相关论文: Inverse homogenization problem for the Drichlet pr…
In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial…
We study the critical problem {equation} {{array}{ll} \Delta ^{2}u=u^{\frac{N+4}{N-4}} & {in}\Omega\setminus \bar{B(\xi_0,\varepsilon)},\medskip u>0&{in}\Omega\setminus \bar{B(\xi_0,\varepsilon)},\medskip u=\Delta u=0 & {on}\partial (\Omega…
We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…
Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness…
We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular…
We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that…
We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$,…
Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-\Delta u = f$ in $\Omega$, $u = 0$ on $\partial\Omega$, $\partial_\nu u = Q$ on $\partial\Omega$. Our main result establishes that if…
We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the…
In a previous article about the homogenization of the classical problem of diff usion in a bounded domain with su ciently smooth boundary we proved that the error is of order $\epsilon^{1/2}$. Now, for an open set with su ciently smooth…
Calculations of the ground state of inhomogeneous many-electron systems involve a solving of the Poisson equation for Coulomb potential and the Schroedinger equation for single-particle orbitals. Due to nonlinearity and complexity this set…
We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of…
In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex…
We study homogenization it its most basic form $$-\left(a\left(\frac{x}{\varepsilon}\right) u_{\varepsilon}'(x)\right)' = f(x) \quad \mbox{for} ~x \in (0,1),$$ where $a(\cdot)$ is a positive $1-$periodic continuous function, $f$ is smooth…
The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = & \lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}% u+\beta\omega_{2}(x)\left\vert u\right\vert…
On a bounded domain $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}_{2,p}$ of functions $u$ whose Kenig-Pipher modified…
We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include…
We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional \[ D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2…
We analyze behavior of weak solutions to compressible fluid flows in a bounded domain in $\mathbb{R}^3$, randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like $\varepsilon^\alpha$, $\alpha > 3$, with…
We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial…