Mathematics

We look for solutions to derivative nonlinear Schrodinger equations built upon solitons. We prove the existence of multi-soliton trainsi.e. solutions behaving at large time as the sum of finite solitons. We also show that one can attach a kink at the begin of the train i.e multi kink-soliton trains. Our proofs proceedby fixed point arguments around the desired profile, using Strichartz estimates.

The exact distributed controllability of the semilinear wave equation ??tt y?�Δy+g(y)=f 1 ? posed over multi-dimensional and bounded domains, assuming that g??C 1 (R) satisfies the growth condition lim?�sup r?��? g(r)/(|r| ln 1/2 |r|)=0 has been obtained by Fu, Yong and Zhang in 2007. The proof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that g ??does not grow faster than β ln 1/2 |r| at infinity for β>0 small enough and that g ??is uniformly Hölder continuous on R with exponent s??0,1] , we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order 1+s after a finite number of iterations.

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov-Poincar'e type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah--Struwe solutions, which satisfy the Strichartz estimates and are coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to time independent function in an appropriate way.

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s -fractional perimeter, defined for 0<s<1 , to the case s?? \,. We show that, as the core-radius vanishes, such core-radius regularized s -fractional perimeters, suitably scaled, ? -converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized s -fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics. Keywords: Fractional perimeters; ? -convergence; Local and nonlocal geometric evolutions; Viscosity solutions; Level set formulation; Fractional mean curvature flow; Dislocation dynamics

In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the L 2 -norm and L ??-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).

We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the α -Hausdorff measure ( α -a.e.). We extend to the fractal setting ( α<n ) a recent counterexample of Bourgain \cite{Bourgain2016}, which is sharp in the Lebesque measure setting ( α=n ). In doing so we recover the necessary condition from \cite{zbMATH07036806} for pointwise convergence~ α -a.e. and we extend it to the range n/2<α??3n+1)/4 .

We present convergence analysis of operator learning in [Chen and Chen 1995] and [Lu et al. 2020], where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection-diffusion equations with or without reaction. We find that the convergence rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.

The present work revisits the classical Wulff problem restricted to crystalline integrands, a class of surface energies that gives rise to finitely faceted crystals. The general proof of the Wulff theorem was given by J.E. Taylor (1978) by methods of Geometric Measure Theory. This work follows a simpler and direct way through Minkowski Theory by taking advantage of the convex properties of the considered Wulff shapes.

We construct counterexamples to inverse problems for the wave operator on domains in R n+1 , n?? , and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On R n+1 the metrics are conformal to the Minkowski metric.