L^p And W^1,p Estimates For Robin Problems Of Second Order Elliptic Equations In Lipschitz Domains

In this paper we study LP and W1,p estimates for Robin problems of second order elliptic equations in bounded Lipschitz domains.By discussing two cases respectively:p≥2 and 1<p<2,we prove that the Robin problem for second order elliptic equation is uniquely solvable for 1<p<2+ε and obtain LP estimate of non-tangential maximal function for the gradient of the solution.For W1,p estimates of Robin problems of second order elliptic equations,we prove that if p>2,then reverse Holder inequalities imply the W1,p estimates.Then we establish reverse Holder inequalities for p=3+ε if n≥3(p=4+ε if n=2)under the assumptions coefficient matrix satisfies symmetric condition,uniformly elliptic condition and belongs to VMO.According to the real variable method and duality,then we prove that the W1,p estimates hold for 3/2-ε<p<3+ε if n≥3(4/3-ε<p<4+ε if n=2).