Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Boundary ε-regularity in optimal transportation

We develop an ε  -regularity theory at the boundary for a general class of Monge–Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on C²$C^2$ uniformly convex domains are C^1,α$C^{1,\alpha }$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x⋅y$-x\cdot y$.     

On the two-phase membrane problem with coefficients below the Lipschitz threshold

We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C1,1$C^{1,1}$-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C¹$C^1$-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C1,1$C^{1,1}$-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.     

Existence of piecewise weak solutions of a discrete Cucker–Smale's flocking model with a singular communication weight

We prove existence of global C¹$C^1$ piecewise weak solutions for the discrete Cucker–Smale's flocking model with a non-Lipschitz communication weight ψ(s)=s^−α$\psi (s)=s^{-\alpha }$, 0<α<1$0<\alpha <1$. We also discuss the possibility of finite in time alignment of the velocities of the particles.     

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u   is a solution of (−Δ)^su=g${(-\mathrm{\Delta })}^{s}u=g$ in Ω  , u≡0$u\equiv 0$ in Rⁿ\Ω${\mathbb{R}}^n\backslash \Omega$, for some s∈(0,1)$s\in (0,1)$ and g∈L^∞(Ω)$g\in L^{\infty }(\Omega )$, then u   is C^s(Rⁿ)$C^s({\mathbb{R}}^n)$ and u/δ^s_|Ω$u/{\delta }^s{|}_{\Omega }$ is C^α$C^{\alpha }$ up to the boundary ∂Ω   for some α∈(0,1)$\alpha \in (0,1)$, where δ(x)=dist(x,∂Ω)$\delta (x)=\mathrm{dist}(x,\partial \Omega )$. For this, we develop a fractional analog of the Krylov boundary Harnack method.     

Absence of robust rigidity for circle maps with breaks

We give examples of analytic circle maps with singularities of break type with the same rotation number and the same size of the break for which no conjugacy is Lipschitz continuous. In the second part of the paper, we discuss a class of rotation numbers for which a conjugacy is C¹$C^1$-smooth, although the numbers can be strongly non-Diophantine (Liouville). For the rotation numbers in this class, we construct examples of analytic circle maps with breaks, for which the conjugacy is not C^1+α$C^{1+\alpha }$ smooth, for any α>0$\alpha >0$.     

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C^1,1$C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C^1,1$C^{1,1}$ open set D$D$ and identify the Martin boundary of D$D$ with respect to the subordinate Brownian motion with the Euclidean boundary.     

Property of positive solutions and its applications for Sturm–Liouville singular boundary value problems

This paper is concerned with the property of the positive solutions for Sturm–Liouville singular boundary value problems with the linear conditions. We obtain a relation between the solutions and Green’s function. It implies a necessary condition for the C¹[0,1]$C^1[0,1]$ positive solutions. We apply the result to conclude that the given equation has no C¹[0,1]$C^1[0,1]$ positive solutions.     

Regularity for solutions of nonlocal parabolic equations II

We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form _ut−Iu=f$u_t-Iu=f$, where the operator I   is not necessarily translation invariant. As a consequence of this and the regularity results for the translation invariant case, we obtain C^1,α$C^{1,\alpha }$ interior estimates in space for nontranslation invariant operators under some hypothesis on the time regularity of the boundary data.     

The wave equation for the Bessel Laplacian

We study radial solutions of the Cauchy problem for the wave equation in the multidimensional unit ball B^d$B^d$, d≥1$d\ge 1$. In this case, the operator that appears is the Bessel Laplacian and the solution u(t,x)$u(t,x)$ is given in terms of a Fourier–Bessel expansion. We prove that, for initial L^p$L^p$ data, the series converges in the L²$L^2$ norm. The analysis of a particular operator, the adjoint of the Riesz transform for Fourier–Bessel series, is needed for our purposes, and may be of independent interest. As applications, certain L^p−L²$L^p-L^2$ estimates for the solution of the heat equation and the extension problem for the fractional Bessel Laplacian are obtained.     

Trace asymptotics for fractional Schrödinger operators

This paper proves an analogue of a result of Bañuelos and Sá Barreto [5] on the asymptotic expansion for the trace of Schrödinger operators on R^d${\mathbb{R}}^d$ when the Laplacian −Δ, which is the generator of the Brownian motion, is replaced by the non-local integral operator (−Δ)^α/2${(-\mathrm{\Delta })}^{\alpha /2}$, 0<α<2$0<\alpha <2$, which is the generator of the symmetric stable process of order α. These results also extend recent results of Bañuelos and Yildirim [6] where the first two coefficients for (−Δ)^α/2${(-\mathrm{\Delta })}^{\alpha /2}$ are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed-stable processes are obtained.     

The Moore–Penrose inverse of the normalized graph Laplacian

We prove a formula that relates the Moore–Penrose inverses of two matrices A,B$A,B$ such that A=N⁻¹BM⁻¹$A=N^{-1}B{M}^{-1}$ and discuss some applications, in particular to the representation of the Moore–Penrose inverse of the normalized Laplacian of a graph. The Laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties. However, our formula applies to asymmetric matrices, so that we can generalize our results for asymmetric Laplacians, whose importance for the study of directed graphs is increasing.     

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

Efficient algorithms for the longest common subsequence problem with sequential substring constraints

In this paper, we generalize the inclusion constrained longest common subsequence (CLCS) problem to the hybrid CLCS problem which is the combination of the sequence inclusion CLCS and the string inclusion CLCS, called the sequential substring constrained longest common subsequence   (SSCLCS) problem. In the SSCLCS problem, we are given two strings A$A$ and B$B$ of lengths m$m$ and n$n$, respectively, formed by alphabet Σ$\Sigma$ and a constraint sequence C$C$ formed by ordered strings (C¹,C²,C³,…,C^l)$(C^1,C^2,C^3,\dots ,C^l)$ with total length r$r$. The problem is that of finding the longest common subsequence D$D$ of A$A$ and B$B$ containing C¹,C²,C³,…,C^l$C^1,C^2,C^3,\dots ,C^l$ as substrings and with the order of the C$C$’s retained. This problem has two variants, depending on whether the strings in C$C$ cannot overlap or may overlap. We propose algorithms with O(mnl+(m+n)(|Σ|+r))$O(mnl+(m+n)(\left|\Sigma \right|+r))$ and O(mnr+(m+n)|Σ|)$O(mnr+(m+n)\left|\Sigma \right|)$ time for the two variants. For the special case with one or two constraints, our algorithm runs in O(mn+(m+n)(|Σ|+r))$O(mn+(m+n)(\left|\Sigma \right|+r))$ or O(mnr+(m+n)|Σ|)$O(mnr+(m+n)\left|\Sigma \right|)$ time, respectively—an order faster than the algorithm proposed by Chen and Chao.     

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.     

To the theory of viscosity solutions for uniformly elliptic Isaacs equations

We show how a theorem about the solvability in C^1,1$C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the C^1+χ$C^{1+\chi }$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in C^γ$C^{\gamma }$ with γ slightly less than 1/2.     

Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H^s(T)×H^r(T)$H^s(\mathbb{T})\times H^r(\mathbb{T})$ to C([0,1],H^s(T)×H^r(T))$C([0,1],H^s(\mathbb{T})\times H^r(\mathbb{T}))$ when s?1$s?1$ and r?0$r?0$. We also demonstrate the nonuniform continuous property in the continuous function space C¹(T)×C¹(T)$C^1(\mathbb{T})\times C^1(\mathbb{T})$.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.