No Arabic abstract
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with $L^2$ boundary data in a periodically perforated Lipschitz domain.
In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.
We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $mathbb{C}$ that depend smoothly on a real parameter $lambda in [0,1]$. We derive the precise regularity of the solutions in all variables, including the parameter $lambda$. More specifically we show that the solution and its derivatives are continuous in all variables, and the Holder norms of the space variables are bounded uniformly in $lambda$.
We study homogenization of a boundary obstacle problem on $ C^{1,alpha} $ domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $gamma$. For any $ epsiloninmathbb{R}_+$, $partial D=Gamma cup Sigma$, $Gamma cap Sigma=emptyset $ and $ S_{epsilon}subset Sigma $ with suitable assumptions, we prove that as $epsilon$ tends to zero, the energy minimizer $ u^{epsilon} $ of $ int_{D} |gamma abla u|^{2} dx $, subject to $ ugeq varphi $ on $ S_{varepsilon} $, up to a subsequence, converges weakly in $ H^{1}(D) $ to $ widetilde{u} $ which minimizes the energy functional $int_{D}|gamma abla u|^{2}+int_{Sigma} (u-varphi)^{2}_{-}mu(x) dS_{x}$, where $mu(x)$ depends on the structure of $S_{epsilon}$ and $ varphi $ is any given function in $C^{infty}(overline{D})$.
We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the $W^1_p$ estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdorff distance, and the semi-convexity condition. For the conormal problem with inhomogeneous boundary conditions, we also assume that the domain is Lipschitz. By using these results, we obtain the $W^1_p$ and weighted $W^1_p$ estimates for the Robin problem in these domains.
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.