Showing papers in "Annales De L Institut Henri Poincare-analyse Non Lineaire in 2014"

Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates

Abstract: This is the first of two articles dealing with the equation ( − Δ ) s v = f ( v ) in R n , with s ∈ ( 0 , 1 ) , where ( − Δ ) s stands for the fractional Laplacian — the infinitesimal generator of a Levy process. This equation can be realized as a local linear degenerate elliptic equation in R + n + 1 together with a nonlinear Neumann boundary condition on ∂ R + n + 1 = R n . In this first article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R . These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform as s ↑ 1 , establishing in the limit the corresponding known results for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

Well-posedness for Hall-magnetohydrodynamics

Abstract: We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.

Constrained energy minimization and orbital stability for the NLS equation on a star graph

Abstract: On a star graph G , we consider a nonlinear Schrodinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as i ∂ t Ψ ( t ) = − Δ Ψ ( t ) − | Ψ ( t ) | 2 μ Ψ ( t ) + α δ 0 Ψ ( t ) , where the strength α of the vertex interaction is negative and the wave function Ψ is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for 0 μ ⩽ 2 the energy at fixed mass is bounded from below and that for every mass m below a critical mass m ⁎ it attains its minimum value at a certain Ψ ˆ m ∈ H 1 ( G ) . Moreover, the set of minimizers has the structure M = { e i θ Ψ ˆ m , θ ∈ R } . Correspondingly, for every m m ⁎ there exists a unique ω = ω ( m ) such that the standing wave Ψ ˆ ω e i ω t is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to α = 0 .

Hardy inequalities on Riemannian manifolds and applications

Abstract: We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator Δ p u : = div ( | ∇ u | p − 2 ∇ u ) . Namely, if ρ is a nonnegative weight such that − Δ p ρ ⩾ 0 , then the Hardy inequality c ∫ M | u | p ρ p | ∇ ρ | p d v g ⩽ ∫ M | ∇ u | p d v g , u ∈ C 0 ∞ ( M ) , holds. We show concrete examples specializing the function ρ. Our approach allows to obtain a characterization of p-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo–Nirenberg inequalities, uncertain principle and first order Caffarelli–Kohn–Nirenberg interpolation inequality.

Bang-bang property for time optimal control of semilinear heat equation

Abstract: This paper studies the bang-bang property for time optimal controls governed by semilinear heat equation in a bounded domain with control acting locally in a subset. Also, we present the null controllability cost for semilinear heat equation and an observability estimate from a positive measurable set in time for the linear heat equation with potential.

Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms

Abstract: We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the W 1 , p norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.

Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

Abstract: In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabolic equations. Sub-linear and super-linear cases are considered.

Simultaneous local exact controllability of 1D bilinear Schrödinger equations

Abstract: We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrodinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay.

Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces

Abstract: We study equivalence between the Poincare inequality and several different relative isoperimetric inequalities on metric measure spaces. We then use these inequalities to establish sufficient conditions for the finite perimeter of sets.

On linear instability of solitary waves for the nonlinear Dirac equation

Abstract: We consider the nonlinear Dirac equation, also known as the Soler model: i ∂ t ψ = − i α ⋅ ∇ ψ + m β ψ − ( ψ ⁎ β ψ ) k β ψ , m > 0 , ψ ( x , t ) ∈ C N , x ∈ R n , k ∈ N . We study the point spectrum of linearizations at small amplitude solitary waves in the limit ω → m , proving that if k > 2 / n , then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with ω sufficiently close to m, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh–Schrodinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov–Kolokolov stability criterion.

Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements

Abstract: Motivated by models of fracture mechanics, this paper is devoted to the analysis of a unilateral gradient flow of the Ambrosio–Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. Solutions of such evolution are constructed by means of an implicit Euler scheme. An asymptotic analysis in the Mumford–Shah regime is then carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set. In the spirit of gradient flows in metric spaces, a notion of curve of maximal unilateral slope is also investigated, and analogies with the unilateral slope of the Mumford–Shah functional are also discussed.

Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping

Abstract: We consider the KdV–Burgers equation u t + u x x x − u x x + λ u + u u x = 0 and its linearized version u t + u x x x − u x x + λ u = 0 on the whole real line. We investigate their well-posedness their exponential stability when λ is an indefinite damping.

Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors

Abstract: This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.

Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations

Abstract: We study weak solutions of the 3D Navier-Stokes equations in whole space with L 2 initial data. It will be proved that r u is locally integrable in space-time for any real such that 1 < < 3, which says that almost third derivative is locally integrable. Up to now, only second derivativer 2 u has been known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-L 4=( +1) loc . These estimates depend only on the L 2 norm of initial data and integrating domains. Moreover, they are valid even for 3 as long as u is smooth. The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study. For the local study, we use De Giorgi method with a new pressure decomposition. To handle non-locality of the fractional Laplacian, we will adopt some properties of the Hardy space and Maximal functions.

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

Abstract: We study the new geometric flow that was introduced in the paper [12] of Topping and the author that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper of Topping and the author [12] where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed.

Counterexample to regularity in average-distance problem

Abstract: The average-distance problem is to find the best way to approximate (or represent) a given measure μ on R d by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure μ, minimize E ( Σ ) = ∫ R d d ( x , Σ ) d μ ( x ) + λ H 1 ( Σ ) among connected closed sets, Σ, where λ > 0 , d ( x , Σ ) is the distance from x to the set Σ, and H 1 is the one-dimensional Hausdorff measure. Here we provide, for any d ⩾ 2 , an example of a measure μ with smooth density, and convex, compact support, such that the global minimizer of the functional is a rectifiable curve which is not C 1 . We also provide a similar example for the constrained form of the average-distance problem.

Feedback stabilization of a simplified 1d fluid-particle system

Abstract: We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ σ 0 An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different

Standing waves for linearly coupled Schrödinger equations with critical exponent

Abstract: We study the following linearly coupled Schrodinger equations: { − e 2 Δ u + a ( x ) u = u p + λ v , x ∈ R N , − e 2 Δ v + b ( x ) v = v 2 ⁎ − 1 + λ u , x ∈ R N , u , v > 0 in R N , u ( x ) , v ( x ) → 0 as | x | → ∞ , where N ⩾ 3 , 2 ⁎ = 2 N N − 2 , 1 p 2 ⁎ − 1 , and a ( x ) , b ( x ) are positive continuous potentials which are both bounded away from 0. Under some assumptions on a ( x ) and λ > 0 , we obtain positive solutions of the coupled system for sufficiently small e > 0 , which have concentration phenomenon as e → 0 . It is interesting that we do not need any further assumptions on b ( x ) .

Multiple brake orbits on compact convex symmetric reversible hypersurfaces in R2n

Abstract: In this paper, we prove that there exist at least [ n + 1 2 ] + 1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2 n for n ⩾ 2 satisfying the reversible condition N Σ = Σ with N = diag ( − I n , I n ) . As a consequence, we show that there exist at least [ n + 1 2 ] + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in R n with n ⩾ 2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n = 3 . As an application, for n = 4 and 5 , we prove that if there are exactly n geometrically distinct closed characteristics on Σ, then all of them are symmetric brake orbits after suitable time translation.

Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations

Abstract: We consider the (KdV)/(KP-I) asymptotic regime for the nonlinear Schrodinger equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg–de Vries equation (in dimension 1) and to the Kadomtsev–Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/(KP-I) equation in the spirit of the work of Lannes and Saut, and then prove a comparison result with quantitative error estimates. For either suitable nonlinearities for (NLS) either a Landau–Lifshitz type equation, we derive a (mKdV)/(mKP-I) equation involving cubic nonlinearity. We then give a partial result justifying this asymptotic limit.

From homogenization to averaging in cellular flows

Abstract: We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A , in a two-dimensional domain with L 2 cells. For fixed A , and L → ∞ , the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞ . In this case the solution equilibrates along stream lines. In this paper, we show that if both A → ∞ and L → ∞ , then a transition between the homogenization and averaging regimes occurs at A ≈ L 4 . When A ≫ L 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A ≪ L 4 , the principal eigenvalue behaves like σ ¯ ( A ) / L 2 , where σ ¯ ( A ) ≈ A I is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent L p → L ∞ estimates for elliptic equations with an incompressible drift. This provides effective sub- and super-solutions for our problem.

Partially hyperbolic geodesic flows

Abstract: We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.

Conditions at infinity for the inhomogeneous filtration equation

Abstract: We investigate existence and uniqueness of solutions to the filtration equation with an inhomogeneous density in R N ( N ⩾ 3 ), approaching at infinity a given continuous datum of Dirichlet type.

Gradient integrability and rigidity results for two-phase conductivities in two dimensions

Abstract: This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u)=0 in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries. We focus also on two-phase conductivities, i.e., conductivities assuming only two matrix values, σ1 and σ2, and study the higher integrability of the corresponding gradient field |∇u| for this special but very significant class. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement described by the sets Ei=σ−1(σi). We find the optimal integrability exponent of the gradient field corresponding to any pair {σ1,σ2} of elliptic matrices, i.e., the worst among all possible microgeometries. We also treat the unconstrained case when an arbitrary but finite number of phases are present.

Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold

Abstract: We prove the existence of new extremal domains for the first eigenvalue of the Laplace–Beltrami operator in some compact Riemannian manifolds of dimension n ⩾ 2 . The volume of such domains is close to the volume of the manifold. If the first eigenfunction ϕ 0 of the Laplace–Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of ϕ 0 . If ϕ 0 is a constant function and n ⩾ 4 , these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of the scalar curvature.

Constant Q-curvature metrics near the hyperbolic metric

Abstract: Let ( M , g ) be a Poincare–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.

Graphs of maps between manifolds in trace spaces and with vanishing mean oscillation

Abstract: We give a positive answer to a question raised by Alberti in connection with a recent result by Brezis and Nguyen. We show the existence of currents associated with graphs of maps in trace spaces that have vanishing mean oscillation. The degree of such maps may be written in terms of these currents, of which we give some structure properties. We also deal with relevant examples.

Duality methods for a class of quasilinear systems

Abstract: Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge–Backlund transformation, underlying symmetries among superficially different forms of the equations.

Distributional Jacobian equal to H1 measure

Abstract: Let 1 ⩽ p 2 . We construct a Holder continuous W 1 , p mapping of a square into R 2 such that the distributional Jacobian equals to one-dimensional Hausdorff measure on a line segment.