We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.
Let $A$ and $B$ be invariant linear operators with respect to a decomposition $\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem $$ u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms of the (global) matrix-valued symbols $σ_A$ and $σ_B$ of $A$ and $B,$ respectively, associated to the decomposition $\{H_{j}\}_{j\in \mathbb{N}}$. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllabi…
In this work, the Riemann-Hilbert problem for the 3-component Manakov system is formulated on the basis of the corresponding $4\times 4$ matrix spectral problem. Furthermore, by applying the nonlinear steepest descent techniques to an associated $4\times 4$ matrix valued Riemann-Hilbert problem, we can find leading-order asymptotics for the Cauchy problem of the 3-component Manakov system.
In a previous article of Dos Santos Ferreira, Kenig, Salo and Uhlmann, anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n \geq 3. In this article we extend this result to the case of unbounded potentials, namely those in Ln/2. In the process, we derive Lp Carleman esti…
We present an inverse scattering transform approach to the Cauchy problem on the line for the Degasperis--Procesi equation $u_t-u_{txx}+3ωu_x+4uu_x=3u_xu_{xx}+uu_{xxx}$ in the form of an associated Riemann-Hilbert problem. This approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used in studying its long-time behavior.
This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &m_t+\left((u^2-u_x^2)m\right)_x=0,&\quad&m:= u-u_{xx},&\quad&t>0,&\;&-\infty<x<+\infty,\\ &u(x,0)=u_0(x),&&&&&&-\infty<x<+\infty, \end{alignat*} in the case when the initial data $u_0(x)$ as well as the solution $u(x,t)$ are assumed to approach a nonzero constant as $x\to\pm\infty$. In a recent paper we developed the Riemann--Hilbert formalism for this problem, which allowed us to represent the solution of the Cauchy problem in terms of the solution of an associated Riemann--Hilbert facto…
Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W^n_2$) under weaker assumptions than those used by X. Zhou [Journal of Functional Analysis 103, 275--293 (1992)]. As an application, a comparison theorem is obtained.
We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = α\partial_x \big( |u|^2 u \big) + β\partial_x \big[ H \big( |u|^2 \big) u \big] , \quad (t, x) \in [0,T] \times \mathbf{T}, \] where the constants $α,β$ are such that $α\in \mathbf{R}$ and $β<0$, and $H$ denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transform technique, which is useful for de…
In this paper, regularity properties, Strichartz type estimates to solutions of multipoınt Cauchy problem for linear and nonlinear abstract wave equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Hilbert space H, in which by choosing H and A we can obtain numerous classis of nonlocal initial value problems for wave equations which occur in a wide variety of physical systems.
We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution. Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give $L^p$-$L^q$ estimates of the solution.
In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+\mathcal H Δu +uu_x &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb T^2,\; t\in\mathbb R,\\ u(x,y,0)&\hspace{-2mm}=u_0(x,y), \end{array} \right\}\,,$$ where $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$ and $Δ$ is the Laplacian with respect to the spatial variables $x$ and $y$, is locally well-posed in the periodic Sobolev space $H^s(\mathbb T^2)$, with $s>7/4$.
We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a H…
We obtain the the existence of global solutions to the Cauchy problem of the Fokas-Lenells (FL) equation on the line \begin{align} &u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0,\nonumber \\ &u(x,t=0)=u_0(x), \nonumber \end{align} where without the small-norm assumption on initial data $u_0(x)\in H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$. Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and the uniqueness of the RH problem is shown via a general vanishing lemma.…
In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+D_x^α u_x +\mathcal Hu_{yy} +uu_x &=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R, u(x,y,0)&=u_0(x,y), \end{array} \right\}\,,$$ where $0<α\leq1$, $D_x^α$ denotes the operator defined through the Fourier transform by \begin{align} (D_x^αf)\widehat{\;}(ξ,η):=|ξ|^α\widehat{f}(ξ,η)\,, \end{align} and $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$, is locally well posed in the Sobolev space $H^s(\mathbb R^2)$…
We consider non-autonomous wave equations \[ \left\{ \begin{aligned} \&\ddot u(t) + \B(t)\dot u(t) + \A(t)u(t) = f(t) \quad t\text{-a.e.}\\ \&u(0)=u_0,\, \dot u(0) = u_1. \end{aligned} \right. \] where the operators $\A(t)$ and $\B(t)$ are associated with time-dependent sesquilinear forms $\fra(t,.,.)$ and $\frb$ defined on a Hilbert space $H$ with the same domain $V$. The initial values satisfy $ u_0 \in V$ and $u_1 \in H$. We prove well-posedness and maximal regularity for the solution both in the spaces $V'$ and $H$. We apply the results to non-autonomous Robin-boundary conditions and also …
With this paper we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincaré group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient Paley-Wiener- and Sobolev-type estimates. These generalized Fourier transforms immediately all…
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet pro…
We develop a scattering theory for the linear wave equation $\Box_g ψ= 0$ on the interior of Reissner-Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution $ψ$ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resu…
This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
