Complementary Extremum Principles for Isoperimetric Optimization Problems

The Coupled Equation Approach to the Numerical Solution of the Biharmonic Equation by finite Differences. I

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints.

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

Completely boundary-free minimum and maximum principles for neutron transport and their least-squares and Galerkin equivalents

The slow progress in the theory of Finsler spaces as compared to Riemann spaces is partly due to lack of information regarding the corresponding local, that is the Minkowskian, geometry. Those Minkowskian features will contribute most to an understanding of Finsler spaces which are not merely verbal generalizations of known euclidean statements.' The purpose of the present note was originally only to show that the isoperimetric problem (for any dimension) in Minkowski spaces leads to such a feature. It turned out, however, that the plane problem can be solved in a general form-no longer significant for Finsler spaces-and then exhibits a phenomenon which is of interest for the theory of isoperimetric problems in the calculus of variations. The result seems to indicate that the standard methods may have followed too closely the pattern of the fixed endpoint problem. For that reason the plane case is here presented separately. The following are the results: let F(x,,y)be continuous, positive for x, y # 0, and positive homogeneous of order 1. The problem, to find among all simple closed curves x (t), y (t) with a given orientation and a given Minkowski length L = 5 F (x, y) dt one which bounds the greatest (euclidean) area, has a unique solution (up to translations) no matter whether the indicatrix C : F(x, y) =1 is convex or not. For non-convex C t-he solution is the same as for the boundary 0 of the convex closure of C as indicatrix and is homothetic to the polar reciprocal (figuratrix) of 0 with respect to the unit circle rotated through ? 7r/2. For Finsler spaces only the case is of interest where C is convex and has the origin as center. Since Finsler or Minkowski area differs from the euclidean area by a constant factor, the solution of the Minkovskian isoperimetric problem is the same as for the above problem. In intrinsic Minkowskian terms it may be described as the curve of length L for which as-new

On the existence in the large of solutions to the one-dimensional, isentropic hydrodynamic equations in a bounded domain.- Initial-boundary value and scattering problems in mathematical physics.- On shape optimization of a turbine blade.- Free boundary problems for the Navier-Stokes equations.- A geometric maximum principle, plateau's problem for surfaces of prescribed mean curvature, and the two dimensional analogue of the catenary.- Finite Elements for the Beltrami operator on arbitrary surfaces.- Comparison principles in capillarity.- Remarks on diagonal elliptic systems.- Quasiconvexity, growth conditions and partial regularity.- The monotonicity formula in geometric measure theory, and an application to a partially free boundary problem.- Isoperimetric problems having continua of solutions.- Harmonic maps - Analytic theory and geometric significance.- Asymptotic behavior of solutions of some quasilinear elliptic systems in exterior domains.- Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems.- Initial boundary value problems in thermoelasticity.- Applications of variational methods to problems in the geometry of surfaces.- Open problems in the degree theory for disc minimal surfaces spanning a curve in ?3.- On a modified version of the free geodetic boundary-value problem.

Construction of the attainability set of a brockett integrator

For calculus of variations problems of Bolza with variable end-points, nonlinear inequality and equality isoperimetric constraints and nonlinear inequality and equality mixed pointwise constraints, sufficient conditions for strong minima are derived. The main novelty of the new sufficiency results presented in this article concerns their applicability to cases in which the derivatives of the extremals to be optimal solutions are not necessarily continuous nor piecewise continuous but only essentially bounded and they do not necessarily satisfy the standard strengthened Legendre condition but only the corresponding necessary condition.

In the recent technical literature a high interest has been devoted to the employment of energy storage at the aim of improving the performances of electrified light transit systems. Some significant results have been obtained for deriving in a rational way a design procedure for ensuring contemporaneously desirable requirements as the energetic efficiency and the reduction of pantograph voltage droops. In particular, it has been verified that the optimization theory is a powerful and proper candidate for determining in a feasible way the optimal characteristics of the storage devices both in stationary and on-board configuration. This problem was approached by exploiting the classical theory of calculus of variations. In this paper, by starting from this approach, a complete analytical solution to the problem of the optimal design storage is afforded. Without loss of generality, the analytical methodology is presented with respect to the stationary case. The formulation as classical isoperimetric problem is developed, based upon a current source for the electrical traction load. This mild assumption allows to obtain a closed analytical form to the storage current law, which in turns results very useful in performing the sensitivity analysis for choosing the free parameters appearing in the methodological framework. A great advantage of the procedure is related to the fact that the analytical expression of the storage currents could be employed for realizing an optimal open-loop law for controlling in real-time the storage device. In the paper in order to show the potentiality of the analytical methodology, a numerical application with respect to realistic light railway vehicles operation is performed. The various simulations confirm the feasibility and the goodness of the proposed methodology

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists of applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler–Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.

The isoperimetric problem is a well-known problem in geometry, and it has a long and rich history (Blasjo 2005). In the plane, the isoperimetric problem consists of finding the simple closed curve of a given perimeter that encloses the greatest area, with the circle being the famous solution. Attempts to solve the isoperimetric problem, as well as other analogous problems in calculus and physics, were undertaken by many great mathematicians in the past whose work ultimately laid the foundation for the elegant branch of analysis known today as the calculus of variations.

We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem.

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

An alternate sufficiency proof for the fixed end-point isoperimetric problem in the calculus of variations is presented. This technique not only shows how the problem need not be transformed into a problem of Lagrange but also shows how we can weaken the classical strengthened condition of Weierstrass. The usefulness of this sufficiency result is illustrated with an example which cannot be transformed into a problem of Lagrange and for which it is possible to apply the alternate sufficiency theorem in order to conclude that a given extremal affords a strict strong minimum. On the other hand, we show that the classical sufficiency theorem does not respond for this case.