Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

In [3], Floer and Weinstein consider the case N = 1 and p = 3. For a given nondegenerate critical point of the potential V, assumed globally bounded, and for 0 < E < inf V, they construct a standing wave provided that h is sufficiently small. This solution concentrates around the critical point as h --+ 0. Their method, based on an interesting Lyapunov-Schmidt reduction, was extended by Oh in [6], [7] to conclude a similar result in higher dimensions, proN+2 vided that 1 < p < g'=l" He restricts himself to potentials with "mild oscillation" at infinity, namely belonging to a Kato class. In case that V is bounded this restriction is not necessary as observed by Wang in [10].

Local mountain passes for semilinear elliptic problems in unbounded domains

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Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆

Semi-classical States for Nonlinear Schrödinger Equations

We construct solutions exhibiting a single spike-layer shape around some point of the boundary as $\var \to 0$ for the problem \left\{ \begin{array}{l} \var^2 \tri u - u + u^p = 0 \quad \mbox{in} \ \Omega, \\ u > 0 \quad \mbox{in} \ \Omega, \\ \frac{\partial u}{\partial \nu} = 0 \quad \mbox{on} \ \partial \Omega, \end{array} \right.\label {1.1} where $ \Omega $ is a bounded domain with smooth boundary in $R^N$, $p > 1 $, and $p< {N+2\over N-2}$ if $N\ge 3$. Our main result states that given a topologically nontrivial critical point of the mean curvature function of $\partial \Omega$, for instance, a possibly degenerate local maximum, local minimum, or saddle point, there is a solution with a single local maximum, which is located at the boundary and approaches this point as $\var\to 0$ while vanishing asymptotically elsewhere.

On the Role of Mean Curvature in Some Singularly Perturbed Neumann Problems

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem where (-Δ)α denotes the fractional Laplacian, α ∈ (0, 1), and B1 denotes the open unit ball centered at the origin in ℝN with N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1 → ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing. Our third goal is to consider radial symmetry of positive solutions for system of the form where α1, α2 ∈(0, 1), the functions f1 and f2 are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1 and g2 are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

Radial symmetry of positive solutions to equations involving the fractional laplacian

http://capde.cmm.uchile.cl/files/2015/07/felmer2011.pdf

Fundamental solutions and liouville type theorems for nonlinear integral operators

We examine the asymptotic behavior of the eigenvalue μ(h) and corresponding eigenfunction associated with the variational problem

$$$$

in the regime h>>1. Here A is any vector field with curl equal to 1. The problem arises within the Ginzburg–Landau model for superconductivity with the function μ(h) yielding the relationship between the critical temperature vs. applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross-section $\Omega\subset\R^{2}$. We first carry out a rigorous analysis of the associated problem on a half-plane and then rigorously justify some of the formal arguments of [BS], obtaining an expansion for μ while also proving that the first eigenfunction decays to zero somewhere along the sample boundary  when Ω is not a disc. For interior decay, we demonstrate that the rate is exponential.

Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity

https://www.ceremade.dauphine.fr/~dolbeaul/Preprints/Fichiers/DFLP-f.pdf

Patricio Felmer

Papers

On the Role of Mean Curvature in Some Singularly Perturbed Neumann Problems

Radial symmetry of positive solutions to equations involving the fractional laplacian

Fundamental solutions and liouville type theorems for nonlinear integral operators

Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity

Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems