相关论文: Donati representation theorem for periodic functio…
We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem which generalizes the well known Stefan model, and includes the classical porous medium equation. It may be represented…
The second gradient model of poromechanics, introduced in Part I, is here linearized in the neighborhood of a prestressed reference configuration to be applied to the one-dimensional consolidation problem originally considered by Terzaghi…
Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…
In this paper, we present a general derivation of multicontinuum equations and discuss cell problems. We present constraint cell problem formulations in a representative volume element and oversampling techniques that allow reducing…
We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of…
Field-theoretic construction of functional representations of solutions of stochastic differential equations and master equations is reviewed. A generic expression for the generating function of Green functions of stochastic systems is put…
Periodic configurations of electrodes, in particular of microelectrodes, have been of interest since the advent of microfabrication. In this report, theory which is common to any periodic cell (or any cell that can be extended periodically)…
We present a general formula for the particular solution of an inhomogeneous linear difference equation with variable coefficients. The answer is expressed as a weighted sum of fundamental solutions of the associated linear difference…
We provide a general treatment of perturbations of a class of functionals modeled on convolution energies with integrable kernel which approximate the $p$-th norm of the gradient as the kernel is scaled by letting a small parameter…
The periodic homogenization problem of integro-differential equations of the alpha stable L{\'e}vy operators is studied in this paper. Thanking to the symmetry of the L{\'e}vy density, we can use the method of the formal asymptotic…
Aspects of the theory of characteristic modes, based on their variational formulation, are presented and an explicit form of a related functional, involving only currents in a spatial domain, is derived. The new formulation leads to deeper…
We present a rather unknown version of the change of variables formula for non-autonomous functions. We will show that this formula is equivalent to Green's Theorem for regions of the plane bounded by the graphs of two continuously…
The inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered. For considered equation the general representation of regular solution in rectangular domain is obtained, and the…
This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We first establish a representation formula for the partial derivatives of the solutions of the homogeneous Dirichlet problem. Along the…
This work gives a mathematical study of tissue dynamics. We combine within-cell genome dynamics and diffusion between cells, where the synthesis of the two gives rise to the emergence of function. We introduce a concept of monotonicity and…
This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green's functions $(G_\varepsilon,\Pi_\varepsilon)$ (and fundamental solutions $(\Gamma_\varepsilon, Q_\varepsilon)$) for the Stokes system with…
In this note we treat the equations of fractional elasticity. After establishing well-posedness, we show a compactness result related to the theory of homogenization. For this, a previous result in (abstract) homogenization theory of…
Based on an assumption on the Hessian of the Green function, we derive some monotonicity formulas on nonparabolic manifolds. This assumption is satisfied on manifolds that meet certain conditions including bounds on the sectional curvature…
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
Partial differential equations are a convenient way to describe reaction- advection-diffusion processes of signalling models. If only one cell type is present, and tissue dynamics can be neglected, the equations can be solved directly.…