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:Yavar Kian Pages: 973 - 990 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Yavar Kian We consider the inverse problem of determining a time-dependent potential q, appearing in the wave equation ∂ t 2 u − Δ x u + q ( t , x ) u = 0 in Q = ( 0 , T ) × Ω with T > 0 and Ω a C 2 bounded domain of R n , n ⩾ 2 , from partial observations of the solutions on ∂Q. More precisely, we look for observations on ∂Q that allows to recover uniquely a general time-dependent potential q without involving an important set of data. We prove global unique determination of q ∈ L ∞ ( Q ) from partial observations on ∂Q. Besides being nonlinear, this problem is related to the inverse problem of determining a semilinear term appearing in a nonlinear hyperbolic equation from boundary measurements.

Authors:Johan Helsing; Hyeonbae Kang; Mikyoung Lim Pages: 991 - 1011 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Johan Helsing, Hyeonbae Kang, Mikyoung Lim We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.

Authors:Marcel Braukhoff Pages: 1013 - 1039 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Marcel Braukhoff In biology, the behaviour of a bacterial suspension in an incompressible fluid drop is modelled by the chemotaxis-Navier–Stokes equations. This paper introduces an exchange of oxygen between the drop and its environment and an additionally logistic growth of the bacteria population. A prototype system is given by { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + n − n 2 , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t = Δ u + u ⋅ ∇ u + ∇ P − n ∇ φ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 in conjunction with the initial data ( n , c , u ) ( ⋅ , 0 ) = ( n 0 , c 0 , u 0 ) and the boundary conditions ∂ c ∂ ν = 1 − c , ∂ n ∂ ν = n ∂ c ∂ ν , u = 0 , x ∈ ∂ Ω , t > 0 . Here, the fluid drop is described by Ω ⊂ R N being a bounded convex domain with smooth boundary. Moreover, φ is a given smooth gravitational potential. Requiring sufficiently smooth initial data, the existence of a unique global classical solution for N = 2 is proved, where ‖ n ‖ L p ( Ω ) is bounded in time for all p < ∞ , as well as the existence of a global weak solution for PubDate: 2017-06-16T16:58:58Z DOI: 10.1016/j.anihpc.2016.08.003

Authors:Peter Constantin; Francisco Gancedo; Roman Shvydkoy; Vlad Vicol Pages: 1041 - 1074 Abstract: Publication date: July–August 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 4 Author(s): Peter Constantin, Francisco Gancedo, Roman Shvydkoy, Vlad Vicol We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class W 2 , p , 1 < p ≤ ∞ . We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

Authors:Nicola Garofalo; Arshak Petrosyan; Camelia A. Pop; Mariana Smit Vega Garcia Pages: 533 - 570 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Nicola Garofalo, Arshak Petrosyan, Camelia A. Pop, Mariana Smit Vega Garcia We establish the C 1 + γ -Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.

Authors:El Haj Laamri; Michel Pierre Pages: 571 - 591 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): El Haj Laamri, Michel Pierre We prove here global existence in time of weak solutions for some reaction–diffusion systems with natural structure conditions on the nonlinear reactive terms which provide positivity of the solutions and uniform control of the total mass. The diffusion operators are nonlinear, in particular operators of the porous media type u i ↦ − d i Δ u i m i . Global existence is proved under the assumption that the reactive terms are bounded in L 1 . This extends previous similar results obtained in the semilinear case when the diffusion operators are linear of type u i ↦ − d i Δ u i .

Authors:Paolo Baroni; Casimir Lindfors Pages: 593 - 624 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Paolo Baroni, Casimir Lindfors We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy–Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary p-Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself.

Authors:Nicola Soave; Alessandro Zilio Pages: 625 - 654 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Nicola Soave, Alessandro Zilio We consider a family of positive solutions to the system of k components − Δ u i , β = f ( x , u i , β ) − β u i , β ∑ j ≠ i a i j u j , β 2 in Ω , where Ω ⊂ R N with N ≥ 2 . It is known that uniform bounds in L ∞ of { u β } imply convergence of the densities to a segregated configuration, as the competition parameter β diverges to +∞. In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of u β in terms of entire solutions to the limit system Δ U i = U i ∑ j ≠ i a i j U j 2 . Moreover, we develop a uniform-in-β regularity theory for the interfaces.

Authors:Damião J. Araújo; Gleydson C. Ricarte; Eduardo V. Teixeira Pages: 655 - 678 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Damião J. Araújo, Gleydson C. Ricarte, Eduardo V. Teixeira This work is devoted to the study of nonvariational, singularly perturbed elliptic equations of degenerate type. The governing operator is anisotropic and ellipticity degenerates along the set of critical points. The singular behavior is of order O ( 1 ϵ ) along ϵ-level layers { u ϵ ∼ ϵ } , and a non-homogeneous source acts in the noncoincidence region { u ϵ > ϵ } . We obtain the precise geometric behavior of solutions near ϵ-level surfaces, by means of optimal regularity and sharp geometric nondegeneracy. We further investigate Hausdorff measure properties of ϵ-level surfaces. The analysis of the asymptotic limits as the ϵ parameter goes to zero is also carried out. The results obtained are new even if restricted to the uniformly elliptic, isotropic setting.

Authors:Alessandro Fonda; Antonio J. Ureña Pages: 679 - 698 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Alessandro Fonda, Antonio J. Ureña We propose an extension to higher dimensions of the Poincaré–Birkhoff Theorem which applies to Poincaré time-maps of Hamiltonian systems. Examples of applications to pendulum-type systems and weakly-coupled superlinear systems are also given.

Authors:Stephan Fackler Pages: 699 - 709 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Stephan Fackler An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous symmetric forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We give a negative answer to this question and discuss the minimal regularity needed for positive results.

Authors:Philip Isett; Sung-Jin Oh Pages: 711 - 730 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Philip Isett, Sung-Jin Oh In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L t ∞ B 3 , ∞ 1 / 3 due to low regularity of the energy profile. The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class C t , x 1 / 5 − ϵ . The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [1].

Authors:V. Sciacca; M.E. Schonbek; M. Sammartino Pages: 731 - 757 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): V. Sciacca, M.E. Schonbek, M. Sammartino We consider the two-dimensional shallow water model derived in [29], describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, working in the whole space R 2 , under suitable conditions on the time dependent forcing term, we prove the L 2 asymptotic decay of the weak solutions.

Authors:Benjamin Dodson; Changxing Miao; Jason Murphy; Jiqiang Zheng Pages: 759 - 787 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Benjamin Dodson, Changxing Miao, Jason Murphy, Jiqiang Zheng We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

Authors:Nicola Zamponi; Ansgar Jüngel Pages: 789 - 792 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Nicola Zamponi, Ansgar Jüngel This note corrects Lemma 7 in [1] on the positive (semi-)definiteness of a certain matrix product, which yields a priori estimates for the cross-diffusion system.

Authors:Agathe Decaster; Dragoş Iftimie Pages: 277 - 291 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Agathe Decaster, Dragoş Iftimie In this paper, we address the problem of determining the asymptotic behaviour of the solutions of the incompressible stationary Navier–Stokes system in the full space, with a forcing term whose asymptotic behaviour at infinity is homogeneous of degree −3. We identify the asymptotic behaviour at infinity of the solution. We prove that it is homogeneous and that the leading term in the expansion at infinity uniquely solves the homogeneous Navier–Stokes equations with a forcing term which involves an additional Dirac mass. This also applies to the case of an exterior domain.

Authors:Henrik Shahgholian; Karen Yeressian Pages: 293 - 334 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Henrik Shahgholian, Karen Yeressian In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary B ∩ { x 1 = 0 } in the obstacle type problem { div ( x 1 a ∇ u ) = χ { u > 0 } in B + , u = 0 on B ∩ { x 1 = 0 } where a < 1 , B + = B ∩ { x 1 > 0 } , B is the unit ball in R n and n ≥ 2 is an integer. Let Γ = B + ∩ ∂ { u > 0 } be the free boundary and assume that the origin is a contact point, i.e. 0 ∈ Γ ‾ . We prove that the free boundary touches the fixed boundary uniformly tangentially at the origin, near to the origin it is the graph of a C 1 function and there is a uniform modulus of continuity for the derivatives of this function.

Authors:Alberto Bressan; Geng Chen Pages: 335 - 354 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Alberto Bressan, Geng Chen The paper is concerned with conservative solutions to the nonlinear wave equation u t t − c ( u ) ( c ( u ) u x ) x = 0 . For an open dense set of C 3 initial data, we prove that the solution is piecewise smooth in the t–x plane, while the gradient u x can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.

Authors:Verena Bögelein; Frank Duzaar; Paolo Marcellini; Stefano Signoriello Pages: 355 - 379 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Verena Bögelein, Frank Duzaar, Paolo Marcellini, Stefano Signoriello In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations { ∂ t u − div D f ( D u ) = 0 in Ω T , u = u o on ∂ P Ω T . The only assumptions needed are the convexity of the generating function f : R n → R , and the classical bounded slope condition on the initial and the lateral boundary datum u o ∈ W 1 , ∞ ( Ω ) . We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – u o could be any Lipschitz initial and boundary datum, vanishing at the boundary ∂Ω, and the boundary may contain flat parts, for instance Ω could be a rectangle in R n .

Authors:Animikh Biswas; Ciprian Foias; Adam Larios Pages: 381 - 405 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Animikh Biswas, Ciprian Foias, Adam Larios In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier–Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.

Authors:Godofredo Iommi; Thomas Jordan; Mike Todd Pages: 407 - 421 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Godofredo Iommi, Thomas Jordan, Mike Todd We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

Authors:Bernard Helffer; Ayman Kachmar Pages: 423 - 438 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Bernard Helffer, Ayman Kachmar We explore the relationship between two reference functions arising in the analysis of the Ginzburg–Landau functional. The first function describes the distribution of superconductivity in a type II superconductor subjected to a constant magnetic field. The second function describes the distribution of superconductivity in a type II superconductor submitted to a variable magnetic field that vanishes non-degenerately along a smooth curve.

Authors:Nicola Abatangelo; Louis Dupaigne Pages: 439 - 467 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Nicola Abatangelo, Louis Dupaigne We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak- L 1 theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method. We finally show the existence of large solutions for some power-like nonlinearities.

Authors:François Hamel; Xavier Ros-Oton; Yannick Sire; Enrico Valdinoci Pages: 469 - 482 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): François Hamel, Xavier Ros-Oton, Yannick Sire, Enrico Valdinoci We consider entire solutions to L u = f ( u ) in R 2 , where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L , we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

Abstract: Publication date: Available online 21 June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Yves Achdou, Alessio Porretta We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton–Jacobi equation and a forward Kolmogorov equation both posed in ( 0 , T ) × ( R N / Z N ) . Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T.

Abstract: Publication date: Available online 20 June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Andrew Lorent, Guanying Peng Let Ω ⊂ IR 2 be a bounded simply-connected domain. The Eikonal equation ∇ u = 1 for a function u : Ω ⊂ IR 2 → IR has very little regularity, examples with singularities of the gradient existing on a set of positive H 1 measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ∇u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles-Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles-Giga functional. Formally if u satisfies the Eikonal equation and if (1) ∇ ⋅ ( Σ ˜ e 1 e 2 ( ∇ u ⊥ ) ) = 0 and ∇ ⋅ ( Σ ˜ ϵ 1 ϵ 2 ( ∇ u ⊥ ) ) = 0 distributionally in Ω , where Σ ˜ e 1 e 2 and Σ ˜ ϵ 1 ϵ 2 are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ∇u is locally Lipschitz continuous outside a locally finite set. Condition (1) is motivated by the zero energy states of the Aviles-Giga functional. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if lim n → ∞ I ϵ n ( u n ) = 0 for some sequence u n ∈ W 0 2 , 2 ( Ω ) and u = lim n → ∞ u n PubDate: 2017-06-21T18:04:38Z

Authors:Isabeau Birindelli; Giulio Galise; Hitoshi Ishii Abstract: Publication date: Available online 26 May 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Isabeau Birindelli, Giulio Galise, Hitoshi Ishii In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.

Authors:Thomas Chen; Younghun Hong; Nataša Pavlović Abstract: Publication date: Available online 24 May 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Thomas Chen, Younghun Hong, Nataša Pavlović In this paper, we study the dynamics of a system of infinitely many fermions in dimensions d ≥ 3 near thermal equilibrium and prove scattering in the case of small perturbation around equilibrium in a certain generalized Sobolev space of density operators. This work is a continuation of our previous paper [18], and extends the important recent result of M. Lewin and J. Sabin in [35] of a similar type for dimension d = 2 . In the work at hand, we establish new, improved Strichartz estimates that allow us to control the case d ≥ 3 .

Authors:Ciprian G. Gal Abstract: Publication date: Available online 19 May 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Ciprian G. Gal We consider a doubly nonlocal nonlinear parabolic equation which describes phase-segregation of a two-component material in a bounded domain. This model is a more general version than the recent nonlocal Cahn-Hilliard equation proposed by Giacomin and Lebowitz [26], such that it reduces to the latter under certain conditions. We establish well-posedness results along with regularity and long-time results in the case when the interaction between the two levels of nonlocality is strong-to-weak.

Authors:B. Dacorogna; W. Gangbo; O. Kneuss Abstract: Publication date: Available online 8 May 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): B. Dacorogna, W. Gangbo, O. Kneuss This manuscript identifies a maximal system of equations which renders the classical Darboux problem elliptic, thereby providing a selection criterion for its well posedness. Let f be a symplectic form close enough to ω m , the standard symplectic form on R 2 m . We prove existence of a diffeomorphism φ, with optimal regularity, satisfying φ ⁎ ( ω m ) = f and 〈 d φ ♭ ; ω m 〉 = 0 . We establish uniqueness of φ when the system is coupled with a Dirichlet datum. As a byproduct, we obtain, what we term symplectic factorization of vector fields, that any map u, satisfying appropriate assumptions, can be factored as: u = χ ∘ ψ with ψ ⁎ ( ω m ) = ω m , 〈 d χ ♭ ; ω m 〉 = 0 and ∇ χ + ( ∇ χ ) t > 0 ; moreover there exists a closed 2-form Φ such that χ = ( δ Φ ⌟ ω m ) ♯ . Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an L 2 -projection problem.

Authors:Yury Grabovsky; Davit Harutyunyan Abstract: Publication date: Available online 5 May 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Yury Grabovsky, Davit Harutyunyan We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.

Authors:Satoshi Masaki; Jun-ichi Segata Abstract: Publication date: Available online 20 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Satoshi Masaki, Jun-ichi Segata In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical L ˆ r space where L ˆ r = { f ∈ S ′ ( R ) ‖ f ‖ L ˆ r = ‖ f ˆ ‖ L r ′ < ∞ } . We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to L ˆ r -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

Authors:Jiaqi Liu; Peter A. Perry; Catherine Sulem Abstract: Publication date: Available online 20 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Jiaqi Liu, Peter A. Perry, Catherine Sulem The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.

Authors:Yujin Guo; Xiaoyu Zeng Abstract: Publication date: Available online 19 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Manuel del Pino; Konstantinos T. Gkikas Abstract: Publication date: Available online 4 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Manuel del Pino, Konstantinos T. Gkikas We consider the parabolic Allen–Cahn equation in R n , n ≥ 2 , u t = Δ u + ( 1 − u 2 ) u in R n × ( − ∞ , 0 ] . We construct an ancient radially symmetric solution u ( x , t ) with any given number k of transition layers between −1 and +1. At main order they consist of k time-traveling copies of w with spherical interfaces distant O ( log t ) one to each other as t → − ∞ . These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: x = − 2 ( n − 1 ) t . More precisely, if w ( s ) denotes the heteroclinic 1-dimensional solution of w ″ + ( 1 − w 2 ) w = 0 w ( ± ∞ ) = ± 1 given by w ( s ) = tanh ( s 2 ) we have u ( x , t ) ≈ ∑ j = 1 k ( − 1 ) j − 1 w ( x − ρ j ( t ) ) − 1 2 ( 1 + ( − 1 ) k ) as t → − ∞ where ρ j ( t ) = − 2 ( n − 1 ) t + 1 2 ( j − k + 1 2 ) log ( t log t ) + O ( PubDate: 2017-04-11T15:39:24Z DOI: 10.1016/j.anihpc.2017.03.005

Authors:Jérôme Droniou; Kyle S. Talbot Abstract: Publication date: Available online 30 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Jérôme Droniou, Kyle S. Talbot This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium, with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with L 2 wells (Amirat and Ziani, 2004 [1]). We give a novel treatment of the diffusion–dispersion term, which requires delicate use of the Aubin–Simon lemma to ensure the strong convergence of the pressure gradient, owing to the troublesome lower-order terms introduced by the localisation procedure.

Authors:Thomas Kappeler; Jan-Cornelius Molnar Abstract: Publication date: Available online 28 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Thomas Kappeler, Jan-Cornelius Molnar In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on H s the KdV2 equation is C 0 -wellposed if s ⩾ 0 and illposed (in a strong sense) if s < 0 .

Authors:Myoungjean Bae; Shangkun Weng Abstract: Publication date: Available online 28 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Myoungjean Bae, Shangkun Weng We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler-Poisson system in a cylinder supplemented with non small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl(=angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler-Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis r = 0 .

Authors:Alessio Porretta; Philippe Souplet Abstract: Publication date: Available online 21 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Manuel Friedrich; Francesco Solombrino Abstract: Publication date: Available online 21 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Manuel Friedrich, Francesco Solombrino In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation ( G S B D ). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the G S B D setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.

Authors:Seonghak Kim; Baisheng Yan Abstract: Publication date: Available online 18 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Seonghak Kim, Baisheng Yan We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona-Malik model in image processing and the other that of Höllig's model related to the Clausius-Duhem inequality in the second law of thermodynamics.