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:Martin Dindoš; Stefanie Petermichl; Jill Pipher Pages: 1155 - 1180 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Martin Dindoš, Stefanie Petermichl, Jill Pipher We prove that a sharp regularity property ( A ∞ ) of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. This equivalence was established in the elliptic case by Kenig, Kirchheim, Pipher and Toro, improving an earlier result of Kenig, Dindos and Pipher for solutions with data in BMO. The connection between regularity of the elliptic measure and certain Carleson measure properties of solutions was established in order to study solvability of boundary value problems for non-symmetric divergence form operators (Kenig, Koch, Pipher, and Toro). The extension to the parabolic setting requires an approach to the key estimates of the aforementioned works that primarily exploits the maximum principle. For various classes of parabolic operators ([24]), this criterion also provides an easier route to establish the solvability of the Dirichlet problem with data in L p for some p, and also to quantify these results in several aspects.

Authors:Roland Donninger; Birgit Schörkhuber Pages: 1181 - 1213 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Roland Donninger, Birgit Schörkhuber We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions d ≥ 5 . We prove that for every p > d + 3 d − 1 there exists an open set of radial initial data in H d + 1 2 × H d − 1 2 such that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions.

Authors:Pierre Berger; Alejandro Kocsard Pages: 1227 - 1253 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Pierre Berger, Alejandro Kocsard We prove that every endomorphism which satisfies Axiom A and the strong transversality conditions is C 1 -inverse limit structurally stable. These conditions were conjectured to be necessary and sufficient. This result is applied to the study of unfolding of some homoclinic tangencies. This also achieves a characterization of C 1 -inverse limit structurally stable covering maps.

Authors:Jaywan Chung; Zihua Guo; Soonsik Kwon; Tadahiro Oh Pages: 1273 - 1297 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Jaywan Chung, Zihua Guo, Soonsik Kwon, Tadahiro Oh We consider the quadratic derivative nonlinear Schrödinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the Cole–Hopf transformation, we prove unconditional global well-posedness in L 2 ( T ) , and more generally in certain Fourier–Lebesgue spaces F L s , p ( T ) , under the mean-zero and smallness assumptions. As a byproduct, we construct an infinite sequence of quantities that are invariant under the dynamics. We also show the necessity of the smallness assumption by explicitly constructing a finite time blowup solution with non-small mean-zero initial data.

Authors:J.A. Carrillo; A. Figalli; F.S. Patacchini Pages: 1299 - 1308 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): J.A. Carrillo, A. Figalli, F.S. Patacchini We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class C 2 and mildly repulsive at the origin; moreover, if the minimizer is global, then its support is finite. In addition, for some class of potentials we prove the validity of a uniform upper bound on the cardinal of the support of a global minimizer. Finally, in the one-dimensional case, we give quantitative bounds.

Authors:Matthieu Alfaro Pages: 1309 - 1327 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Matthieu Alfaro We consider the nonlocal diffusion equation ∂ t u = J ⁎ u − u + u 1 + p in the whole of R N . We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel J near the origin, which is linked to the tails of J. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation ∂ t u = Δ u + u 1 + p . On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of J. As an application of the result in population dynamics models, we discuss the hair trigger effect for ∂ t u = J ⁎ u − u + u 1 + p ( 1 − u ) .

Authors:Chang-Shou Lin; Shusen Yan Pages: 1329 - 1354 Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Chang-Shou Lin, Shusen Yan This is the first part of our comprehensive study on the structure of doubly periodic solutions for the Chern–Simons–Higgs equation with a small coupling constant. We first classify the bubbling type of the blow-up point according to the limit equations. Assuming that all the blow-up points are away from the vortex points, we prove the non-coexistence of different bubbling types in a sequence of bubbling solutions. Secondly, for the CS type bubbling solutions, we obtain an existence result without the condition on the blow-up set as in [4]. This seems to be the first general existence result of the multi-bubbling CS type solutions which is obtained under nearly necessary conditions. Necessary and sufficient conditions are also discussed for the existence of bubbling solutions blowing up at vortex points.

Authors:Pengfei Zhang Pages: 793 - 816 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Pengfei Zhang In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that C ∞ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is Herman's result on Diophantine invariant curves that we use to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.

Authors:Lorenzo Brasco; Berardo Ruffini Pages: 817 - 843 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Lorenzo Brasco, Berardo Ruffini For a general open set, we characterize the compactness of the embedding for the homogeneous Sobolev space D 0 1 , p ↪ L q in terms of the summability of its torsion function. In particular, for 1 ≤ q < p we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a torsional Hardy inequality that we investigate in detail.

Authors:Herbert Koch; Angkana Rüland; Wenhui Shi Pages: 845 - 897 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Herbert Koch, Angkana Rüland, Wenhui Shi This article deals with the variable coefficient thin obstacle problem in n + 1 dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and W 1 , p metrics with p ∈ ( n + 1 , ∞ ] . In this framework, we prove the C 1 , α regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal C 1 , min { 1 − n + 1 p , 1 2 } regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in [18] and the present article, we then also discuss the case of non-zero and interior thin obstacles. We obtain the optimal regularity of the solutions and the regularity of the regular free boundary for W 1 , p metrics and W 2 , p obstacles with p ∈ ( 2 ( n + 1 ) , ∞ ] .

Authors:L. Caffarelli; D. De Silva; O. Savin Pages: 899 - 932 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): L. Caffarelli, D. De Silva, O. Savin We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the Hölder continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain C 1 , γ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.

Authors:Heiner Olbermann Pages: 933 - 959 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Heiner Olbermann We prove that the Brouwer degree deg ( u , U , ⋅ ) for a function u ∈ C 0 , α ( U ; R n ) is in L p ( R n ) if 1 ≤ p < n α d , where U ⊂ R n is open and bounded and d is the box dimension of ∂U. This is supplemented by a theorem showing that u j → u in C 0 , α ( U ; R n ) implies deg ( u j , U , ⋅ ) → deg ( u , U , ⋅ ) in L p ( R n ) for the parameter regime 1 ≤ p < n α d , while there exist convergent sequences u j → u in C 0 , α ( U ; R n ) such that ‖ deg ( u j , U , ⋅ ) ‖ L p → ∞ for the opposite regime p > n α d .

Authors:Suleyman Ulusoy Pages: 961 - 971 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Suleyman Ulusoy We analyze an equation that is gradient flow of a functional related to Hardy–Littlewood–Sobolev inequality in whole Euclidean space R d , d ≥ 3 . Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of “free-energy solutions”, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a critical value. Actually, there is a critical value of a parameter in the equation below which there is a global-in-time energy solution and above which there exist blowing-up energy solutions.

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:Juan Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Juan Casado-Díaz Given two isotropic homogeneous materials represented by two constants 0 < α < β in a smooth bounded open set Ω ⊂ R N , and a positive number κ < Ω , we consider here the problem consisting in finding a mixture of these materials α χ ω + β ( 1 − χ ω ) , ω ⊂ R N measurable, with ω ≤ κ , such that the first eigenvalue of the operator u ∈ H 0 1 ( Ω ) → − div ( ( α χ ω + β ( 1 − χ ω ) ) ∇ u ) reaches the minimum value. In a recent paper, [6], we have proved that this problem has not solution in general. On the other hand, it was proved in [1] that it has solution if Ω is a ball. Here, we show the following reciprocate result: If Ω ⊂ R N is smooth, simply connected and has connected boundary, then the problem has a solution if and only if Ω is a ball.

Authors:Nan Abstract: Publication date: September–October 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 5 Author(s): Nan Lu We consider a class of nonlinear Klein–Gordon equations u t t = u x x − u + f ( u ) and obtain a family of small amplitude periodic solutions, where the temporal and spatial period have different scales. The proof is based on a combination of Lyapunov–Schmidt reduction, averaging and Nash–Moser iteration.

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.

Authors:Huyuan Chen; Patricio Felmer; Jianfu Yang Abstract: Publication date: Available online 16 August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Huyuan Chen, Patricio Felmer, Jianfu Yang In this paper, we study the elliptic problem with Dirac mass (1) { − Δ u = V u p + k δ 0 in R N , lim x → + ∞ u ( x ) = 0 , where N > 2 , p > 0 , k > 0 , δ 0 is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in R N ∖ { 0 } , with non-empty support and satisfying 0 ≤ V ( x ) ≤ σ 1 x a 0 ( 1 + x a ∞ − a 0 ) , with a 0 < N , a 0 < a ∞ and σ 1 > 0 . We obtain two positive solutions of (1) with additional conditions for parameters on a ∞ , a 0 , p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.

Authors:Andy Hammerlindl Abstract: Publication date: Available online 11 August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Andy Hammerlindl We show that certain derived-from-Anosov diffeomorphisms on the 2-torus may be realized as the dynamics on a center-stable or center-unstable torus of a 3-dimensional strongly partially hyperbolic system. We also construct examples of center-stable and center-unstable tori in higher dimensions.

Authors:Mramor Abstract: Publication date: Available online 9 August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Blaž Mramor We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence.

Authors:Stephen Cameron; Luis Silvestre; Stanley Snelson Abstract: Publication date: Available online 27 July 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Stephen Cameron, Luis Silvestre, Stanley Snelson We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.

Authors:Klemens Fellner; Evangelos Latos; Bao Quoc Tang Abstract: Publication date: Available online 25 July 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Klemens Fellner, Evangelos Latos, Bao Quoc Tang We consider a model system consisting of two reaction–diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division. Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction–diffusion system.

Authors:Qing Han; Guofang Wang Abstract: Publication date: Available online 25 July 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Qing Han, Guofang Wang In this paper we prove that any smooth surfaces can be locally isometrically embedded into C 2 as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

Authors:Francis J. Chung; Petri Ola; Mikko Salo; Leo Tzou Abstract: Publication date: Available online 14 July 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Francis J. Chung, Petri Ola, Mikko Salo, Leo Tzou In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim–Uhlmann and Kenig–Sjöstrand–Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.

Authors:Matías G. Delgadino; Scott Smith Abstract: Publication date: Available online 5 July 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Matías G. Delgadino, Scott Smith This work focuses on drift-diffusion equations with fractional dissipation ( − Δ ) α in the regime α ∈ ( 1 / 2 , 1 ) . Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some β ∈ ( 0 , 1 ) , the C β norm of the solution depends only on the size of the drift in critical spaces of the form L t q ( BMO x − γ ) with q > 2 and γ ∈ ( 0 , 2 α − 1 ] , along with the L x 2 norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.