Weighted energy-dissipation functionals for doubly nonlinear evolution

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler-Lagrange problems. Moreover, we check that minimizers of the WED functionals converge to the corresponding limiting doubly nonlinear evolution. Finally, we present a discussion on the functional convergence of sequences of WED functionals and present some application of the abstract theory to nonlinear PDEs.     

Hamiltonian Stationary Lagrangian Surfaces in Euclidean Four-Space and Related Minimization Problems. Part II: Some Minimization Results

We study some minimization problems for Hamiltonian stationaryLagrangian surfaces in R⁴. We show that the flat Lagrangian torusS ¹ × S ¹ minimizes the Willmore functional among Hamiltonianstationary tori of its isotopy class, which gives a new proof of thefact that it is area minimizing in the same class. Considering theLagrangian flat cylinder as a torus in some quotient space R⁴/v Z, we show that it is also area minimizing in its isotopy class.     

Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

Generalized Cauchy equations on groups

It is well known ([1], [3]) that any measurable solution of the Cauchy functional equationf(x+y)=f(x)+f(y) must actually be continuous. The same is true of some other functional equations likef(x+y)=f(x)f(y),f(x+y)f(x–y)=f(x) ² –f(y) ², etc. (cf. [1]). In this note we prove a general result of this type for functional equations on groups.     

A new Pythagorean functional equation

The purpose of this paper is to solve the following Pythagorean functional equation:(e ^p(x,y) ) ² ) = q(x,y) ² + r(x, y) ², where each ofp(x,y), q(x, y) andr(x, y) is a real-valued unknown harmonic function of the real variablesx, y on the wholexy-planeR ².The result is as follows.     

Existence result for a new variational problem in one-dimensional segmentation theory

In order to study some segmentation problems in dimension one, we propose a new functional, whose leading term includes the second order derivative of the unknown function. We prove an existence result for the associated minimization problem, by relying on the compactness and the lower semicontinuity of the functional with respect to theL ¹-convergence.     

VORONOI POLYHEDRA OF UNIT BALL PACKINGS WITH SMALL SURFACE AREA

In this note, first, we give a very short new proof of the theorem which yields a lower bound for the surface area of Voronoi cells of unit ball packings in E ^d and implies Rogers' upper bound for the density of unit ball packings in E ^d for all d ≥ 2. Second we sharpen locally a classical result of Gauss by finding the locally smallest surface area Voronoi cells of lattice unit ball packings in E ³. This revised version was published online in August 2006 with corrections to the Cover Date.     

Affinely Regular Polygons as Extremals of Area Functionals

For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.     

Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional

Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C^∞, and let g_ij = δ_ij denote the flat metric on \input amssym ${\Bbb R}^2$ . Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W^2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$ . In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C³ away from a certain singular set Σ and C^∞ away from a larger singular set Σ ∪ Σ₀. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ₀. Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates. © 2010 Wiley Periodicals, Inc.     

On the L 1-lower semicontinuity of certain convex functionals defined on BV-vector valued functions

The aim of this work is to provide an L ¹-lower semicontinuity result for an integral convex functional defined on BV-vector valued functions, without any coerciveness and any continuity assumption on the integrand f(x, ξ) with respect to the variable x.      

Multiple solutions of nonlinear elliptic systems

We proved a multiplicity result for a nonlinear elliptic system in R^N. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions and the topology of the set of the global maxima of the coefficients.     

Monotone trajectories of differential inclusions and functional differential inclusions with memory

The paper gives a necessary and sufficient condition for the existence of monotone trajectories to differential inclusionsdx/dt ∈S[x(t)] defined on a locally compact subsetX ofR ^p, the monotonicity being related to a given preorder onX. This result is then extended to functional differential inclusions with memory which are the multivalued case to retarded functional differential equations. We give a similar necessary and sufficient condition for the existence of trajectories which reach a given closed set at timet=0 and stay in it with the monotonicity property fort≧0.     

Determination of Spherical Area Measures by Means of Dilation Volumes

For a large class of compact sets X in ℝ^d which can be represented as finite unions of sets of positive reach, the spherical area measure of order d — 1 is determined by the volumes of dilations of X with infinitesimal multiples of all rotations of a suitable convex test set. For spherical area measures of lower orders, a similar result is obtained by approximating the set X by the closures of the complements of close parallel bodies.     

Regularity of solutions for some variational problems subject to a convexity constraint

We first study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [3]. We then give various extensions of this result. In Particular, we consider a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [4]. © 2001 John Wiley & Sons, Inc.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.     

Topological and analytical properties of Sobolev bundles,I: The critical case

We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W ^2,m/2 defined over M ^m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.      

Upper Bound Estimates of the Cramér Functionals for Markov Processes

We prove that the Cramér functional of a Markov process can be controlled by a function of an integral functional if the transition semigroup is uniformly integrable in L ^p . As an application of this result, a general large deviation upper bound is obtained. Then, the notation of F-Sobolev inequality is extended to general Markov processes by replacing the Dirichlet form with the Donsker–Varadhan entropy. As the other application, it is proved that the uniform integrability of a transition semigroup implies a F-Sobolev inequality.     

On some equations in prime rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let T : R → R be an additive mapping satisfying the relation T(x ³) = T(x)x ² − xT(x)x + x ² T(x) for all x ∈ R. In this case T is of the form 4T(x) = qx + xq, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.     

On random surface area

Consider a random smooth Gaussian field G(x): F ? \mathbbR F \to \mathbb{R} , where F is a compact set in \mathbbR^d {\mathbb{R}^d} . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.     

Pairs of positive solutions for nonlinear elliptic equations with the <Emphasis Type="Italic">p</Emphasis>-Laplacian and a nonsmooth potential

In this paper we examine a nonlinear elliptic problem driven by the p-Laplacian differential operator and with a potential function which is only locally Lipschitz, not necessarily C¹ (hemivariational inequality). Using the nonsmooth critical point theory of Chang, we obtain two strictly positive solutions. One solution is obtained by minimization of a suitable modification of the energy functional. The second solution is obtained by generalizing a result of Brezis-Nirenberg about the local C¹₀-minimizers versus the local H¹₀-minimizers of a C¹-functional. Mathematics Subject Classification (2000) 35J50, 35J85, 35R70