The strong minimum principle for quasisuperminimizers of non-standard growth

We prove the strong minimum principle for non-negative quasisuperminimizers of the variable exponent Dirichlet energy integral under the assumption that the exponent has modulus of continuity slightly more general than Lipschitz. The proof is based on a new version of the weak Harnack estimate.     

Asymptotically regular problems I: Higher integrability

We consider weak solutions u of non-linear systems of partial differential equations. Assuming that the system exhibits a certain kind of elliptic behavior near infinity we prove higher integrability results for the gradient Du. In particular, we establish Hölder continuity of u in low dimensions. Moreover, we obtain analogous results for vectorial minimizers of multi-dimensional variational integrals. Finally, we discuss an extension to minimizing sequences and applications to generalized minimizers.     

On W-solvability for special vectorial Hamilton-Jacobi systems

We study the solvability of special vectorial Hamilton-Jacobi systems of the form F(Du(x))=0 in a Sobolev space. In this paper we establish the general existence theorems for certain Dirichlet problems using suitable approximation schemes called W^1,p-reduction principles that generalize the similar reduction principle for Lipschitz solutions. Our approach, to a large extent, unifies the existing methods for the existence results of the special Hamilton-Jacobi systems under study. The method relies on a new Baire's category argument concerning the residual continuity of a Baire-one function. Some sufficient conditions for W^1,p-reduction are also given along with certain generalization of some known results and a specific application to the boundary value problem for special weakly quasiregular mappings.     

HILBERT SPACES ASSOCIATED WITH SECOND ORDER ELLIPTIC OPERATORS AND APPLICATIONS TO BOUNDARY VALUE PROBLEMS AND VARIATIONAL INEQUALITIES

Abstract

Boundary value problems and variational inequalities, associated with second order elliptic operators, will be studied in a Hilbert space framework. In this space, functions will have (at least) locally square integrable derivatives of order up to two. Also the conormal derivative, extended by continuity, will be square integrable on the boundary of the region considered. Criteria for approximating elements of the Hilbert space by smooth functions will be given and thus closed convex sets, associated with inequalities on the boundary, exist.

The idea of the present approach originated from the method suggested by Lions and Magenes, for putting some regular elliptic problems in the variational setting. The differential equation is multiplied by Qv, with Q some operator and v a function and the result is integrated as required.     

Limiting properties of the second order Ginzburg–Landau minimizers

The author studies the weak convergence for the gradient of the minimizers for a second order energy functional when the parameter tends to 0. And this paper is also concerned with the location of the zeros and the blow-up points of the gradient of the minimizers of this functional. Finally, the strong convergence of the gradient of the radial minimizers is obtained.     

A critical point theorem via the Ekeland variational principle

The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais-Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.     

An upper gradient approach to weakly differentiable cochains

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheim?s theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen–Koskela?s concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with p-integrable upper gradient in n-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey–Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.     

A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a G-closure problem. Under convexity and p  -growth conditions (p>1$p>1$), it is proved that all such possible effective energy densities obtained by a Γ-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.     

On the Jordan–Kinderlehrer–Otto variational scheme and constrained optimization in the Wasserstein metric

We prove the monotonicity of the second-order moments of the discrete approximations to the heat equation arising from the Jordan–Kinderlehrer–Otto (JKO) variational scheme. This issue appears in the study of constrained optimization in the 2-Wasserstein metric performed by Carlen and Gangbo for the kinetic Fokker–Planck equation. As an alternative to their duality method, we provide the details of a direct approach, via Lagrange multipliers. Estimates for the fourth-order moments in the constrained case, which are essential to the subsequent alternate analysis, are also obtained. Partial support provided by NSF grant DMS 0305794.     

The optimal shape of a dendrite sealed at both ends

In this paper, we are interested in the geometric structures which appear in nature. We consider the example of a nerve fiber and we suppose that shapes in nature arise in order to optimize some criterion. Then, we try to solve the problem consisting in searching the shape of a nerve fiber for a given criterion. The first considered criterion represents the attenuation in space of the electrical message troughout the fiber and seems to be relevant. Our second criterion represents the attenuation in time of the electrical message and doesn't provide a realistic shape. We prove that the associated optimization problem has no solution.     

Closed loop control for evolution inclusions

The existence of a closed loop control problem for quasi-variational inequalities is established. Feedback laws are given.     

Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians,eikonal equations,and shape-from-shading

We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation that is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem.     

An inverse problem of identifying the coefficient of parabolic equation

This paper studies an inverse problem of identifying the coefficient of parabolic equation when the final observation is given, which has important application in a large fields of applied science. Based on the optimal control framework, the existence and necessary condition of the minimum for the control functional are established. Since the optimal control problem is nonconvex, one may not expect a unique solution. However, in this paper the solution is proved to be locally unique. After the necessary condition is transformed into an elliptic bilateral variational inequality, an algorithm and some numerical experiments are proposed in the paper. The numerical results show that the algorithm designed in this paper is stable and that the coefficient is recovered very well.     

Shooting with degree theory: Analysis of some weighted poly-harmonic systems

In this paper, the author establishes the existence of positive entire solutions to a general class of semilinear poly-harmonic systems, which includes equations and systems of the weighted Hardy–Littlewood–Sobolev type. The novel method used implements the classical shooting method enhanced by topological degree theory. The key steps of the method are to first construct a target map which aims the shooting method and the non-degeneracy conditions guarantee the continuity of this map. With the continuity of the target map, a topological argument is used to show the existence of zeros of the target map. The existence of zeros of the map along with a non-existence theorem for the corresponding Navier boundary value problem imply the existence of positive solutions for the class of poly-harmonic systems.     

ADJOINT PROBLEMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

Abstract

A general theory of adjoint variational problems is formulated for essentially arbitrary Lagrangians involving m independent and n dependent variables, together with the first derivatives of the latter, This approach contains as a special case the theory of Haar [4], in which the Lagrangian may depend solely on the derivatives of a single dependent function of two arguments. Because of the eventual occurrence of possibly incompatible sets of integrability conditions, the basic theory is developed against the background of non-integrable m-dimensional subspaces, which is in sharp contrast to the traditional approach to the calculus of variations. Relatively self-adjoint Lagrangians are defined and completely characterized in terms of an arbitrary Riemannian metric. In the course of the general theory certain geometric object fields are encountered in a very natural manner, some of which had arisen previously in the canonical formalism proposed by Caratheodory [2]. Accordingly the analysis of the present paper may serve to shed some light on this conceptually extremely difficult formalism.     

Remarks on the method of modulus of continuity and the modified dissipative Porous Media Equation

We employ Besov space techniques and the method of modulus of continuity to obtain the global well-posedness of the modified Porous Media Equation.     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.