Using a calibration method we prove that, if $\Gamma \subset \Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma$ and regular outside, then the function $u_{\beta }$ which solves$$\left\{\begin{array}{cc}\Delta u_{\beta }=\beta (u_{\beta }-g)\hfill & \text{in}\Omega \setminus \Gamma \hfill \\ {\partial }_{\nu }{u}_{\beta }=0\hfill & \text{on}\partial \Omega \cup \Gamma \hfill \end{array}\right.$$is in turn discontinuous along $\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional$${\int }_{\Omega \setminus S_u}{\left|\nabla u\right|}^{2}dx+{\mathscr{H}}^{n-1}\left(S_u\right)+\beta {\int }_{\Omega \setminus S_u}{(u-g)}^{2}dx,$$over $SBV\left(\Omega \right)$, for $\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and...

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set R(t) ⊂ ℝ2, described as the reachable set for a differential inclusion. To restrict its growth, a barrier Γ can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification...

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized method is given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility...

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized method is given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility...

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...

We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power ${(-A)}^{\beta }$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already...

MSC 2010: 26A33, 70H25, 46F12, 34K37 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler–Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation....