Showing papers in "Annales De L Institut Henri Poincare-analyse Non Lineaire in 2016"

Local behavior of fractional p-minimizers

Abstract: We extend the De Giorgi–Nash–Moser theory to nonlocal, possibly degenerate integro-differential operators.

Spectral analysis of semigroups and growth-fragmentation equations

Abstract: The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson-Phillips series in the spirit of those used in [87,82,48,81]. (2) On the other hand, we present the semigroup spectral analysis for three important classes of ''growth-fragmentation" equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick-Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated remarkable profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in \cite{MR2114128,MR2536450} for the cell division model with (almost) constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in \cite{MR2162224,MR2114413} which have been established under similar assumptions to those made in the present work.

Sard property for the endpoint map on some Carnot groups

Abstract: In Carnot–Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

Instability of point defects in a two-dimensional nematic liquid crystal model

Abstract: We study a class of symmetric critical points in a variational 2D Landau–de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3 × 3 matrices. These critical points play the role of topological point defects carrying a degree k 2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when | k | ≥ 2 .

Cloaking using complementary media in the quasistatic regime

Abstract: Cloaking using complementary media was suggested by Lai et al. in [8] . The study of this problem faces two difficulties. Firstly, this problem is unstable since the equations describing the phenomenon have sign changing coefficients, hence the ellipticity is lost. Secondly, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this paper, we give a proof of cloaking using complementary media for a class of schemes inspired from [8] in the quasistatic regime. To handle the localized resonance, we introduce the technique of removing localized singularity and apply a three spheres inequality. The proof also uses the reflecting technique in [11] . To our knowledge, this work presents the first proof on cloaking using complementary media.

Lagrangian flows for vector fields with anisotropic regularity

Abstract: We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov-Poisson equation with measure density. The proof exploits an anisotropic variant of the argument in [Crippa-De Lellis, Bouchut-Crippa] and suitable estimates for the difference quotients in such anisotropic context. In contrast to regularization methods, this approach gives quantitative estimates in terms of the given regularity bounds. From such estimates it is possible to recover the well posedness for the ordinary differential equation and for Lagrangian solutions to the continuity and transport equations.

Strong maximum principle for Schrödinger operators with singular potential

Abstract: We prove that for every p>1 and for every potential V∈Lp, any nonnegative function satisfying −Δu+Vu≥0 in an open connected set of RN is either identically zero or its level set {u=0} has zero W2,p capacity. This gives an affirmative answer to an open problem of Benilan and Brezis concerning a bridge between Serrin–Stampacchia's strong maximum principle for p>N2 and Ancona's strong maximum principle for p=1. The proof is based on the construction of suitable test functions depending on the level set {u=0}, and on the existence of solutions of the Dirichlet problem for the Schrodinger operator with diffuse measure data.

Long time behavior of solutions of a reaction–diffusion equation on unbounded intervals with Robin boundary conditions ☆

Abstract: We study the long time behavior, as $t\to\infty$, of solutions of $$ \left\{ \begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \ t >0,\\ u(0,t) = b u_x(0,t), & t>0,\\ u(x,0) = u_0 (x)\geqslant 0 , & x\geqslant 0, \end{array} \right. $$ where $b\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\{ \sigma \phi \}_{\sigma >0}$, where $\phi$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $\sigma^*$ such that the solution converges uniformly to 0 for any $0 \sigma^*$. In the threshold case $\sigma= \sigma^*$, the profile of the solution approaches the symmetrically decreasing ground state with some shift, which may be either finite or infinite. In the latter case, the shift evolves as $C \ln t$ where~$C$ is a positive constant we compute explicitly, so that the solution is traveling with a pulse-like shape albeit with an asymptotically zero speed. Depending on $b$, but also in some cases on the choice of the initial datum, we prove that one or both of the situations may happen.

Annealed estimates on the Green functions and uncertainty quantification

Abstract: We prove Lipschitz bounds for linear elliptic equations in divergence form whose measurable coefficients are random stationary and satisfy a logarithmic Sobolev inequality, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. This improves the celebrated De Giorgi–Nash–Moser theory in the large (that is, away from the singularity) for this class of coefficients. This regularity result is obtained as a corollary of optimal decay estimates on the derivative and mixed second derivative of the elliptic Green functions on Rd. As another application of these decay estimates we derive optimal estimates on the fluctuations of solutions of linear elliptic PDEs with “noisy” diffusion coefficients.

Qualitative analysis of rupture solutions for a MEMS problem

Abstract: We prove sharp Holder continuity and an estimate of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent

Stringent error estimates for one-dimensional, space-dependent 2 × 2 relaxation systems

Abstract: Sharp and local L 1 a posteriori error estimates are established for so-called “well-balanced” BV (hence possibly discontinuous) numerical approximations of 2 × 2 space-dependent Jin–Xin relaxation systems under sub-characteristic condition. According to the strength of the relaxation process, one can distinguish between two complementary regimes: 1) a weak relaxation, where local L 1 errors are shown to be of first order in Δx and uniform in time, 2) a strong relaxation, where numerical solutions are kept close to entropy solutions of the reduced scalar conservation law, and for which Kuznetsov's theory indicates a behavior of the L 1 error in t ⋅ Δ x . The uniformly first-order accuracy in weak relaxation regime is obtained by carefully studying interaction patterns and building up a seemingly original variant of Bressan–Liu–Yang's functional, able to handle BV solutions of arbitrary size for these particular inhomogeneous systems. The complementary estimate in strong relaxation regime is proven by means of a suitable extension of methods based on entropy dissipation for space-dependent problems. Preliminary numerical illustrations are provided.

High magnetic field equilibria for the Fokker–Planck–Landau equation

Abstract: The subject matter of this paper concerns the equilibria of the Fokker–Planck–Landau equation under the action of strong magnetic fields. Averaging with respect to the fast cyclotronic motion when the Larmor radius is supposed to be finite leads to an integro-differential version of the Fokker–Planck–Landau collision kernel, combining perpendicular space coordinates (with respect to the magnetic lines) and velocity. We determine the equilibria of this gyroaveraged Fokker–Planck–Landau kernel and derive the macroscopic equations describing the evolution around these equilibria, in the parallel direction.

A Dirichlet problem involving the divergence operator

Abstract: We consider the problem { div u + 〈 a ; u 〉 = f in Ω u = u 0 on ∂ Ω . We show that if curl a ( x 0 ) ≠ 0 for some x 0 ∈ Ω , then the problem is solvable without restriction on f . We also discuss the regularity of the solution.

Optimal magnetic Sobolev constants in the semiclassical limit

Abstract: This paper is devoted to the semiclassical analysis of the best constants in the magnetic Sobolev embeddings in the case of a bounded domain of the plane carrying Dirichlet conditions. We provide quantitative estimates of these constants (with an explicit dependence on the semiclassical parameter) and analyze the exponential localization in L ∞ -norm of the corresponding minimizers near the magnetic wells.

Quasi-static damage evolution and homogenization: A case study of non-commutability

Abstract: In this paper we consider a family of quasi-static evolution problems involving oscillating energies E e and dissipations D e . Even though we have separate Γ-convergence of E e and D e , the Γ-limit F of the sum does not agree with the sum of the Γ-limits. Nevertheless, F can still be viewed as the sum of an internal energy and a dissipation, and the corresponding quasi-static evolution is the limit of the quasi-static evolutions related to E e and D e . This result contributes to the analysis of the interaction between Γ-convergence and variational evolution, which has recently attracted much interest both in the framework of energetic solutions and in the theory of gradient flows.