Authors:Sun-Sig Byun; Jihoon Ok; Jung-Tae Park Pages: 1639 - 1667 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Sun-Sig Byun, Jihoon Ok, Jung-Tae Park We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderón–Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal function of the associated measure, up to a correct power, under minimal regularity requirements on the nonlinearity, the variable exponent and the boundary of the domain.

Authors:Tomoyuki Miyaji; Yoshio Tsutsumi Pages: 1707 - 1725 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Tomoyuki Miyaji, Yoshio Tsutsumi We show the existence of global solution and the global attractor in L 2 ( T ) for the third order Lugiato–Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the L 2 ( T ) norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the L 2 norm conservation is useful for the third order Lugiato–Lefever equation unlike the KdV and the cubic NLS equations. Therefore, it seems important and natural to construct the global attractor in L 2 ( T ) . For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.

Authors:Ricardo Alonso; Thierry Goudon; Arthur Vavasseur Pages: 1727 - 1758 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Ricardo Alonso, Thierry Goudon, Arthur Vavasseur We investigate the large time behavior of the solutions of a Vlasov–Fokker–Planck equation where particles are subjected to a confining external potential and a self-consistent potential intended to describe the interaction of the particles with their environment. The environment is seen as a medium vibrating in a direction transverse to particles' motion. We identify equilibrium states of the model and justify the asymptotic trend to equilibrium. The analysis relies on hypocoercivity techniques.

Authors:Lassaad Aloui; Moez Khenissi; Luc Robbiano Pages: 1759 - 1792 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Lassaad Aloui, Moez Khenissi, Luc Robbiano We prove, under the exterior geometric control condition, the Kato smoothing effect for solutions of an inhomogeneous and damped Schrödinger equation on exterior domains.

Authors:Huy Quang Nguyen Pages: 1793 - 1836 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Huy Quang Nguyen This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. Local well-posedness for this problem with Lipschitz initial velocity was established by Alazard–Burq–Zuily [1]. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for initial data 1 4 -derivative less regular than the aforementioned threshold, which corresponds to the gain of Hölder regularity of the semi-classical Strichartz estimate for the fully nonlinear system. In order to obtain this Cauchy theory, we establish global, quantitative results for the paracomposition theory of Alinhac [5].

Authors:Christian Seis Pages: 1837 - 1850 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Christian Seis In this work, we provide stability estimates for the continuity equation with Sobolev vector fields. The results are inferred from contraction estimates for certain logarithmic Kantorovich–Rubinstein distances. As a by-product, we obtain a new proof of uniqueness in the DiPerna–Lions setting. The novelty in the proof lies in the fact that it is not based on the theory of renormalized solutions.

Authors:Jürgen Jost; Lei Liu; Miaomiao Zhu Pages: 1851 - 1882 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Jürgen Jost, Lei Liu, Miaomiao Zhu We show the existence of a global weak solution of the heat flow for Dirac-harmonic maps from compact Riemann surfaces with boundary when the energy of the initial map and the L 2 -norm of the boundary values of the spinor are sufficiently small. Dirac-harmonic maps couple a second order harmonic map type system with a first-order Dirac type system. The heat flow that has been introduced in [9] and that we investigate here is novel insofar as we only make the second order part parabolic, but carry the first order part along the resulting flow as an elliptic constraint. Of course, since the equations are coupled, both parts then change along the flow. The solution is unique and regular with the exception of at most finitely many singular times. We also discuss the behavior at the singularities of the flow. As an application, we deduce some existence results for Dirac-harmonic maps. Since we may impose nontrivial boundary conditions also for the spinor part, in the limit, we shall obtain Dirac-harmonic maps with nontrivial spinor part.

Authors:Marta Lewicka; Annie Raoult; Diego Ricciotti Pages: 1883 - 1912 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Marta Lewicka, Annie Raoult, Diego Ricciotti We study the elastic behaviour of incompatibly prestrained thin plates of thickness h whose internal energy E h is governed by an imposed three-dimensional smooth Riemann metric G only depending on the variable in the midsurface ω. It is already known that h − 2 inf E h converges to a finite value c when the metric G restricted to the midsurface has a sufficiently regular immersion, namely W 2 , 2 ( ω , R 3 ) . The obtained limit model generalizes the bending (Kirhchoff) model of Euclidean elasticity. In the present paper, we deal with the case when c equals 0. Then, equivalently, three independent entries of the three-dimensional Riemann curvature tensor associated with G are null. We prove that, in such regime, necessarily inf E h ≤ C h 4 . We identify the Γ-limit of the scaled energies h − 4 E h and show that it consists of a von Kármán-like energy. The unknowns in this energy are the first order incremental displacements with respect to the deformation defined by the bending model and the second order tangential strains. In addition, we prove that when inf h − 4 E h → 0 , then G is realizable and hence min E h = 0 for every h.

Authors:Alessio Porretta; Philippe Souplet Pages: 1913 - 1923 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Alessio Porretta, Philippe Souplet We consider the diffusive Hamilton–Jacobi equation, with superquadratic Hamiltonian, homogeneous Dirichlet conditions and regular initial data. It is known from [4] (Barles–DaLio, 2004) that the problem admits a unique, continuous, global viscosity solution, which extends the classical solution in case gradient blowup occurs. We study the question of the possible loss of boundary conditions after gradient blowup, which seems to have remained an open problem till now. Our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena. For any smooth bounded domain, we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary, as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup. Actually, we show that the latter possibility is rather exceptional. More generally, we show that the set of the points where boundary conditions are lost, can be prescribed to be arbitrarily close to any given open subset of the boundary.

Authors:Jianfeng Cheng; Lili Du; Yongfu Wang Pages: 1355 - 1386 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Jianfeng Cheng, Lili Du, Yongfu Wang The free streamline theory in hydrodynamics is an important and difficult issue not only in fluid mechanics but also in mathematics. The major purpose in this paper is to establish the well-posedness of the impinging jets in steady incompressible, rotational, plane flows. More precisely, given a mass flux and a vorticity of the incoming flows in the inlet of the nozzle, there exists a unique smooth impinging plane jet. Moreover, there exists a smooth free streamline, which goes to infinity and initiates at the endpoint of the nozzle smoothly. In addition, asymptotic behavior in upstream and downstream, uniform direction and other properties of the impinging jet are also obtained. The main ingredients of the mathematic analysis in this paper are based on the modified variational method developed by H. W. Alt, L. A. Caffarelli and A. Friedman in the elegant works [1,17], which has been shown to be powerful to deal with the steady irrotational flows with free streamlines.

Authors:Serena Dipierro; Enrico Valdinoci Pages: 1387 - 1428 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Serena Dipierro, Enrico Valdinoci We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.

Authors:Chenjie Fan Pages: 1429 - 1482 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Chenjie Fan We study the focusing mass-critical nonlinear Schrödinger equation, and construct certain solutions which blow up at exactly m points according to the log–log law.

Authors:Ting-Ying Chang; Florica C. Cîrstea Pages: 1483 - 1506 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Ting-Ying Chang, Florica C. Cîrstea We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form (0.1) − div ( A ( x ) ∇ u p − 2 ∇ u ) + b ( x ) h ( u ) = 0 in B 1 ∖ { 0 } , where B r denotes the open ball with radius r > 0 centred at 0 in R N ( N ≥ 2 ) . We assume that A ∈ C 1 ( 0 , 1 ] , b ∈ C ( B 1 ‾ ∖ { 0 } ) and h ∈ C [ 0 , ∞ ) are positive functions associated with regularly varying functions of index ϑ, σ and q at 0, 0 and ∞ respectively, satisfying q > p − 1 > 0 and ϑ − σ < p < N + ϑ . We prove that the condition b ( x ) h ( Φ ) ∉ L 1 ( B 1 / 2 ) is sharp for the removability of all singularities at 0 for the positive solutions of (0.1), where Φ denotes the “fundamental solution” of − div ( A ( x ) ∇ u p − 2 ∇ u ) = δ 0 (the Dirac mass at 0) in B 1 , subject to Φ ∂ B 1 = 0 . If b ( x ) h ( Φ ) ∈ L 1 ( B 1 / 2 PubDate: 2017-10-18T17:15:27Z DOI: 10.1016/j.anihpc.2016.12.001

Authors:Toshiaki Hishida; Ana Leonor Silvestre; Takéo Takahashi Pages: 1507 - 1541 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Toshiaki Hishida, Ana Leonor Silvestre, Takéo Takahashi Consider a rigid body S ⊂ R 3 immersed in an infinitely extended Navier–Stokes fluid. We are interested in self-propelled motions of S in the steady state regime of the system rigid body-fluid, assuming that the mechanism used by the body to reach such a motion is modeled through a distribution of velocities v ⁎ on ∂ S . If the velocity V of S is given, can we find v ⁎ that generates V' We show that this can be solved as a control problem in which v ⁎ is a six-dimensional control such that either Supp v ⁎ ⊂ Γ , an arbitrary nonempty open subset of ∂Ω, or v ⁎ ⋅ n ∂ Ω = 0 . We also show that one of the self-propelled conditions implies a better summability of the fluid velocity.

Authors:Kari Astala; Albert Clop; Daniel Faraco; Jarmo Jääskeläinen; Aleksis Koski Pages: 1543 - 1559 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Kari Astala, Albert Clop, Daniel Faraco, Jarmo Jääskeläinen, Aleksis Koski We provide Schauder estimates for nonlinear Beltrami equations and lower bounds of the Jacobians for homeomorphic solutions. The results were announced in [1] but here we give detailed proofs.

Authors:L.J. Díaz; K. Gelfert; M. Rams Pages: 1561 - 1598 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): L.J. Díaz, K. Gelfert, M. Rams We study transitive step skew-product maps modeled over a complete shift of k, k ≥ 2 , symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak ⁎ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.

Authors:Charles Baker; Huy The Nguyen Pages: 1599 - 1610 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Charles Baker, Huy The Nguyen We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature smoothly evolve by mean curvature flow to round points.

Authors:Yujin Guo; Xiaoyu Zeng Pages: 1611 - 1632 Abstract: Publication date: November–December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 6 Author(s): Yujin Guo, Xiaoyu Zeng We consider ground states of pseudo-relativistic boson stars with a self-interacting potential K ( x ) in R 3 , which can be described by minimizers of the pseudo-relativistic Hartree energy functional. Under some assumptions on K ( x ) , minimizers exist if the stellar mass N satisfies 0 < N < N ⁎ , and there is no minimizer if N > N ⁎ , where N ⁎ is called the critical stellar mass. In contrast to the case of the Coulomb-type potential where K ( x ) ≡ 1 , we prove that the existence of minimizers may occur at N = N ⁎ , depending on the local profile of K ( x ) near the origin. When there is no minimizer at N = N ⁎ , we also present a detailed analysis of the behavior of minimizers as N approaches N ⁎ from below, for which the stellar mass concentrates at a unique point.

Authors:Paul M.N. Feehan; Camelia A. Pop Pages: 1075 - 1129 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Paul M.N. Feehan, Camelia A. Pop We prove local supremum bounds, a Harnack inequality, Hölder continuity up to the boundary, and a strong maximum principle for solutions to a variational equation defined by an elliptic operator which becomes degenerate along a portion of the domain boundary and where no boundary condition is prescribed, regardless of the sign of the Fichera function. In addition, we prove Hölder continuity up to the boundary for solutions to variational inequalities defined by this boundary-degenerate elliptic operator.

Authors:Wolansky Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): G. Wolansky The object of this paper is to study estimates of ϵ − q W p ( μ + ϵ ν , μ ) for small ϵ > 0 . Here W p is the Wasserstein metric on positive measures, p > 1 , μ is a probability measure and ν a signed, neutral measure ( ∫ d ν = 0 ). In [16] we proved uniform (in ϵ) estimates for q = 1 provided ∫ ϕ d ν can be controlled in terms of ∫ ∇ ϕ p / ( p − 1 ) d μ , for any smooth function ϕ. In this paper we extend the results to the case where such a control fails. This is the case where, e.g., μ has a disconnected support, or the dimension d of μ (to be defined) is larger or equal to p / ( p − 1 ) . In the latter case we get such an estimate provided 1 / p + 1 / d ≠ 1 for q = min ( 1 , 1 / p + 1 / d ) . If 1 / p + 1 / d = 1 we get a log-Lipschitz estimate. As an application we obtain Hölder estimates in W p for curves of probability measures which are absolutely continuous in the total variation norm. In case the support of μ is disconnected (corresponding to d = ∞ ) we obtain sharp estimates for q = 1 / p (“optimal teleportation”): lim ϵ → 0 ϵ − 1 / p W p ( μ , μ + ϵ ν ) = ‖ ν ‖ μ where ‖ ν ‖ μ is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of μ, and the weights of the measure ν in each connected component of this support.

Authors:Andrej Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Andrej Zlatoš We study reaction–diffusion equations in one spatial dimension and with general (space- or time-) inhomogeneous mixed bistable–ignition reactions. For those satisfying a simple quantitative hypothesis, we prove existence and uniqueness of transition fronts, as well as convergence of “typical” solutions to the unique transition front (the existence part even extends to mixed bistable–ignition–monostable reactions). These results also hold for all pure ignition reactions without the extra hypothesis, but not for all pure bistable reactions. In fact, we find examples of either spatially or temporally periodic pure bistable reactions (independent of the other space–time variable) for which we can prove non-existence of transition fronts. The latter are the first such examples in periodic media which are non-degenerate in a natural sense, and they also prove a conjecture from [7].

Authors:Haïm Brezis; Petru Mironescu Abstract: Publication date: Available online 23 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Haïm Brezis, Petru Mironescu We investigate the validity of the Gagliardo–Nirenberg type inequality (1) ‖ f ‖ W s , p ( Ω ) ≲ ‖ f ‖ W s 1 , p 1 ( Ω ) θ ‖ f ‖ W s 2 , p 2 ( Ω ) 1 − θ , with Ω ⊂ R N . Here, 0 ≤ s 1 ≤ s ≤ s 2 are non negative numbers (not necessarily integers), 1 ≤ p 1 , p , p 2 ≤ ∞ , and we assume the standard relations s = θ s 1 + ( 1 − θ ) s 2 , 1 / p = θ / p 1 + ( 1 − θ ) / p 2 for some θ ∈ ( 0 , 1 ) . By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s 1 , s 2 , s are integers. It turns out that (1) holds for “most” of values of s 1 , … , p 2 , but not for all of them. We present an explicit condition on s 1 , s 2 , p 1 , p 2 which allows to decide whether (1) holds or fails.

Authors:Eduard Feireisl; Václav Mácha; Šárka Nečasová; Marius Tucsnak Abstract: Publication date: Available online 21 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Eduard Feireisl, Václav Mácha, Šárka Nečasová, Marius Tucsnak We consider a system modelling the motion of a piston in a cylinder filled by a viscous heat conducting gas. The piston is moving longitudinally without friction under the influence of the forces exerted by the gas. In addition, the piston is supposed to be thermally insulating (adiabatic piston). This fact raises several challenges which received a considerable attention, essentially in the statistical physics literature. We study the problem via the methods of continuum mechanics, specifically, the motion of the gas is described by means of the Navier–Stokes–Fourier system in one space dimension, coupled with Newton's second law governing the motion of the piston. We establish global in time existence of strong solutions and show that the system stabilizes to an equilibrium state for t → ∞ .

Authors:Mitia Duerinckx; Julian Fischer Abstract: Publication date: Available online 16 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Mitia Duerinckx, Julian Fischer We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in [24] as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.

Authors:A. Avila; P. Hubert Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): A. Avila, P. Hubert In this paper, we give a geometric criterion ensuring the recurrence of the vertical flow on Z d -covers of compact translation surfaces ( d ≥ 2 ). We prove that the linear flow in the windtree model is recurrent for every pair of parameters and almost every direction.

Authors:Andrei V. Faminskii Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Andrei V. Faminskii Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are established.

Authors:Inwon Kim; Olga Turanova Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Inwon Kim, Olga Turanova We study a model introduced by Perthame and Vauchelet [19] that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We adopt the viscosity solution approach to establish an optimal uniform convergence result of the tumor density as well as the pressure in the incompressible limit. The system lacks standard maximum principle, and thus modification of the usual approach is necessary.

Authors:Denis Serre Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Denis Serre We consider d × d tensors A ( x ) that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of ( det A ) 1 d − 1 . We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc. We deduce an a priori estimate for a new quantity, namely the space–time integral of ρ 1 n p , where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

Authors:Dongxiang Chen; Yuxi Wang; Zhifei Zhang Abstract: Publication date: Available online 14 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Dongxiang Chen, Yuxi Wang, Zhifei Zhang In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class 2 − θ for any θ > 0 . This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like e k t for the linearized Prandtl equation around a non-monotonic shear flow.

Authors:Christoph Scheven; Thomas Schmidt Abstract: Publication date: Available online 14 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Christoph Scheven, Thomas Schmidt We investigate the Dirichlet minimization problem for the total variation and the area functional with a one-sided obstacle. Relying on techniques of convex analysis, we identify certain dual maximization problems for bounded divergence-measure fields, and we establish duality formulas and pointwise relations between (generalized) BV minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of BV minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as BV obstacles, thin obstacles, and boundary obstacles, and they include information on exceptional sets and up to the boundary. As a side benefit, in some cases we also obtain assertions on the limit behavior of p-Laplace type obstacle problems for p ↘ 1 . On the technical side, the statements and proofs of our results crucially depend on new versions of Anzellotti type pairings which involve general divergence-measure fields and specific representatives of BV functions. In addition, in the proofs we employ several fine results on (BV) capacities and one-sided approximation.

Authors:Andrea Giorgini; Maurizio Grasselli Hao Abstract: Publication date: Available online 6 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Andrea Giorgini, Maurizio Grasselli, Hao Wu The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.

Authors:Wei Zhang; Jiguang Bao Abstract: Publication date: Available online 20 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Wei Zhang, Jiguang Bao We classify all solutions to − u t det D 2 u = f ( x ) in R − n + 1 , where f ∈ C α ( R n ) is a positive periodic function in x. More precisely, if u is a parabolically convex solution to above equation, then u is the sum of a convex quadratic polynomial in x, a periodic function in x and a linear function of t. It can be viewed as a generalization of the work of Gutiérrez and Huang in 1998. And along the line of approach in this paper, we can treat other parabolic Monge-Ampère equations.

Authors:Pavel Gurevich; Sergey Tikhomirov Abstract: Publication date: Available online 20 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Pavel Gurevich, Sergey Tikhomirov We address the question: Why may reaction-diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this' To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order a t − 1 / 2 as t → ∞ and explicitly find the rate a.

Authors:Dongho Chae; Jörg Wolf Abstract: Publication date: Available online 18 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Dongho Chae, Jörg Wolf We prove the existence of a forward discretely self-similar solutions to the Navier-Stokes equations in R 3 × ( 0 , + ∞ ) for a discretely self-similar initial velocity belonging to L l o c 2 ( R 3 ) .

Authors:Maria Alessandra Ragusa; Atsushi Tachikawa Abstract: Publication date: Available online 6 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Maria Alessandra Ragusa, Atsushi Tachikawa In the paper [1] “Boundary regularity of minimizers of p ( x ) -energy functionals”, some modifications are needed. 1. The exponent p 2 = p 2 ( 2 R ) in the statement of Theorem 2.6 should be p 2 ( ρ ) . According to this correction, we should modify the proof of Theorem 3.2. 2. In Theorem 1.1, the domain Ω is assumed to have the Lipschitz boundary ∂Ω. However, we need to assume that ∂Ω is in the class C 1 .

Authors:Michael Borghese; Robert Jenkins; Kenneth D.T.-R. McLaughlin Abstract: Publication date: Available online 2 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Michael Borghese, Robert Jenkins, Kenneth D.T.-R. McLaughlin We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the Image 2 generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion of the solution ψ ( x , t ) in any fixed space-time cone C ( x 1 , x 2 , v 1 , v 2 ) = { ( x , t ) ∈ R 2 : x = x 0 + v t with x 0 ∈ [ x 1 , x 2 ] , v ∈ [ v 1 , v 2 ] } up to an (optimal) residual error of order O ( t − 3 / 4 ) . In each cone C the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions as one moves through the cone. Our results require that the initial data possess one L 2 ( R ) moment and (weak) derivative and that it not generate any spectral singularities.

Authors:Yong Liu; Kelei Wang; Juncheng Wei Abstract: Publication date: Available online 2 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Yong Liu, Kelei Wang, Juncheng Wei From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem [43] is optimal.

Authors:Hongjie Dong; Hong Zhang Abstract: Publication date: Available online 2 October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Hongjie Dong, Hong Zhang We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order σ ∈ ( 0 , 2 ) with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined C σ + α estimate in [9].

Authors:Darvas Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Tamás Darvas Suppose ( X , ω ) is a compact Kähler manifold. We introduce and explore the metric geometry of the L p , q -Calabi Finsler structure on the space of Kähler metrics H . After noticing that the L p , q -Calabi and L p ′ -Mabuchi path length topologies on H do not typically dominate each other, we focus on the finite entropy space E Ent , contained in the intersection of the L p -Calabi and L 1 -Mabuchi completions of H and find that after a natural strengthening, the L p -Calabi and L 1 -Mabuchi topologies coincide on E Ent . As applications to our results, we give new convergence results for the Kähler–Ricci flow and the weak Calabi flow.

Authors:Hui Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Hui Yu We prove a W σ , ϵ -estimate for a class of nonlocal fully nonlinear elliptic equations by following Fanghua Lin's original approach [8] to the analogous problem for second order elliptic equations, by first proving a potential estimate, then combining this estimate with the ABP-type estimate by N. Guillen and R. Schwab to control the size of the superlevel sets of the σ-order derivatives of solutions.

Authors:Horatio Boedihardjo Abstract: Publication date: Available online 22 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Horatio Boedihardjo Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.

Authors:Mahir Hadžić; Andreas Seeger; Charles K. Smart; Brian Street Abstract: Publication date: Available online 22 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Mahir Hadžić, Andreas Seeger, Charles K. Smart, Brian Street We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem.

Authors:Susanna Terracini; Stefano Vita Abstract: Publication date: Available online 12 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Susanna Terracini, Stefano Vita For a competition-diffusion system involving the fractional Laplacian of the form − ( − Δ ) s u = u v 2 , − ( − Δ ) s v = v u 2 , u , v > 0 in R N , with s ∈ ( 0 , 1 ) , we prove that the maximal asymptotic growth rate for its entire solutions is 2s. Moreover, since we are able to construct symmetric solutions to the problem, when N = 2 with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when N ≥ 3 . Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.

Authors:Thomas Alazard Abstract: Publication date: Available online 12 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Thomas Alazard Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the points of contact between the free surface and the vertical walls. The proof relies on the multiplier technique, the Craig-Sulem-Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.

Authors:A. Korepanov; Z. Kosloff; I. Melbourne Abstract: Publication date: Available online 8 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): A. Korepanov, Z. Kosloff, I. Melbourne We prove statistical limit laws for sequences of Birkhoff sums of the type ∑ j = 0 n − 1 v n ∘ T n j where T n is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T n is replaced by a fixed transformation T, and which is particularly effective in the case when T n varies with n. In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards. As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

Authors:Chen Abstract: Publication date: Available online 5 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Xi Chen In the present paper, we investigate global-in-time Strichartz estimates without loss on non-trapping asymptotically hyperbolic manifolds. Due to the hyperbolic nature of such manifolds, the set of admissible pairs for Strichartz estimates is much larger than usual. These results generalize the works on hyperbolic space due to Anker-Pierfelice and Ionescu-Staffilani. However, our approach is to employ the spectral measure estimates, obtained in the author's joint work with Hassell, to establish the dispersive estimates for truncated / microlocalized Schrödinger propagators as well as the corresponding energy estimates. Compared with hyperbolic space, the crucial point here is to cope with the conjugate points on the manifold. Additionally, these Strichartz estimates are applied to the L 2 well-posedness and L 2 scattering for nonlinear Schrödinger equations with power-like nonlinearity and small Cauchy data.

Authors:Pierpaolo Esposito Abstract: Publication date: Available online 5 September 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Pierpaolo Esposito Entire solutions of the n−Laplace Liouville equation in R n with finite mass are completely classified.