On scale invariant bounds for Green's function for second order elliptic equations with lower order coefficients and applications

Abstract

We construct Green's functions for elliptic operators of the form Lu = − div(A∇u + bu) + c∇u+du in domains Ω ⊆ R n , under the assumption d ≥ div b, or d ≥ div c. We show that, in the setting of Lorentz spaces, the assumption b−c ∈ L n,1 (Ω) is both necessary and optimal to obtain pointwise bounds for Green's functions. We also show weak type bounds for Green's functions and their gradients. Our estimates are scale invariant and hold for general domains Ω ⊆ R n . Moreover, there is no smallness assumption on the norms of the lower order coefficients. As applications we obtain scale invariant global and local boundedness estimates for subsolutions to Lu ≤ − div f + g in the case d ≥ div c.

Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains

“…Proof. The proof of the first part can be found in [Cos2,Lemma 4.2(i)] and of the second one in [Sak,Lemma 2.2…”

“…It is also well-posed if (??) holds by the embedding theorem for Lorentz-Sobolev spaces (see [Sak,p.6 (Ω) functions. When we write Lu = f − divg, we mean that it holds "in the weak sense", i.e.,…”

“…Proof. Here we follow the scheme of the proof of [Sak,Lemma 2.2]. Since both u and w belong to Y 1,2 0 (Ω), we can use (2.29) and (2.27) to deduce that (2.31)…”

Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data

“…In [26], Sakellaris, considered boundary value problems for (1.2) on bounded domains with Dirichlet and regularity boundary data, and the additional condition that A is Hölder continuous. Sakellaris, then extended this to arbitrary domains in [27], where the estimates on the Green functions are in Lorentz spaces and are scale invariant. Also, in [25], Mourgoglou proves well posedness for the Dirichlet problem in unbounded domains, with coefficients in a local Stummel-Kato class.…”