anisotropic surface energies, where we showthat the flat configuration is always a local minimizer of the total energy. Following theapproach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strainedelastic films: second order minimality conditions and qualitative properties of solutions.Preprint], we obtain these results by means of a minimality criterion based on thepositivity of the second variation.

The ill-posed problem of solving linear equations in the space of vector-valued finiteRadon measures with Hilbert space data is considered. Approximate solutions are obtainedby minimizing the Tikhonov functional with a total variation penalty. The well-posednessof this regularization method and further regularization properties are mentioned.Furthermore, a flexible numerical minimization algorithm is proposed which convergessubsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparsedeconvolution demonstrate the applicability for a finite-dimensional discrete data spaceand infinite-dimensional solution space.

The aim of this paper is to establish necessary optimality conditions for optimal controlproblems governed by steady, incompressible Navier-Stokes equations with shear-dependentviscosity. The main difficulty derives from the fact that equations of this type mayexhibit non-uniqueness of weak solutions, and is overcome by introducing a family ofapproximate control problems governed by well posed generalized Stokes systems and bypassing to the limit in the corresponding optimality conditions.

We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.

In this paper, a lower bound is established for the local energy of partial sum of eigenfunctions for Laplace-Beltrami operators (in Riemannian manifolds with low regularity data) with general boundary condition. This result is a consequence of a new pointwise and weighted estimate for Laplace-Beltrami operators, a construction of some nonnegative function with arbitrary given critical point location in the manifold, and also two interpolation results for solutions of elliptic equations with lateral Robin boundary conditions.

In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. Theproof is based on a reduction argument and the Goh condition for minimality of singularcurves. The Goh condition is deduced from a reformulation and a calculus of the end-pointmapping which boils down to the graded structures of Carnot groups.

In this paper we deal with the null controllability problem for the heat equation with amemory term by means of boundary controls. For each positive final time Tand when the control is acting on the whole boundary, we prove that there exists a set ofinitial conditions such that the null controllability property fails.

We consider the linear wave equation with Dirichlet boundary conditions in a bounded interval, and with a control acting on a moving point. We give sufficient conditions on the trajectory of the control in order to have the exact controllability property.