We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $Ω$. Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general.
The paper considers direct and inverse elastic scattering from a cavity in homogeneous medium with Dirichlet and Neumann boundary conditions. For direct scattering, existence and uniqueness are derived by variation approach. For inverse scattering, Fr$\acute{\rm e}$chet derivatives of the solution operators are investigated, which give local stability for Dirichlet case.
We show weak-strong uniqueness and stability results for the motion of a two or three dimensional fluid governed by the Navier-Stokes equation interacting with a flexible, elastic plate of Koiter type. The plate is situated at the top of the fluid and as such determines the variable part of a time changing domain (that is hence a part of the solution) containing the fluid. The uniqueness result is a consequence of a stability estimate where the difference of two solutions is estimated by the distance of the initial values and outer forces. For that we introduce a methodology that overcomes the…
In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity. Here it is our main objective to establish uniqueness criteria, which when present, guarantee the uniqueness of the corresponding global minimizer. Then various applications and generalisations are discussed one of which is the construction of a counterexample to regularity.
In this paper, we give a positive answer to a longstanding open problem for determining the shape of an obstacle from the knowledge of the far field pattern for the scattering of time-harmonic elastic wave. We show that the elastic far field pattern by an incoming plane wave with a fixed frequency, a fixed incident direction and a fixed polarization determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely. The boundary condition on $\partial D$ is either the Dirichlet, or the Neumann, or the Robin one.
The $L^2$--gradient flow of the elastic energy of networks leads to a Willmore type evolution law with nontrivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.
We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method, a new algorithm which determines the amplitudes and the wave speed in the theory of isotropic elastic Cosserat materials is described. Moreover, the method allows to prove the existence and uniqueness of a subsonic solution of the secular equation, …
We study the 2+1 dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction proposed by Xu and Xiang (SIAM J. Appl. Math. 69, 1393-1414, 2009). The long-range interaction term and the two length scales in this model makes PDE analysis challenging. Moreover, unlike in the 1+1 dimensional case, there is a nonconvexity contribution in the total energy in the 2+1 dimensional case, and it is not easy to prove that the solution is always in the well-posed regime during the evolution. In this paper, we propose a modified 2+1 dimensional continuum m…
Single-axis high-energy X-ray diffraction tomography for elastic residual strain: uniqueness and stability of solutions in the presence of equilibrium constraints
The University of Newcastle（纽卡斯尔大学）；University of Manchester（曼彻斯特大学）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）
In this paper, we consider a nonlinear Fuchsian type partial differential equation of the second order in the complex domain. Under a very weak assumption, we show the uniqueness of the solution. The result is applied to the problem of analytic continuation of local holomorphic solutions of this equation.
Existence and uniqueness of solutions for a class of models for stress-modulated growth is proven in one spatial dimension. The model features the multiplicative decomposition of the deformation gradient $F$ into an elastic part $F_e$ and a growth-related part $G$. After the transformation due to the growth process, governed by $G$, an elastic deformation described by $F_e$ is applied in order to restore the Dirichlet boundary conditions and therefore the current configuration might be stressed with a stress tensor $S$. The growth of the material at each point in the reference configuration is…
In this article we consider existence and uniqueness of the solutions to a large class of stochastic partial differential of form $\partial_t u = L_x u + b(t,u)+σ(t,u)\dot{W}$, driven by a Gaussian noise $\dot{W}$, white in time and spatial correlations given by a generic covariance $γ$. We provide natural conditions under which classical Picard iteration procedure provides a unique solution. We illustrate the applicability of our general result by providing several interesting particular choices for the operator $L_x$ under which our existence and uniqueness result hold. In particular, we sho…
In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated $O(h^3)$ model. Main issue is the suitable treatment of the curved reference configu…
We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction app…
We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularizing terms.
We prove existence and uniqueness for solutions to equilibrium problems for free-standing, traction-free, non homogeneous crystals in the presence of plastic slips. Moreover we prove that this class of problems is closed under G-convergence of the operators. In particular the homogenization procedure, valid for elliptic systems in linear elasticity, depicts the macroscopic features of a composite material in the presence of plastic deformation.
We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^α u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d \partial_i(a_{ij}(x)\partial_j) + \sum{j=1}^d b_j(x)\partial_j + c(x)$. We establish the uniqueness for an inverse problem of determining an order $α$ of fractional derivatives by data $u(x_0,t)$ for $0<t<T$ at one point $x_0$ in a spatial domain $\OOO$. The uniqueness holds even under assumption that $\OOO$ and $A$ are unknown, provided that the initial value does not change signs and is not identically…
We consider the 3d cubic nonlinear Schrödinger equation (NLS) with a strong 2d harmonic potential. The model is physically relevant to observe the lower-dimensional dynamics of the Bose-Einstein condensate, but its ground state cannot be constructed by the standard method due to its supercritical nature. In Bellazzini-Boussaïd-Jeanjean-Visciglia \cite{BBJV}, the authors constructed a proper ground state introducing a constrained energy minimization problem. In this paper, we further investigate the properties of the ground state. First, we show that as the partial confinement is increased, the…
This paper is concerned with the variational problem for the elastic energy defined on symmetric graphs under the unilateral constraint. Assuming that the obstacle function satisfies the symmetric cone condition, we prove (i) uniqueness of minimizers, (ii) loss of regularity of minimizers, and give (iii) complete classification of existence and non-existence of minimizers in terms of the size of obstacle. As an application, we characterize the solution of the obstacle problem as equilibrium of the corresponding dynamical problem.
Sharp non-uniqueness of weak solutions to the semilinear wave equation
School of Mathematical Sciences & Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学科学学院与当代应用数学上海市重点实验室）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
