Given two annuli

The lectures focused on the Calculus of Variations. As part of optimization theory, the Calculus of Variations originated in 1696 when Johann Bernoulli posed the brachistochrone problem. This problem related to the curve between two points along which a ball would require minimal time of travel to reach the bottom. This problem was solved by many different mathematicians of the time, some of wh...

We consider the following obstacle problem for Monge-Ampere equation detDu = fχ{u>0} and discuss the regularity of the free boundary ∂{u = 0}. We prove that ∂{u = 0} is C if f is bounded away from 0 and ∞, and it is C if f ≡ 1.

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group. Mathematics Subject Classification (1991): 49L25, 35J70, 35J67, 22E30.

Abstract. With a map f : Ω → R,Ω ⊂ R, that belongs to the John Ball classAp,q(Ω) where n − 1 < p < n and q ≥ p/(p − 1) one can associate a set valued map F whose values F (x) ⊂ R are subsets ofR describing the topological character of the singularity of f at x ∈ Ω. Šverak conjectured that Hn−1(F (S)) = 0, where S is the set of points at which f is not continuous andHn−1 is the Hausdorff measure...

In this article, we present a detailed study of the complex calculus of variations introduced in [4]. This calculus is analogous to the conventional calculus of variations, but is applied here to C functions in C. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, i...

The calculus of variations is a branch of mathematical analysis that studies extrema and critical points of functionals (or energies). Here, by functional we mean a mapping from a function space to the real numbers. One of the first questions that may be framed within this theory is Dido’s isoperimetric problem (see Subsection 2.3): to find the shape of a curve of prescribed perimeter that maxi...

In this paper we study a classical optimization problem in heat conduction, which may briefly be described as follows: given a surface ∂D in R, and a positive function φ defined on it (the temperature distribution), we want to surround ∂D with a prescribed volume of insulating material so as to minimize the loss of heat in a stationary situation. Mathematically speaking, we want to find a funct...

where C is some constant depending only on (M, g), and k. He also announced in the same paper the same result for general manifolds, without the locally conformally flat assumption. The proof of this claim has not been made available. For general manifolds of dimension n = 3, a proof was given by Li and Zhu in [73]; while for n = 4, a combination of the results of Li and Zhang [70] and Druet [4...