No Arabic abstract
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 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.
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.
We obtain the maximal regularity for the mixed Dirichlet-conormal problem in cylindrical domains with time-dependent separations, which is the first of its kind. The boundary of the domain is assumed to be Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,ldots,d-2$, with respect to the Hausdorff distance. We consider solutions in both $L_p$-based Sobolev spaces and $L_{q,p}$-based mixed-norm Sobolev spaces.
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 prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.