Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals

We consider local minimizers of variational integrals , where F is of anisotropic (p, q)-growth with exponents . If F is in a certain sense decomposable, we show that the dimensionless restriction together with the local boundedness of u implies local integrability of for all exponents . More precisely, the initial exponents for the integrability of the partial derivatives can be increased by two, at least locally. If n = 2, then we use these facts to prove -regularity of u for any exponents .     

Boundary regularity for almost minimizers of quasiconvex variational problems

We consider boundary regularity for almost minimizers of quasiconvex variational integrals with polynomial growth of order p ≥ 2, and obtain a general criterion for an almost minimizer to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, the proof yields directly the optimal regularity for an almost minimizer in this neighbourhood.     

Twodimensional anisotropic variational problems

We consider local minimizers a domain in , of the variational integral with integrand f of upper (lower) growth rate q (s). We show using a lemma due to Frehse and Seregin that u has H?lder continuous first derivatives provided that . Received: 2 October 2001 / Accepted: 25 October 2001 / Published online: 28 February 2002     

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics

In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.     

Manifold constrained variational problems

The integral representation for the relaxation of a class of energy functionals where the admissible fields are constrained to remain on a -dimensional manifold is obtained. If is a continuous function satisfying for and for all then where is open, bounded, and is the tangential quasiconvexification of f at belong to the tangent space to at Received October 10, 1998 / Accepted December 1, 1998     

On a variational problem with non-differentiable constraints

We study the regularity of minimizers and critical points of the Dirichlet energy under an integral constraint given by a non-differentiable function. We obtain existence of a Lipschitz continuous minimizer for a relaxed problem. In two dimensions, some regularity can also be proved for critical points.     

Comparison estimates in anisotropic variational problems

A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.     

Regularity properties of optimal controls for problems with time-varying state and control constraints

In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.     

Lipschitz continuity and local semiconcavity for exit time problems with state constraints

In the classical time optimal control problem, it is well known that the so-called Petrov condition is necessary and sufficient for the minimum time function to be locally Lipschitz continuous. In this paper, the same regularity result is obtained in the presence of nonsmooth state constraints. Moreover, for a special class of control systems we obtain a local semiconcavity result for the constrained minimum time function.     

Regularization of convex variational problems with applications to generalized Newtonian fluids

We study variational problems with convex integrands of very general structure by introducing certain regularizations leading to particular minimizers. In a second part we apply the method to stationary generalized Newtonian fluids which gives the existence of solutions under weak hypotheses on the dissipative potential. In particular the potential is not required to be a strictly convex function.Received: 29 July 2003     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature

We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of constant mean curvature in ann-dimensional sphereB _R, 1>R>R ₀ ⁽ⁿ⁾ , (R ₀ ⁽ⁿ⁾ being a constant depending only onn), without imposing boundary conditions or bounds of any sort.

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Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB _R, 1>R>R ₀ ⁽ⁿ⁾, (R ₀ ⁽ⁿ⁾ essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.

     

The Euler and Weierstrass conditions for nonsmooth variational problems

Necessary conditions are developed for a general problem in the calculus of variations in which the Lagrangian function, although finite, need not be Lipschitz continuous or convex in the velocity argument. For the first time in such a broadly nonsmooth, nonconvex setting, a full subgradient version of Euler's equation is derived for an arc that furnishes a local minimum in the classical weak sense, and the Weierstrass inequality is shown to accompany it when the arc gives a local minimum in the strong sense. The results are achieved through new techniques in nonsmooth analysis.This research was supported in part by funds from the U.S.-Israel Science Foundation under grant 90-00455, and also by the Fund for the Promotion of Research at the Technion under grant 100-954 and by the U.S. National Science Foundation under grant DMS-9200303.This article was processed by the author using the style filepljourlm from Springer-Verlag.     

Global higher integrability for parabolic quasiminimizers in nonsmooth domains

We study the global higher integrability of the gradient of a parabolic quasiminimizer with quadratic growth conditions. We show that if the lateral boundary satisfies a capacity density condition and if boundary and initial values are smooth enough, then quasiminimizers globally belong to a higher Sobolev space than assumed a priori. We derive estimates near the lateral and the initial boundaries.     

On two-dimensional parametric variational problems

Let be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that is a closed rectifiable Jordan curve in . We then prove that there is a conformally parametrized minimizer of in the class of surfaces of the type of the disk B which are bounded by . An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on -conformal mappings. In addition we show that the minimizer of is H?lder continuous in B, and even in if satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one. Received July 20, 1998 / Accepted October 23, 1998     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

A regularity theory for variational integrals with L\ln L-Growth

In the present paper we study regularity for local minimizers of the convex variational integral defined on certain classes of vector–valued functions . The underlying energy spaces are natural from the point of view of existence theory. We then show that local minimizers are of class apart from a closed singular set with vanishing Lebesgue measure, provided . For twodimensional problems we obtain smoothness in the interior of . Received June 21, 1996 / In revised form December 2, 1996 / Accepted December 17, 1996     

Reparametrizations and approximate values of integrals of the calculus of variations

We prove an approximation result, that implies the non-occurrence of the Lavrentiev phenomenon.