New Weighted Integral Inequalities for the 3D Linear Elastostatic System

Integral inequalities that concern the weighted positivity of a differential operator havemany important applications in the qualitative theory of elliptic boundary valueproblems. In this study, we focus on the 3D linear elastostatic (Lame) system for anisotropic, homogeneous elastic medium and propose to extend our previous resultsobtained in (Luo and Maz'ya 2010, Luo and Maz'ya 2015) to a larger range of elasticparameters, by considering a new class of weight functions first proposed andsystematically studied in (Mayboroda and Maz'ya 2009, Mayboroda and Maz'ya 2014).The weighted inequalities, if successfully established, will lead to many new interestingresults, including Wiener type criteria for the regularity of a boundary point, andpointwise estimates for Green's functions and solutions of the 3D Lame system inarbitrary domains. If a uniform, positive lower bound for the constant occurring in theweighted inequality can be obtained, the results may even be extended to the 3D linearhydrostatic (Stokes) system, which would settle many important problems in the theoryof the Navier-Stokes equations in general domains.?