Regularity Of Local Minimizers Research Articles

Partial regularity for minimizers of a class of discontinuous Lagrangians

We study a one-dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio–Baradat–Brenier, of the discrete Monge–Ampère gravitational model, which describes the motion of interacting particles whose dynamics is ruled by the optimal transport problem. The more general action-type functional we consider contains a discontinuous potential term related to the descending slope of the opposite squared distance function from a generic discrete set in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{d}$$\\end{document}. We exploit the underlying geometrical structure provided by the associated Voronoi decomposition of the space to obtain C1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,1}$$\\end{document}-regularity for local minimizers out of a finite number of shock times.

A Condition for the H\xf6lder Regularity of Local Minimizers of a Nonlinear Elastic Energy in Two Dimensions

We prove the local Hölder continuity of strong local minimizers of the stored energy functionalE(u)=∫Ωλ|∇u|2+h(det∇u)dx\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$E(u)=\\int_{\\Omega}\\lambda |\\nabla u|^{2}+h({\\rm det} \\nabla u)\\,{\\rm d}x$$\\end{document}subject to a condition of ‘positive twist’. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as {sto 0+}, linearly as {s to +infty}, and satisfies {h(s)=+infty} if {s leqq 0}. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed Hölder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term {int_{Omega} h({rm det} nabla u),{rm d}x} can have by analysing the regularity of local minimizers in the class of ‘shear maps’. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are Hölder continuous.

A Hölder continuity result for a class of obstacle problems under non standard growth conditions

AbstractWe prove C0, α regularity for local minimizers u of functionals with p(x)‐growth of the type in the class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$K :=\lbrace w \in W^{1,p(\cdot )}(\Omega ;{\mathbb R}): w \ge \psi \rbrace$\end{document}, where the exponent function p : Ω → (1, ∞) is assumed to be continuous with a modulus of continuity satisfying and 1 < γ1 ⩽ p(x) ≤ γ2 < +∞. Moreover, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\psi \in W^{1,1}_{\textnormal {loc}}(\Omega )$\end{document} is a given obstacle function, whose gradient Dψ belongs to a Morrey space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L_{\textnormal {loc}}^{q,\lambda }(\Omega )$\end{document} with n − γ1 < λ < n and q > γ2. We do not assume any quantitative continuity of the integrand function f.

Regularity theory for the Möbius energy

The Möbius energy, defined 1991 by O'Hara, is the most prominent example of a knot energy. In this text we will focus on the regularity of local minimizers (within a prescribed knot class) whose arc-length parametrization was shown to be $C^{1,1}$ by Freedman, He, and Wang. Later on, He improved this result to $C^\infty$ regularity. In this text we will briefly outline the main ideas of these two steps which require completely different approaches involving techniques from geometry and analysis. Moreover we explain how to rigorously derive the first variation of the Möbius energy and fix a gap in He's treatise.

On Regularity of Local Minimizers of one Class of Quadratic Nondifferentiable Functionals

We establish the partial Holder continuity of a minimizing functions in a neighborhood of the interface and partial Holder continuity of the derivatives up to the boundary of the interface. It is shown, that the singular sets are absent in the case of two spatial variables. Bibliography: 9 titles.

Partial regularity for variational integrals with $(s,\mu ,q)$ -Growth

We introduce integrands $f: {\mathbb R}^{nN}\to{\mathbb R}$ of $(s,\mu,q)$ –type, which are, roughly speaking, of lower (upper) growth rate $s\geq 1 (q > 1$ ) satisfying in addition $D^{2}f(Z)\geq \lambda \left(1+|Z|^2\right)^{-\mu/2}$ for some $\mu \in {\mathbb R}$ . Then, if $q < 2-\mu +s\frac{2}{n}$ , we prove partial $C^{1}$ –regularity of local minimizers $u\in W^{1}_{1,loc}(\Omega ,{\mathbb R}^{N})$ by the way including integrands f being controlled by some N–function and also integrands of anisotropic power growth. Moreover, we extend the known results up to a certain limit and present examples which are not covered by the standard theory.