Global regularity for almost minimizers of nonconvex variational problems

We prove some global, up to the boundary of a domain $\Omega \subset {\mathbb{R}}^{n}We prove some global, up to the boundary of a domain , continuity and Lipschitz regularity results for almost minimizers of functionals of the form The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be H?lder continuous with respect to its first two arguments.      

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.     

Lipschitz regularity for constrained local minimizers of convex variational integrals with a wide range of anisotropy

We establish interior gradient bounds for functions ${u \in W^1_{1, {\rm loc}} (\Omega)}$ which locally minimize the variational integral ${J [u, \Omega] = \int_\Omega h \left( |\nabla u| \right) dx}$ under the side condition ${u \ge \Psi}$ a.e. on Ω with obstacle ${\Psi}$ being locally Lipschitz. Here h denotes a rather general N-function allowing (p, q)-ellipticity with arbitrary exponents 1 < p ≤ q < ∞. Our arguments are based on ideas developed in Bildhauer et al. (Z Anal Anw 20:959–985, 2001) combined with techniques originating in Fuchs (2011).     

Lipschitz regularity for some asymptotically subquadratic problems

We establish a local Lipschitz regularity result for local minimizers of variational integrals under the assumption that the integrand becomes appropriately elliptic at infinity. The exponent that measures the ellipticity of the integrand is assumed to be less than two.     

Global stress regularity of convex and some nonconvex variational problems

We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains. These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a difference quotient technique. In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems. The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case).      

Lipschitz regularity for scalar minimizers of autonomous simple integrals

We prove Lipschitz regularity for a minimizer of the integral , defined on the class of the AC functions having x(a)=A and x(b)=B. The Lagrangian may have L(s,) nonconvex (except at ξ=0), while may be non-lsc, measurability sufficing for ξ≠0 provided, e.g., L^**() is lsc at (s,0) s. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient φ/ρ() (and not of L^**() itself, as usual), where φ(s)+ρ(s)h(ξ) approximates the bipolar L^**(s,ξ) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented.     

Partial regularity of minimizers of polyconvex variational integrals

We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral , where stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C ², strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000): 49N60, 35J50     

Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.      

Symmetry of minimizers for some nonlocal variational problems

We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the existence of some integral identities. We study the identities that lead to symmetry results, the functionals that can be considered and the function spaces that can be used. Then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of Laplacian, for the generalized Choquard functional and for the standing waves of the Davey-Stewartson equation.     

Partial regularity for minimizers of convex integrals with L log L-growth

We prove -partial regularity of minimizers with for a class of convex integral functionals with nearly linear growth whose model is In this way we extend to any dimension n a previous, analogous, result in [FS] valid only in the case Received December 1998     

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Local minimizers of one-dimensional variational problems and obstacle problems

In this Note we suggest a direct approach to study local minimizers of one-dimensional variational problems. To cite this article: M.A. Sychev, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth

We prove a small excess regularity theorem for almost minimizers of a quasi-convex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof. Mathematics Subject Classification (2000) 49N60, 26B25     

Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian

This paper proves new results of existence of minimizers for the nonconvex integral , among the AC functions with x(a)=A, x(b)=B. Our Lagrangian L() is e.g. lsc with superlinear growth, assuming +∞ values freely. We replace convexity by almost convexity, a hypothesis which in the radial superlinear case L(s,ξ)=f(s,|ξ|) is automatically satisfied provided f(s,) is convex at zero.     

A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems

We study two-point Lagrange problems for integrands :

 

Under very weak regularity hypotheses [ is Hölder continuous and locally elliptic on each compact subset of ] we obtain, when is of superlinear growth in , a characterization of problems in which the minimizers of (P) are -regular for all boundary data. This characterization involves the behavior of the value function : defined by . Namely, all minimizers for (P) are -regular in neighborhoods of and if and only if is Lipschitz continuous at . Consequently problems (P) possessing no singular minimizers are characterized in cases where not even a weak form of the Euler-Lagrange equations is available for guidance. Full regularity results for problems where is nearly autonomous, nearly independent of , or jointly convex in are presented.

     

Regularity for minimizers of some variational problems in plasticity theory

A variational problem for a functional depending on the symmetric part of the gradient of the unknown vectorvalued function is considered. We assume that the integrand of the problem has power growth with exponent less than two. We prove the existence of summable second derivatives near a flat piece of the boundary. In the two-dimensional case, Hölder continuity up to the boundary of the strain and stress tensors is established. Bibliography: 6 titles.     

Existence and symmetry of minimizers for nonconvex radially symmetric variational problems

We study functionals of the form