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Concept# Regular singular point

Summary

In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.
Formal definitions
More precisely, consider an ordinary linear differential equation of n-th order
f^{(n)}(z) + \sum

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We consider the singular set in the thin obstacle problem with weight vertical bar x(n +1)vertical bar(a) for a epsilon (-1, 1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single C-2 manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a C-1,C-alpha manifold for some alpha > 0 if a < 0. In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension-2 lower-dimensional obstacle problem (or fractional thin obstacle problem) when a < 0, whereas second blow-ups at singular points in the top stratum are global, homogeneous, and a-harmonic polynomials when a >= 0. To do so, we establish regularity results for this codimension-2 problem, which we call the very thin obstacle problem. Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many C-2 manifolds, up to a lower-dimensional subset.

We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as t→−∞ to asymptotic dynamics as t→+∞. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.

2021We study the regularity of the regular and of the singular set of the obstacle problem in any dimension. Our approach is related to the epiperimetric inequality of Weiss (Invent Math 138:23–50, Wei99a), which works at regular points and provides an alternative to the methods previously introduced by Caffarelli (Acta Math 139:155–184, Caf77). In his paper, Weiss uses a contradiction argument for the regular set and he asks the question if such epiperimetric inequality can be proved in a direct way (namely, exhibiting explicit competitors), which would have significant implications on the regularity of the free boundary in dimension d > 2. We answer positively the question of Weiss, proving at regular points the epiperimetric inequality in a direct way, and more significantly we introduce a new tool, which we call logarithmic epiperimetric inequality. It allows to study the regularity of the whole singular set and yields an explicit logarithmic modulus of continuity on the C1 regularity, thus improving previous results of Caffarelli and Monneau and providing a fully alternative method. It is the first instance in the literature (even in the context of minimal surfaces) of an epiperimetric inequality of logarithmic type and the first instance in which the epiperimetric inequality for singular points has a direct proof. Our logarithmic epiperimetric inequality at singular points has a quite general nature and will be applied to provide similar results in different contexts, for instance for the thin obstacle problem.

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