Variable Hardy spaces associated with Schrodinger operators on strongly Lipschitz domains with their applications to regularity for inhomogeneous Dirichlet problems

Let n > 3, Q be a strongly Lipschitz domain of R', and p(") : > (0, 1] a variable exponent function satisfying the globally log -Holder continuous condition. Assume that 112, = A + V is a Schrodinger operator on L2(Q) with the Dirichlet boundary condition, where A denotes the Laplace operator and the nonnegative potential V belongs to the reverse Holder class RHqo (DI') for some q0 E (n/2, col In this article, the authors first introduce the variable Hardy space I-11;,()(Q) associated with /12 on Q, via the Lusin area function associated with 42, and the "geometrical" variable Hardy space 1-11;,( r(Q), via the variable Hardy space If()(Rn) associated with the Schrodinger operator LR : = A + V on R.', and then prove that Ir()(Q) = HPL( r(Q) with equivalent quasi -norms. As an appliL ogn cation, the authors show that, when Q is a bounded, simply connected, and semiconvex domain of R.' and the nonnegative potential V belongs to the reverse Holder class RI-Igo (Rn) for some q0 E (max{n/2, 2}, co], the operators VLQ-1 and V2LQ-1 are bounded from I-11;,( 1(Q) to the variable Lebesgue space LP(')(Q), or to itself. As a corollary, the second-order regularity for the inhomogeneous Dirichlet problems of the corresponding Schrodinger equations in the scale of variable Hardy spaces HP( (Q) is obtained.