We consider the problem min∫double-struckR12|trueγ̇|2+W(γ)normaldt among curves connecting two given wells of W≥0, and we reduce it, following a standard method, to a geodesic problem of the form min∫01K(γ)|trueγ̇|normaldt with K=2W. We then prove existence of curves minimizing this new action just by proving that the distance induced by K is proper (i.e., its closed balls are compact). The assumptions on W are minimal, and the method seems robust enough to be applied in the future to some PDE problems. Copyright © 2016 John Wiley & Sons, Ltd.

No abstract

No abstract

We study the one‐dimensional symmetry of solutions to the nonlinear Stokes equation {left−Δu+∇W(u)=∇pin ℝd,left∇⋅u=0in ℝd, which are periodic in the d − 1 last variables (living on the torus 𝕋d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋d−1, i.e., E()u=∫Ω12∇u2+W()uitalicdx,1em∇⋅u=0. Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x1 →  ± ∞ is one‐dimensional; i.e., u depends only on the x1‐variable. In particular, this class includes in dimension d = 2 the nonlinearities W=12w2 with w being a harmonic function or a solution to the wave equation, while in dimension d ≥ 3, this class contains a perturbation of the Ginzburg‐Landau potential as well as potentials W having d + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x1 →  ± ∞. © 2019 Wiley Periodicals, Inc.

No abstract

Given a connected Riemannian manifold  , an -dimensional Riemannian manifold  which is either compact or the Euclidean space, ∈ [1, +∞) and ∈ (0, 1], we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space , (,  ) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach-Steinhaus uniform boundedness principle in linear Banach spaces.The space, the energy and the topology on this space are independent of the embedding and can be defined intrinsically [30].1.1. Extension of traces. We first consider relationships between a qualitative and quantitative properties for the problem of surjectivity of the trace. In the setting of linear Sobolev spaces, given ∈ (0, 1), ∈ (1, +∞) and a manifold  which is either compact or the Euclidean space, the classical trace theory states that the restriction of continuous functions in ( × [0, +∞), ℝ) has a linear continuous extension to the trace operator tr ∶ +1∕ , ( × (0, +∞), ℝ) → , (, ℝ) and that the latter trace operator is surjective [1, Theorem 7.39;32, Chapter 10;61, Theorem 2.7.2]. A. Monteil is a postdoctoral researcher (chargé de recherches) by the Fonds de la Recherche Scientifique-FNRS; J. Van Schaftingen was supported by the Mandat d'Impulsion Scientifique F.4523.17, "Topological singularities of Sobolev maps" of the Fonds de la Recherche Scientifique-FNRS.

We are interested in some energy functionals concentrated on the discontinuity lines of divergence-free 2D vector fields valued in the circle S 1 . This kind of energy has been introduced first by P. Aviles and Y. Giga in [4]. They show in particular that, with the cubic cost function f (t) = t 3 , this energy is lower semicontinuous. In this paper, we construct a counter-example which excludes the lower semicontinuity of line energies for cost functions of the form t p with 0 < p < 1. We also show that, in this case, the viscosity solution corresponding to a certain convex domain is not a minimizer.

Models for branched networks are often expressed as the minimization of an energy M α over vector measures concentrated on 1-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation M α ε , introduced by Edouard Oudet and Filippo Santambrogio, which is defined over H 1 vector measures. These energies induce some pseudo-distances between L 2 functions obtained through the minimization problem min{M α ε (u) :We prove some uniform estimates on these pseudo-distances which allow us to establish a Γ-convergence result for these energies with a divergence constraint.