Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations

Let Ω ⊂ R N ( N ≥ 3 ) be a C 2 bounded domain and Σ ⊂ Ω be a compact, C 2 submanifold in R N without boundary, of dimension k with 0 ≤ k < N - 2 . Denote d Σ ( x ) : = dist ( x , Σ ) and L μ : = Δ + μ d Σ - 2 in Ω \ Σ , μ ∈ R . The optimal Hardy constant H : = ( N - k - 2 ) / 2 is deeply involved in the study of the Schrödinger operator L μ . The Green kernel and Martin kernel of - L μ play an important role in the study of boundary value problems for nonhomogeneous linear equations involving - L μ . If μ ≤ H 2 and the first eigenvalue of - L μ is positive then the existence of the Green kernel of - L μ is guaranteed by the existence of the associated heat kernel. In this paper, we construct the Martin kernel of - L μ and prove the Representation theory which ensures that any positive solution of the linear equation - L μ u = 0 in Ω \ Σ can be uniquely represented via this kernel. We also establish sharp, two-sided estimates for Green kernel and Martin kernel of - L μ . We combine these results to derive the existence, uniqueness and a priori estimates of the solution to boundary value problems with measures for nonhomogeneous linear equations associated to - L μ .