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Register a new userOscillatory integrals and periodic homogenization of Robin boundary value problems
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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.
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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 many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: begin{equation*} x(t)+lambda x(t)=h(t)+varepsilon f(x(t)),hspace{.1in}tin(0,pi) end{equation*} subject to non-local boundary conditions begin{equation*} x(0)=h_1+varepsiloneta_1(x)text{ and } x(pi)=h_2+varepsiloneta_2(x). end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.
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.
We study the large time behaviour of the solution of linear dispersive partial differential equations posed on a finite interval, when at least one of the prescribed boundary conditions is time periodic. We use the Q equation approach, pioneered in Fokas & Lenells 2012 and applied to linear problems on the half-line in Fokas & van der Weele 2021, to characterise necessary conditions for the solution of such problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new results for the third order case.
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 $C^{infty}(overline{D})$.
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