This paper records for the Hamiltonian H = 1/2 |p|² + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.

The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume: If |A| = |B| > 0 and |A + B|^1/n = (2+δ)|A|^1/n for some small δ, then, up to a translation, both A and B are close (in terms of δ) to a convex set K. Although this result was already proved by the authors in a previous paper, the present paper provides a more elementary proof that the authors believe has its own interest. Also, the result here provides a stronger estimate for the stability exponent than the previous result of the authors.

This paper deals with representations of groups by “affine” automorphisms of compact, convex spaces, with special focus on “irreducible” representations: equivalently “minimal” actions. When the group in question is PSL(2, R), the authors exhibit a one-one correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called “linear Stone-Weierstrass” for group actions on compact spaces. If it holds for the “universal strongly proximal space” of the group (to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Itô or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next, in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.

The author proves C ¹ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients.

In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known conjecture due to Polya, its connections to Weyl’s asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies.

Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.

The author surveys some recent progress on the Toda system on a twodimensional surface Σ, arising in models from self-dual non-abelian Chern-Simons vortices, as well as in differential geometry. In particular, its variational structure is analysed, and the role of the topological join of the barycentric sets of Σ is shown.

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from ℂ ⁿ to ℂ ⁿ are obtained using Brouwer degree.

The author considers mass critical nonlinear Schrödinger and Korteweg-de Vries equations. A review on results related to the blow-up of solution of these equations is given.

The authors show that for any ε ∈]0, 1[, there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ⊂ ℝ⁵ which belongs to C ^1,ε(B) \C ^1,ε+(B).

Negative index materials are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1946 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. Mathematically, the study of negative index materials faces two difficulties. Firstly, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. 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 survey, the author discusses recent mathematics progress in understanding properties of negative index materials and their applications. The topics are reflecting complementary media, superlensing and cloaking by using complementary media, cloaking a source via anomalous localized resonance, the limiting absorption principle and the well-posedness of the Helmholtz equation with sign changing coefficients.

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t−(3/2)log t+x _∞, the solution of the equation converges as t → +∞ to a translate of the traveling wave corresponding to the minimal speed c _* = 2. The constant x _∞ depends on the initial condition u(0, x). The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments.

This paper presents the fundamental optical concepts of designing multifocal ophthalmic lenses and the mathematical methods associated with them. In particular, it is shown that the design methodology is heavily based on differential geometric ideas such as Willmore surfaces. A key role is played by Hamilton’s eikonal functions. It is shown that these functions capture all the information on the local blur and distortion created by the lenses. Along the way, formulas for computing the eikonal functions are derived. Finally, the author lists a few intriguing mathematical problems and novel concepts in optics as future projects.

This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of ℝ ^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.

The author applies the arguments in his PKU Master degree thesis in 1988 to derive a third derivative estimate, and consequently, a C ^2,α-estimate, for complex Monge-Ampere equations in the conic case. This C ^2,α-estimate was used by Jeffres-Mazzeo-Rubinstein in their proof of the existence of Kähler-Einstein metrics with conic singularities.

CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three. While the sign of this operator is important in the embedding problem, the kernel of this operator is also closely connected with the stability of CR structures. The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed. The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem. The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension, and it is closely connected with the pluriharmonic functions. The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms. Finally, an isoperimetric constant determined by the Q-prime curvature integral is discussed.