Scattering theory for nonlinear Schrödinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x|⁻²$a{|x|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with _Pa=−Δ+a|x|⁻²$P_a=-\mathrm{\Delta }+a{|x|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space H¹(Rⁿ)$H^1({\mathbb{R}}^n)$.     

Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation _∂tu+(−Δ)^σ/2u=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}u=f$ and its elliptic counterpart hv+(−Δ)^σ/2v=f$hv+{(-\mathrm{\Delta })}^{\sigma /2}v=f$, h>0$h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, _∂tu+(−Δ)^σ/2A(u)=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}A(u)=f$, but only when the nondecreasing function A:_R+→_R+$A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)^σ/2v=f$B(v)+{(-\mathrm{\Delta })}^{\sigma /2}v=f$ when B(v)$B(v)$ is a convex nonnegative function for v>0$v>0$ with B(0)=0$B(0)=0$, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2$\sigma =2$.     

Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D)$P(D)$ be a nonnegative homogeneous elliptic operator of order 2m   with real constant coefficients on Rⁿ${\mathbb{R}}^n$ and V   be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e^−tH$e^{-tH}$ generated by H=P(D)+V$H=P(D)+V$ with Kato type perturbing potential V  , which naturally generalizes the classical result for Schrödinger semigroup e^−t(Δ+V)$e^{-t(\mathrm{\Delta }+V)}$ as V∈_K2(Rⁿ)$V\in K_2({\mathbb{R}}^n)$, the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e^−tP(D)$e^{-tP(D)}$ on L¹(Rⁿ)$L^1({\mathbb{R}}^n)$. As a consequence of the Gaussian upper bound, the L^p$L^p$-spectral independence of H is concluded.     

Two-colorings with many monochromatic cliques in both colors

Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k  -cliques is always less than _ckn^k$c_k{n}^k$, where _ck∈(0,1)$c_k\in (0,1)$ is the unique root of the equation z^k=(1−z)^k+kz(1−z)^k−1$z^k={(1-z)}^k+kz{(1-z)}^{k-1}$. On the other hand, we construct a coloring in which there are at least _ckn^k−O(n^k−1)$c_k{n}^k-O(n^{k-1})$ red k-cliques and at least the same number of blue k-cliques.     

Global unique continuation from a half space for the Schrödinger equation

We obtain a global unique continuation result for the differential inequality |(i_∂t+Δ)u|?|V(x)u|$|(i{\partial }_t+\mathrm{\Delta })u|?|V(x)u|$ in Rⁿ⁺¹${\mathbb{R}}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials V(x)$V(x)$ in Rⁿ${\mathbb{R}}^n$. Our method is based on a new type of Carleman estimates for the operator i_∂t+Δ$i{\partial }_t+\mathrm{\Delta }$ on Rⁿ⁺¹${\mathbb{R}}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Generics for computable Mathias forcing

We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n  -generic with n≥2$n\ge 2$ then it satisfies the jump property G⁽ⁿ⁻¹⁾_≡TG^′⊕∅⁽ⁿ⁾$G^{(n-1)}{\equiv }_T{G}^{\prime }\oplus {\varnothing }^{(n)}$. We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.     

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let Ω   be a smooth bounded simply connected domain in R²${\mathbb{R}}^2$. We investigate the existence of critical points of the energy _Eε(u)=1/2_∫Ω|∇u|²+1/(4ε²)_∫Ω(1−|u|²)²$E_{\epsilon }(u)=1/2{\int }_{\Omega }{|\mathrm{\nabla }u|}^2+1/(4{\epsilon }^2){\int }_{\Omega }{(1-{|u|}^2)}^2$, where the complex map u has modulus one and prescribed degree d on the boundary. Under suitable nondegeneracy assumptions on Ω, we prove existence of critical points for small ε. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disk. Next, we prove that critical points exist in “most” of the domains.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Dual Lukacs regressions for non-commutative variables

Dual Lukacs type characterizations of random variables in free probability are studied here. First, we develop a freeness property satisfied by Lukacs type transformations of free-Poisson and free-binomial non-commutative variables which are free. Second, we give a characterization of non-commutative free-Poisson and free-binomial variables by properties of first two conditional moments, which mimic Lukacs type assumptions known from classical probability. More precisely, our result is a non-commutative version of the following result known in classical probability: if U, V   are independent real random variables, such that E(V(1−U)|UV)$\mathbb{E}(V(1-U)|UV)$ and E(V²(1−U)²|UV)$\mathbb{E}(V^2{(1-U)}^2|UV)$ are non-random then V has a gamma distribution and U has a beta distribution.     

Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

We prove Liouville type results for non-negative solutions of the differential inequality _Δφu?f(u)?(|_∇0u|)${\mathrm{\Delta }}_{\phi }u?f(u)?(|{\mathrm{\nabla }}_{0}u|)$ on the Heisenberg group under a generalized Keller–Osserman condition. The operator _Δφu${\mathrm{\Delta }}_{\phi }u$ is the φ  -Laplacian defined by _div0(|_∇0u|⁻¹φ(|_∇0u|)_∇0u)${\mathrm{div}}_0({|{\mathrm{\nabla }}_{0}u|}^{-1}\phi (|{\mathrm{\nabla }}_{0}u|){\mathrm{\nabla }}_{0}u)$ and φ, f and ? satisfy mild structural conditions. In particular, ? is allowed to vanish at the origin. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality.     

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.