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

We consider elliptic operators in divergence form with lower order terms of the form Lu=-div(A·∇u+bu)-c·∇u-du\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Lu = -{{\textrm{div}}}(A \cdot \nabla u + b u ) - c \cdot \nabla u - du$$\end{document}, in an open set Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^n$$\end{document}, n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document}, with possibly infinite Lebesgue measure. We assume that the n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b,c∈Llocn,∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b, c \in L^{n,\infty }_{\text {loc}}({\Omega })$$\end{document} and d∈Llocn2,∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \in L_{\text {loc}}^{\frac{n}{2}, \infty }(\Omega )$$\end{document}, or |b|2,|c|2,|d|∈Kloc(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|b|^2, |c|^2, |d| \in \mathcal {K}_{\text {loc}}(\Omega )$$\end{document}, where Kloc(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_{\text {loc}}(\Omega )$$\end{document} stands for the local Stummel–Kato class. Let KDini(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}_{\text {Dini}}}(\Omega )$$\end{document} be a variant of K(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}(\Omega )$$\end{document} satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu=f-divg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Lu = f - {{\textrm{div}}}g$$\end{document}, where f and |g|2∈KDini(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|g|^2 \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$\end{document} if, for q∈[n,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in [n, \infty )$$\end{document}, any of the following assumptions holds: (i) |b|2,|d|∈KDini(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|b|^2, |d| \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$\end{document} and either c∈Llocn,q(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \in L^{n,q}_{\text {loc}}(\Omega )$$\end{document} or |c|2∈Kloc(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|c|^2 \in \mathcal {K}_{\text {loc}}(\Omega )$$\end{document}; (ii) divb+d≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textrm{div}}}b +d \le 0$$\end{document} and either b+c∈Llocn,q(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b+c \in L^{n,q}_{\text {loc}}(\Omega )$$\end{document} or |b+c|2∈Kloc(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|b+c|^2 \in \mathcal {K}_{\text {loc}}(\Omega )$$\end{document}; (iii) -divc+d≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{{\textrm{div}}}c + d \le 0$$\end{document} and |b+c|2∈KDini(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|b+c|^2 \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$\end{document}. We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming -divc+d≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{{\textrm{div}}}c +d \le 0$$\end{document}, we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis |b+c|2∈K′(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|b+c|^2 \in \mathcal {K}'(\Omega )$$\end{document}, we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

The objective of the current manuscript is to generalize the standard theory of elliptic PDE of the form −divA∇u = 0 in open sets Ω ⊂ R n , n ≥ 3, with possibly infinite Lebesgue measure, to equations of the form (1.1) under the aforementioned standing assumptions. In particular, we aim to show scale invariant a priori local estimates (Caccioppoli inequality, local boundedness and weak Harnack inequality), interior and boundary regularity for solutions of (1.1), the weak maximum principle, well-posedness of the Dirichlet and obstacle problems, and finally, we construct the Green's function for our operator satisfying several quantitative estimates. It is important to highlight that neither the bilinear form associated with the elliptic equation is coercive, nor the norms of the coefficients are small, which is one of the main technical difficulties.
We would like to point out that we will only state the theorems in the main body of the paper, just before their proofs. Nevertheless, the reader can find a detailed description of our results in the introduction.
Let us give a brief overview of our results. Under our standing assumptions for the coefficients, in section 3.1 we prove the standard interior and boundary Caccioppoli's inequality under either negativity condition (Theorems 3.1, 3.2, and 3.3), while in section 5, we show the well-posedness of the generalized Dirichlet problem ( 5.3) satisfying the estimate (5.14), as well as the weak maximum principle (Theorem 5.1). The maximum principle allows us to solve the obstacle problem in bounded domains (Theorem 5.6), which gives us for instance that the minimum of two subsolutions is a subsolution. In section 3.2, we show that if (1. 6) is satisfied, the refined Caccioppoli inequality holds in the interior and the boundary (Theorems 3.4 and 3.6), which leads to the local boundedness of subsolutions (Theorem 4.5) and weak Harnack inequality for non-negative supersolutions (Theorem 4.6) both in the interior and at the boundary. For the same results under the assumption (1.7), we define a scale invariant Dini-type Stummel-Kato class K Dini,1/2 (Ω) (see (2.16)) and require |b + c| 2 to belong there. We shall also discuss an example of an operator that demonstrates the necessity of an additional condition on b + c for local boundedness to hold. In section 4.2 we prove interior and boundary regularity for solutions under the bd negativity condition, assuming additionally that |b| 2 and |d| are in K Dini,1/2 (Ω). The same argument works under the cd negativity condition assuming |b + c| 2 ∈ K Dini,1/2 (Ω). Finally, using the aforementioned results, and assuming either negativity condition and |b + c| 2 ∈ K Dini,1/2 (Ω), we construct the Green's functions for both L and its adjoint L t , satisfying scale invariant quantitative estimates like global pointwise and weak bounds. All the estimates are scale invariant and independent of the Lebesgue measure of the domain.
We now briefly review the history of work in this area for linear elliptic equations in divergence form with merely bounded leading coefficients and singular lower order terms. The generalized Dirichlet problem in the Sobolev space W 1,2 is well-posed if there exists a unique u ∈ W 1,2 (Ω) such that Lu = f + divg and u− φ ∈ W 1,2 0 (Ω) for fixed φ ∈ W 1,2 (Ω) and f, g i ∈ L 2 (Ω). Moreover, there exists a constant C φ,f,g so that the global estimate u W 1,2 (Ω) C φ,f,g holds. For operators without lower order terms this problem has a long history and we refer to [GiTr,p.214] and the references therein for details. In bounded domains, in the presence of lower order terms, Ladyzhenskaya and Ural'tseva [LU] and Stampacchia [St2] proved well-posedness of the generalized Dirichlet problem assuming conditions related to the coercivity of the operator or smallness of the norms of the lower order coefficients. This was quite restrictive as, for example, the "bad" terms coming from the lower order coefficients can be absorbed in view of smallness. Gilbarg and Trudinger [GiTr] gave an extension of the previous results replacing the smallness conditions by the assumptions b, c, d ∈ L ∞ (Ω) assuming either (1. 6) or (1.7). In fact, they only need b, c ∈ L s (Ω) and d ∈ L s/2 (Ω), for some s > n. Recently, Kim and Sakellaris [KSa], generalized it to operators whose coefficients are in the critical Lebesgue space (as in our standing assumptions). Unfortunately, in all those results, the implicit constant in the global estimate depends on the Lebesgue measure of Ω and thus, they cannot be extended to unbounded domains by approximation. On the other hand, in unbounded domains with possibly infinite Lebesgue measure, already in 1976, Bottaro and Marina [BM] proved that, if b, c ∈ L n (Ω), d ∈ L n/2 (Ω) + L ∞ (Ω), and divb + d ≤ µ < 0, then the generalized Dirichlet problem is well-posed. To our knowledge, this was the first paper establishing well-posedness in such generality.
The local pointwise estimates find their roots in De Giorgi's celebrated paper [DeG] on the Hölder continuity of solutions of elliptic equations of the form −divA∇u = 0, where Theorems 4.5 (i) and 4.12 were proved in this special case (see also [Na]). A few years later, Moser gave a new proof of De Giorgi's theorem in [Mos1]. The same results were extended in equations of the form (1.1) by Morrey when b, c ∈ L q and d ∈ L q/2 , for q > n and Stampacchia [St1] (in more special cases). Moser also established the weak Harnack inequality for solutions of −divA∇u = 0 in [Mos2], while Stampacchia [St2] proved all the a priori estimates for equations of the form (1.1) with c ∈ L n and |b| 2 , d ∈ L s , s > n/2, assuming that (1. 6) holds and the radius of the balls are sufficiently small so that the respective norms of the lower order coefficients on those balls are small themselves. Under our standing assumptions, Kim and Sakellaris [KSa] also established local boundedness for subsolutions of the equation (1.1) satisfying either (1. 6) or (1.7) and b + c ∈ L s , s > n (with implicit constants dependent on the Lebesgue measure of Ω). They also constructed a counterexample showing that if (1.7) holds, it is necessary to have an additional hypothesis on b + c (see [KSa,Lemma 7.4]).
Proving the boundary regularity of solutions to the generalized Dirichlet problem with data φ ∈ W 1,2 (Ω) ∩ C(Ω) has been an important problem in the area and stems back to the work of Wiener for the Laplace operator [Wi]. Wiener characterized the points ξ ∈ ∂Ω that a solution converges continuously to the boundary in terms of the capacity of the complement of the domain in the balls centered at ξ. The proof was tied to the pointwise bounds of the Green's function and so were its generalizations to elliptic equations. In particular, Littman, Stampacchia and Weinberger [LSW] constructed the Green's function in a bounded domain for equations −divA∇u = 0, where A is real and symmetric, proving such a criterion and later, Grüter and Widman [GW] extended their results to operators with possibly non-symmetric A. For equations with lower order coefficients in bounded domains, Stampacchia [St2] showed a Wiener-type criterion in sufficiently small balls centered at the boundary of Ω. On the other hand, Kim and Sakellaris [KSa] succeeded to construct the Green's function with pointwise bounds (which was their main goal) following the method of Grüter and Widman, assuming either (1.7) and b+c ∈ L n , or (1. 6) and b+c ∈ L s , s > n. This is the best known result in this setting in domains with finite Lebesgue measure. In this case though, the construction of the Green's function was not used to conclude boundary regularity. For elliptic systems in unbounded domains, Hofmann and Kim constructed the Green's function assuming that their solutions satisfy the interior a priori estimates of De Giorgi/Nash/Moser. They also showed boundary Hölder continuity of the solution of the Dirichlet problem with C α (Ω) data under the stronger assumption of Lebesgue measure density condition of the complement of Ω in the balls centered at ∂Ω (see also [KK]). Recently, Davey, Hill and Mayboroda [DHMa] extended [HK] to systems with lower order terms in b ∈ L q , c ∈ L s and d ∈ L t/2 , with min{q, s, t} > n, whose associated bilinear form is coercive.
Let us now discuss our methods. Inspired by the treatment of the Dirichlet problem in [BM] and specifically the use of Lemma 2.16, we are able to extend their results to operators with either negativity assumption (as opposed to −divb+d ≤ µ < 0) by requiring solvability in the Sobolev space Y 1,2 instead of W 1,2 with non-divergence interior data in L 2n n+2 instead of L 2 . This is the "correct" Sobolev space in unbounded domains and had already appeared in [MZ] and in connection with the Green's function in [HK]. The main difficulty lies on the fact that when we are proving the global bounds for the solution of the Dirichlet problem, we arrive to an estimate where the term b + c L n (Ω) ∇u 2 L 2 (Ω) should be absorbed. But unless one has smallness of b+c L n (Ω) this is impossible. To deal with this issue, we use Lemma 2.16 and split the domain in a finite number of subsets Ω i where the norm L n (Ω i ) norm of b+c becomes small. We also write u as a finite sum of u i so that ∇u i ⊂ Ω i and, loosely speaking, the term above can be hidden. An iteration argument is then required, which concludes the desired result. The change of data is taken care of by approximating both the domain and the non-divergence data f ∈ L 2n n+2 by sequences of bounded domains Ω k ⊂ Ω and functions f j ∈ L 2 ∩ L 2n n+2 , solve the Dirichlet problem there, and use the global bounds to pass to the limits. It is exactly at this point that we are benefited by the domain-independence of the constants in the global estimates. The same considerations apply to prove the weak maximum principle for subsolutions with either negativity condition, which, in turn, allows us to solve the unilateral variational poblem and thus, the obstascle problem in bounded domains. As a corollary we obtain that the minimum of two subsolutions of the inhomogeneouus equation Lu = f − divg is also a subsolution. Moving further to the proof of Caccioppoli inequality, some serious difficulties arise. Up to now, Caccioppoli's inequality was unknown with so general conditions, since it could be solved only for balls r ≤ 1 and then rescale. This resulted to the appearance of the Lebesgue measure in the constants and so, it could not serve our purpose for scale invariant estimates. To overcome this important obstacle we had to make a technically challenging adaptation of the method that solves the Dirichlet problem.
To prove local boundedeness, weak Harnack inequality, interior and boundary regularity, we adjust the arguments of Gilbarg and Trudinger [GiTr,. Although, to do so, we are required to prove some refined versions of Caccioppoli inequality (Theorems 3. 4-3.6), which in [GiTr] were immediate. This turns out to be an even more demanding task than the proof of Caccioppoli inequality itself. Once we obtain them, we show Lemma 4.1, which is the building block of a Moser-type iteration argument. For this lemma, we need an embedding inequality (see Corollary 2.6) with constants independent of the domain, which we prove, since we were not able to find it in the literature (with constants independent of the domain). The use of the Stummel-Kato class K(Ω) as an appropriate class of functions for the interior data and the lower order coefficients is not new and has its roots to Scrödinger operators with singular potentials (see [Ku] and the references therein). Although, in our case, due to the counterexample of Kim and Sakellaris [KSa] (see Example 4.8), |b + c| 2 should be in appropriate subspace of it satisfying a Dini-type scale invariant condition. In fact, a Dini condition was imposed in [RZ] to prove local boundedness of subsolutions for certain quasi-linear equations, but their constants depend on Ω. Our iteration argument in the proof of Theorem 4.5 follows their ideas, but to get scale invariant estimates, it is necessary to come up with the condition (2.3) and deal with certain technical details that were not evident to us in [RZ]. In Example 4.9, we also show that a negativity condition is necessary to obtain local boundedness.
Regarding interior and boundary regularity, as is customary, we go through an application of the weak Harnack inequality. But for this, we need the positivity condition to hold which would force us to assume L1 = 0, or equivalently −divb + d = 0. But since this would lead to a significant restriction on the class of operators our theorems would apply, we incorporate −div(bu) and du to the interior data −divg and f respectively. The "new" equation now has the form for which it is true that L1 = 0. The price we have to pay is to impose the additional assumptions |b| 2 and |d| ∈ K Dini,1/2 (Ω). Of course, we require u to be locally bounded as well and thus, we need to assume either (1. 6) or |b + c| 2 ∈ K Dini,1/2 (Ω) and (1.7). It is interesting to see that the proof of Theorem 4.13 is quite laborious as it requires a modification of the original argument in [GiTr] (which is not obvious without the capacity density condition) and a new way of handling the second term Σ 2 in the iteration scheme. To our knowledge, this is the first Wiener-type criterion for boundary regularity of solutions for equations with lower order coefficients with so general assumptions. Moreover, the interior regularity is also new if (1.6) holds, while, if |b + c| 2 ∈ K Dini,1/2 (Ω) and (1.7) are satisfied, it is possibly new as well in the case that the radii of the balls we consider are not small. Let us comment here that one could try to prove boundary regularity following [GW] or even [HeKM], but in both cases, there would only be treated solutions of equations with no right hand-side and b i = d = 0, 1 ≤ i ≤ n. This is because of the need of lower poitwise bounds for the Green's function or a equivalently a Harnack inequality, which in this situation holds just for equations of the form Lu = −divA∇u − c∇u = 0.
Finally, having proved all the results above, we are in a position to construct the Green's function associated with L when |b + c| 2 ∈ K Dini,1/2 (Ω) and either negativity condition holds. We use the method of Hofmann and Kim [HK] along with its variant of Kang and Kim [KK], where the main ingredients are the well-posedness of the Dirichlet problem, local boundedness, Caccioppoli's inequality, and maximum principle, while, for the approximating operators, we also use the interior continuity for solutions of equations with lower order coefficients that satisfy |b| 2 , |c| 2 , |d| ∈ K Dini,1/2 (Ω). We would not need an approximation argument if it wasn't for the lack of continuity in the general case. This creates some trouble in the proof of G(x, y) = G t (y, x), where G t stands for the Green's function associated with L t , the formal adjoint of L. It is important to point out that the pointwise bounds for G do not hold unless local boundedness of subsolutions of L t u = 0 is true; in view of Example 4.8, an additional condition on b + c is necessary. In our case, this will be |b + c| 2 ∈ K Dini,1/2 (Ω) as before. Remark that, since Ω may have infinite Lebesgue measure, we can assume Ω = R n and construct the fundamental solution.
Very recently, Giorgos Sakellaris informed us about an interesting result he obtained, which is very related to our work. His primary goal was to construct Green?s functions for elliptic operators of the form (1.1) in general domains under either negativity condition that satisfy scale invariant pointwise bounds. Then, he applies them to obtain global and local boundedness for solutions to equations with interior data in the case (1.7). To do this, it was required b + c to be in a scale invariant space, which for the author was the Lorentz space L n,1 (Ω) (as opposed to |b + c| 2 ∈ K Dini,1/2 (Ω) we identified). His method is totally different than ours and is based on delicate estimates for decreasing rearrangements. In fact, he first proves the existence of Green's functions via various approximations and then uses their properties to obtain a priori estimates; our method follows the exact opposite direction. Our paper and [Sak] are rather complementary since, apart from the major differences in the approach, the conditions |b + c| 2 ∈ K Dini,1/2 and |b + c| ∈ L n,1 are not comparable. We would like to note here that Sakellaris observed that, due to a Lorentz-Sobolev embedding theorem and density, (1. 6) or (1.7) can be applied assuming that b, c ∈ L n,∞ (Ω), d ∈ L n/2,∞ (Ω), and b + c ∈ L n (Ω). Using this, all of our proofs carry out unchanged and our standing assumptions can be replaced by those more general ones (see [Sak,Lemmas 2.2 and 3.4

]).
Acknowldegements. We would like to thank Giorgos Sakellaris for making his paper available to us and for helpful discussions. We are also grateful to him for spotting a gap in our previous proof of ( 6.11 |f (y)| |x − y| n−2 dy, for r > 0, ϑ Ω (f, r) = 0 and ϑ Ω (f, r) < ∞, for each r ∈ (0, diam(Ω)), then we say that f is in the Stummel-Kato class K(Ω). In the case that Ω is unbounded we for any bounded open set so that Ω ⊂ Ω, then we say that f ∈ K loc (Ω). For q ≥ 1, we will write f ∈ K Dini,q (Ω) if f ∈ K(Ω) and satisfies the following scale invariant Dini condition: Recall that a function f ∈ L 1 loc (Ω) is in the Morrey space M λ (Ω), if and notice also that if ϑ Ω (f, t) ≈ t ε , for some ε > 0, then (2.3) is satisfied. So, if M λ (Ω) stands for the Morrey space with λ = n − 2 + ε and p = n + ε, it holds L p (Ω) ⊂ K Dini,q (Ω) and M λ (Ω) ⊂ K Dini,q (Ω).
It is easy to see that there exists a dimensional constant C db > 0 so that Therefore, there exists a constant c > 0 so that The following considerations can be found in [Ku,p.416] and are based on an inequality proved by Simon in [Sim,p.455]. Assume that f ∈ K(Ω) and let (2. 6) ψ ∈ C ∞ c (R n ), 0 ≤ ψ ≤ 1, ψ = 0 in R n \ B(0, 1), and ψ = 1.
Lemma 2.2. If f ∈ K(B r ), then, there exists a constant c 1 > 0 depending only on n such that for any r > 0 and u ∈ W 1,2 (B r ), it holds (2.10) Br Proof. This inequality can be found in the proof of Lemma 2.1 in [Ku] (display (12), p. 416). It is stated with slightly different assumptions but an inspection of the proof reveals that the inequality above is also true. For a similar inequality see Lemma 7.3 in [Sch].
Note that if we set f = f δ in (2.10) and use (2.8), we can see that where c 1 is independent of δ. Let us set µ(r) = r r+1 and note that it is increasing and satisfies lim r→0 + µ(r) = 0 and lim r→+∞ µ(r) = 1. If we define ϑ ′ Ω (f, r) = µ(r)ϑ Ω (f, r), then ϑ ′ Ω (f, r) is increasing and satisfies the same properties as ϑ Ω (f, r). Therefore, it is invertible with continuous and increasing inverse ϑ ′ −1 Ω (f, r). It is clear that ϑ ′ Ω (f, ·) also satisfies the doubling condition (2.5) with slightly different constant. Note that Lemma 2.3. If f ∈ K(R n ), then, there exists a constant c 2 > 0 depending only on n such that for any ε > 0 and u ∈ W 1,2 (R n ), it holds Proof. Fix ε > 0 and choose r > 0 small enough so that ϑ ′ R n (r) = (N c 1 ) −1 ε, where c 1 is the constant in (2.10). We cover R n with balls B(z j , r), with center all the points z j so that nz j /r have integer coordinates. It is clear that each point x ∈ R n is contained in at most N balls B(z j , 2r), where N is a positive constant depending only on the dimension n. Thus, using (2.10), we have that which, if we set c 2 = N c 1 , implies (2.13).
An immediate corollary of the latter theorem that we will use in Section 4 is the following: Corollary 2. 4. If f ∈ K(Ω), then, there exists a constant c 2 > 0 depending only on n such that for any ε > 0 and u ∈ W 1,2 0 (Ω), it holds (2.14) Remark 2. 5. In view of (2.11), it is easy to see that (2.13) and (2.15) still hold if we replace f by f δ on the left hand-side and keep the same term on the right hand-side.
This last remark, combined with (2.9) and (the proofs of) Lemmas 2.2 and 2.3, and Corollary 2.6, leads to the following corollary which will be crucial in an approximation argument we will need later.
By Sobolev embedding theorem, it is clear that W 1,2 0 (Ω) ⊂ Y 1,2 0 (Ω), while if Ω has finite Lebesgue measure they are in fact equal. See, for instance, Theorem 1.56 and Corollary 1.57 in [MZ]. Moreover, Y 1,2 0 (R n ) = Y 1,2 (R n ) (see e.g. Lemma 1.76 in [MZ]). In the sequel we will require a notion of supremum and infimum of a function in Y 1,2 (Ω) at the boundary of an open set Ω ⊂ R n since such a function is not necessarily continuous all the way to the boundary. Let Y denote either Y 1.2 (Ω) or W 1.2 (Ω) and Y 0 be either Y 1.2 0 (Ω) or W 1.2 0 (Ω). Definition 2.9. Given a function u ∈ Y , we say that u ≤ 0 on ∂Ω if u + ∈ Y . If u is continuous in a neighborhood of ∂Ω then u ≤ 0 on ∂Ω in the Sobolev sense if u ≤ 0 in the pointwise sense. In the same way u ≥ 0 if −u ≤ 0 and u ≤ w if u − w ≤ 0. We define the boundary supremum and infimum of u as Definition 2. 10. Let E ⊂ Ω and u ∈ Y . We say that u ≤ 0 on E if u + is the limit in Y -norm of a sequence of C ∞ c (Ω \ E). Then u ≥ 0 and u ≤ v can be defined naturally.
Moreover, if Ω has finite Lebesgue measure.
If E = ∂Ω the two definitions above coincide.
We record some results for Sobolev functions we will need later. Their proofs can be found in [MZ] and/or in [HeKM] for functions in W 1.2 (Ω) or W 1.2 0 (Ω). Although, one can make the obvious modifications to prove them for Y 1.2 Proof. The fact that u is a constant can be found in Corollary 1.42 in [MZ], while the second part can proved by a slight modification of the proof of Lemma 1.17 in [HeKM].
Lemma 2.12 (Corollary 1.43 in [MZ] In particular, ∇u = ∇v a.e. on the set {x ∈ Ω : Theorem 2.13 (Theorem 1.74 in [MZ]). Let Ω ⊂ R n be an open set and let f be a Lipschitz function such that f (0) = 0.
Lemma 2.14. Let Ω ⊂ R n be an open set and let f : The following lemma will be used repeatedly in this manuscript and was essentially proved in [BM]. Although the required modifications are minor, we present the proof for the reader's convenience 1 .
For (6) and (7), we may rewrite u j = u k j ,∞ − u k j−1,∞ , in view of which, we have In the case i = κ, we have yielding (6). By definition, ∇u k i ,∞ = ∇u, when |u| > k i (i.e., in the support of u i ), while u i = 0, whenever |u| ≤ k i . and so, (7) follows from (2.19). Since {∇u i = 0} ⊂ Ω i we can use (6) to get This concludes the proof of the lemma.
Let Ω ⊂ R n be open and E ⊂ Ω. If we set The following properties of capacity verify that it is a Choquet capacity and satisfies the axioms considered by Brelot. A proof can be found for instance in Theorem 2.3 in [MZ].

Standard Caccioppoli inequality.
Theorem 3.1 (Caccioppoli inequality I). Assume that u ∈ Y 1,2 loc (Ω) is either a solution or a non-negative subsolution of (3.1) and (1.6) holds. Then, for any non-negative function Proof. We will only treat the case that u is a non-negative subsolution of (3.1) as the proof when u is a solution is almost identical and is omitted. Notice that if K := supp η, which is a compact subset of Ω, we may find a bounded open set Ω ′ such that K ⊂ Ω ′ ⋐ Ω, and since u ∈ Y 1,2 loc (Ω), it holds that u ∈ Y 1,2 (Ω ′ ). Working in Ω ′ instead of Ω, we may assume, without loss of generality, that u ∈ Y 1,2 (Ω). Moreover, u is clearly a subsolution in any open subset of Ω. For simplicity, let us preserve the notation Ω instead of Ω ′ .
After the initial reductions above we proceed to the proof of (3.2). Apply Lemma 2.16 to the function u, for p = n, h = b + c, and a = λ 8Cs , where C s is the constant in Sobolev's inequality, to find Ω i ⊂ Ω and u i ∈ Y 1,2 (Ω), 1 ≤ i ≤ κ, satisfying (1)-(8). Note that (5) tells us that u i and u have the same sign, and so, the functions η 2 u i ∈ Y 1,2 0 (Ω) are non-negative. Thus, using that u is a subsolution for (3.1) we have where in the last inequality we used (5) and (1.6). In view of (3) and (6), the latter inequality can be written as while by Hölder's's inequality If we use Hölder's, Sobolev's, and Young's inequalities, along with the fact that for any Once more, by Hölder's, Sobolev's, and Young's inequalities, we obtain Choosing δ = λ 32 in (3.7), we can combine (3.3), (3.4), (3.5), and (3.6) and infer that which implies that there exist positive constants C 1 , C 2 and C 3 depending on λ, Λ and C s so that Note that the constant the sum is multiplied with is indeed 1, which is convenient in the iteration argument below. If we denote C 0 := max(C 1 , C 2 , C 3 ), x j := η∇u j L 2 , and y 0 := u∇η L 2 + η∇u 1/2 and use that (4), the latter inequality can be written By induction, we get Then, by (3.10) and the induction hypothesis, which, in light of (6), (3.12) and Young's inequality (with a small constant), implies that This concludes our proof since κ depends only b + c L n , λ, Λ, and C s .
The proofs of Theorems 3.1 and 3.2 can easily be adapted to prove the following Caccioppoli inequality at the boundary.
Theorem 3.3 (Caccioppoli inequality at the boundary). If B r is a ball such that B r ∩Ω = ∅, set Ω r = B r ∩ Ω and assume that u ∈ Y 1,2 (Ω r ) vanishing on ∂Ω ∩ B r in the sense of definition 2. 10. If either (1.6) or (1.7) holds and u is either a solution or a non-negative subsolution of (3.1) in Ω r , then, for any non- where the implicit constant depends only on n, λ, Λ, b + c L n (Ω) , and C s . Proof. We may apply Lemma 2.16 with p = n, h = b + c, and a = λ 8Cs to find Ω i ⊂ Ω r (x) and u i ∈ Y 1,2 (Ω r ) that vanishes on B r ∩ ∂Ω, for 1 ≤ i ≤ κ, satisfying (1)-(8). Using that η 2 u i ∈ Y 1,2 0 (Ω r (x)) and non-negative, along with the fact that u is either a solution or a non-negative subsolution of (3.1) in Ω r , we may proceed as in the proofs of Theorems 3.1 and 3.2 to obtain (3.20). We skip the details.

Proof.
We first assume that u is a non-negative supersolution of (3.1) and β < −1.
For k > 0 we define the auxiliary function It is clear that w ∈ Y 1,2 (Ω r ) vanishing on ∂Ω ∩ B r and so, we can apply Lemma 2.16 to w and Ω r with p = n, h = b + c, and a = λ 8Cs , where C s is the constant in Sobolev's inequality, to find w i ∈ Y 1,2 (Ω r ) that vanishes on ∂Ω ∩ B r and Ω i ⊂ Ω r , 1 ≤ i ≤ κ, so that (1)-(8) hold.

(3.24)
Recalling that β < −1 and using (3.24), (1.2) andū > 0, Since u is a supersolution of (3.1), β + 1 < 0, and divb − d ≥ 0, we obtain we may write I 3 as the sum of three integrals I 31 , I 32 , I 33 that correspond to the terms on the right hand-side of the latter equality. So, by Young's inequality (for ε small enough to be chosen) along with (3.23) and (3.24), we get Moreover, by (7), β + 1 2 If we apply Hölder's, Sobolev's and Young's inequalities, Let us set and also, Then, using this notation and choosing α and ε small enough, depending on λ, Λ, b + c L n (Ω) , and C s , we can collect the inequalities (3.35)-(3.34) and find a constant C 0 (depending on λ, Λ and C s ) so that By the induction argument that appeared in the proof of Theorem 3.1 we can show that where C 1 depends on λ, Λ, b + c L n (Ω) and C s . We may choose ε small enough compared to C −2 ! and use Young's inequality with ε to deduce |β + 1|x 0 ≤ C 2 (z 0 + y 0 ).
in order to show (3.21). The details are omitted.
We turn our attention to the case that u is a non-negative supersolution of (3.1) and β ∈ [−1, 0). For k > 0 we define the auxiliary function Since w ∈ Y 1,2 (Ω) and vanishes on ∂Ω ∩ B r , we apply Lemma 2.16 as in the previous case to w and Ω r , for p = n, h = b + c, and a small enough depending on λ, β, C s (to be picked later), to find w i ∈ Y 1,2 (Ω) that also vanishes on ∂Ω ∩ B r and Ω i ⊂ Ω r , 1 ≤ i ≤ m, satisfying (1)-(8). By (5) we see that η 2 w i ∈ Y 1,2 0 (Ω) is non-negative and we may use it as a test function. Therefore, (3.35) where in the last inequality we used (1.6).
At this point let us recall (3.24) and also record that Therefore, by (3.37) and β < 0, (3.35) can be written We apply Hölder's inequality along with (3.24) and (3.36) to get It is easy to see that by Young's inequality, So, it only remains to handle I 2 . At this point we cannot use (6) or (7) as in previous arguments. The reason why is that we do not have u and u i but rather two different functions u and w i . Although, we can recall that {x ∈ Ω r : w i = 0} = ∪ i j=1 Ω j and thus, Using (3.24), (3.36), b+c L n (Ω j ) ≤ a for any j ∈ {1, 2, · · · m}, and w iū where in the last step we used Sobolev inequality. In the same way, we can show that Also, for β ∈ [−1, 0), it holds β−1 2β > 0 and β+1 2 > 0. Thus, by (3.36), which, in turn, implies that With this notation, we can write which, in combination with inequalities (1.2) and (3.38)-(3.44), implies . Therefore, if we choose α small enough (depending linearly on |β|), by Young's inequality, we can find a positive constant C 0 depending only on λ, Λ, and C s so that The proof of (3.21) is concluded by the same iteration argument as in the proof of Theorem 3.1 along with the fact that ∪ κ i=1 Ω i = Ω r obtaining The case β > 0 and u positive subsolution of (3.1) is almost identical and so, we will not repeat it.
The analogue of Theorem 3.4 for the case −divc + d ≥ 0 (or −divc + d ≤ 0) will be a lot easier to prove, as one does not need to handle the L n -norm of b + c in a delicate way as before, but rather to incorporate |b + c| 2 into the interior data side. It may look surprising bearing in mind the special case β = 1 we proved in Theorem 3.2, but (3.21) cannot hold in this case. The reason why is that it is the main ingredient of the proof of local boundedness, weak Harnack inequality and maximum principle and, by Example (4.8), we know that if b + c does not have any additional hypothesis solutions may not be locally bounded.
Because β < 0 and {x ∈ Ω r : w = 0} = Ω m r , the latter inequality can be written as Note that if we use 0 ≤ u ≤ū, then I 1 , I 3 and I 4 can be bounded as in (3.39) and (3.40). So, it only remains to handle I 2 . But as we do not need to use Lemma 2.16 it will be fairly easy to do so. Indeed, which, in light of Young's inequality with ε small (to be picked), w ≤ū β χ Ω m r and β ∈ [−1, 0), implies If we choose ε small enough we conclude our result. We may handle the case β < −1 and β ≥ 0 for subsolutions in a similar fashion adapting the argument in the proof of Theorem 3. 4. We omit the routine details.
Moreover, the proofs of Theorems 3.4 and 3.5 can easily be adapted to get a refined version of Theorems 3.1 and 3.2. We only state the first one.

LOCAL ESTIMATES AND REGULARITY OF SOLUTIONS UP TO THE BOUNDARY
In this part we will present the iterating method of Moser to obtain the following results: • Local boundedness for subsolutions; • Weak Harnack inequality for supersolutions; • Hölder continuity in the interior for solutions; • A Wiener criterion for continuity of solutions at the boundary.
We will follow the proofs from Gilbarg and Trudinger [GiTr,, although the details are quite different. 4.1. Local boundedness and weak Harnack inequality. We set γ = β +1, and for r > 0, whereū is either the one given in Theorem 3.4 or in Theorem 3.6, with k = k(r 0 ). Here B r is a ball of radius r > 0 which is either centered at the boundary (as in Theorem 3.4) or is such that B r ⊂ Ω (as in Theorem 3.6). We will handle both cases simultaneously and so, it should be understood from the context what kind of balls we are referring to.
We are now ready to prove the local boundedness of subsolutions.
Definition 4. 4. We will say that the condition (N) is satisfied if one the following conditions hold: (1) divb + d ≤ 0 and b + c ∈ L n (Ω), or (2) −divc + d ≤ 0 and |b + c| 2 ∈ K Dini,1/2 (Ω). Analogously, we will say that the condition (P) is satisfied if we reverse the inequalities in condition (N). Here, (N) and (P) stand for the negativity and positivity condition respectively.
In the next theorem we borrow ideas from [RZ], although some details seem to be different in our case. For example, we had to introduce the auxiliary modulus ϑ Ωr to be able to use Lemma 2.7, define the appropriate Dini condition that gives constants independent of Ω and be careful with the computations. Theorem 4.5 (Local boundedness). Let B r a ball such that B r ∩ Ω = ∅ and assume that f, |g| 2 ∈ K Dini,1/2 (Ω). If σ ∈ (0, 1), then the following hold: (1) If u is a subsolution of (3.1) in B r ∩ Ω and the condition (N) holds, then
If we apply (4.12) to u r in the domain D r = 1 r Ω, by the same change of variables, we obtain the following estimate: Remark that the implicit constants do not depend on r.
We turn our attention to the weak Harnack inequality.
Example 4.8. Let us now refer to the counterexample constructed in [KSa,Lemma 7.4]. In particular, the authors defined the operator x |x| 2 | ln |x|| and δ > 0, and showed that the solution u = | ln |x|| δ ∈ Y 1,2 (B(0, e −1 )) does not satisfy (4.7) around 0. They proved that b ∈ L q (B(0, e −1 )) for any q > n but not in L n (B(0, e −1 )). It is not hard to see that |b| 2 ∈ K(B(0, e −1 )) but not in K Dini,1/2 (B(0, e −1 )), and thus, assuming |b + c| 2 ∈ K(Ω) is not enough to establish local boundedness. A modification of this example shows that (4.14) does not hold either when |b + c| 2 ∈ K(Ω). It is important to note that, since δ can be taken as small as we want, this example shows that assuming the norms to be small is not enough either. Example 4.9. Adjusting the previous example we can find an operator which does not satisfy neither (1.6) nor (1.7), for which there exists a non-bounded solution in the ball B(0, e −1 ). Indeed, let (4.19) − It is not hard to see that d ≥ 0 is in the Lorentz space L n/2,q (B(0, e −1 )), for any q > 1. But notice that u = | ln |x|| ∈ Y 1,2 (Ω) is a solution of ( 4.19) and u ∈ Y 1,2 (B(0, e −1 )).
Since u fails to be bounded around 0, the necessity of either (1. 6) or (1.7) to prove local boundedness is established. It is interesting to see that d is not in K(B(0, e −1 )) (and thus, it is not in L n/2,1 (B(0, e −1 )) either). As in the previous example, it is easy to see that having small norm does not help either.

Interior and boundary regularity.
Theorem 4.10. Let u be a supersolution of (3.1) in Ω with sup Ω u < ∞ and assume that the condition (P) holds, then u has a lower semi-continuous representative satisfying Proof. We follow the proof of [HeKM,Theorem 3.66]. Fix a ball B r centered at x ∈ Ω so that B 2r ⊂ Ω and apply Theorem 4.6 (i) to u − m r , where m r = ess inf Br u. Then, we have Since C is either a constant independent of r and (m r/2 − m r ) + k(r) → 0 as r → 0, by taking limits in the inequality above as r → 0, we obtain We advise the reader that in the rest of this section we will not use that b ∈ L n (Ω) and d ∈ L n/2 (Ω), but only c ∈ L n (Ω).
Proof. Fix r 0 > 0 such that B r 0 ⊂ Ω and assume that u is a weak solution of the equation It is easy to see that u is also a weak solution of the equation in B 2r 0 . Note that L1 = 0 and since d = b i = 0, i = 1, . . . , n, we can use Theorems 4.5 and 4.6 with k as in (4.24) to get (4.29) sup Br (u + k(r)) − B 2r (u + k(r)) inf Br (u + k(r)), for any r ≤ 2r 0 . and since M r − u and u − m r are non-negative solutions of (4.28) in B r 0 , by (4.14) for r ≤ r 0 , we obtain
The last goal of this section is to develop of a Wiener-type criterion for boundary regularity of solutions. We will follow the proof of Theorem 8.30 in [GiTr]. Several modifications are required in our case and in particular, we would like to draw the reader's attention to the iteration argument at the end of the proof. In [GiTr] it is claimed that the inequality (8.81) on p. 208 can be iterated to produce the desired oscillation bound at the boundary. Unless the CDC is satisfied, it is not clear to us that the second term on the right hand-side of that inequality will converge after infinitely many iterations. In fact, the exact term one picks up after m iterations is  It seems that if we do not have additional information about the behavior of the sequence a k = χ(r/2 k ), we could choose different sequences a k so that S m either converges or diverges or even have multiple limit points. We resolve this issue by incorporating this term into the main oscillation term.
Proof. If we set B r = B r (ξ) we record that u is a solution of Lu = f − divg in B r ∩ Ω and thus, a solution of (4.28). With the same notation as above, one can prove that for η ∈ C ∞ c (B r ), (4.31) η∇u − m L 2 (Br ) (η/r + |∇η|)(u − m + k) L 2 (Br ) . This follows easily by inspection of the proofs of Theorem 3.4 and Lemma 4.1.
Since b(r) ≤ 1 − k 0 for all r ∈ (0, r 0 ), 1 − t ≤ e −t and b(r) ≤ γ(r), we have Arguing similarly, we get  which can be used in (4.38) along with (1 − k 0 )γ(r) ≤ b(r) to obtain for some constant C ′ > 1 that depends on α and the constant C in the definition of γ(r).
To state the second important corollary of 4.12, we need the following definitions.
If Ω ⊂ R n is an open set and for ξ ∈ ∂Ω it holds that for some c ∈ (0, 1) independent of r, we say that ∂Ω has the capacity density condition (CDC) at ξ. If this holds for every ξ ∈ ∂Ω and a uniform constant c, we say that ∂Ω has the capacity density condition.
As a corollary of the previous theorem we obtain the following Wiener-type criterion for continuity of solutions up to the boundary as well as a modulus of continuity under the CDC. Corollary 4. 16. If we assume that the condition (N) is satisfied, then Theorems 4.12, 4.13, and (4.15) hold with constants independent of r. (i) If (1.6) holds then
A direct consequence of the weak maximum principles proved above is the following comparison principle: Let Ω ⊂ R n be an open and connected set and either (1. 6) or (1.7) holds. If u ∈ Y 1,2 (Ω) is a supersolution of (3.1) and v ∈ Y 1,2 (Ω) is a subsolution of (3.1) such that (v − u) + ∈ Y 1,2 0 (Ω), then we have that v ≤ u in Ω.
and is obviously well-defined. So, when we say that (5.4) holds "in the weak sense", we mean that Well-posedness of the Dirichlet problem (5.4) with solutions u ∈ W 1,2 0 (Ω) instead of u ∈ Y 1,2 0 (Ω) in unbounded domains was shown in [BM,Theorem 1.4] for data f, g ∈ L 2 (Ω), but with a stronger negativity assumption than divb + d ≤ 0. Namely, it was assumed that there exists µ < 0 such that divb+ d ≤ µ. This was necessary exactly because they required the solutions to be in W 1,2 0 (Ω) as opposed to Y 1,2 0 (Ω). It is worth mentioning that (1.7) was not treated at all.
In the following theorem we follow the proof of [BM,Theorem 1.4] adjusting the arguments to the weaker negativity assumption divb + d ≤ 0 and the Sobolev space Y 1,2 0 (Ω). Theorem 5.3. If f ∈ L 2 * (Ω) and g i ∈ L 2 (Ω) for 1 ≤ i ≤ n, and either (1. 6) or (1.7) holds, then the Dirichlet problem (5.4) has a unique solution u ∈ Y 1,2 0 (Ω) satisfying where the implicit constant depends only on λ, b + c L n (Ω) , and C s . Proof. To demonstrate that (5.6) holds assuming that such a solution exists, it is enough to repeat the argument in the proof of Theorem 5.1 applying Lemma 2.16 to u ∈ Y 1,2 0 (Ω). The difference is that we should use that u is a solution of (3.1) instead of a subsolution of Lu = 0 and thus, we pick up two terms related to the interior data exactly as in the proofs of Theorems 3.1 and 3.2. Similar (but easier) manipulations along with the same induction argument conclude (5.6). We omit the details.
By (5.9), its associated bilinear form is clearly coercive and bounded in V . As f ∈ H and g ∈ H, by Lax-Milgram's theorem, there exists a unique solution to the problem and so, L σ has a bounded inverse L −1 σ : V * → V . If J : V → V * is an embedding given by (5.11) Jv = Ω uv, v ∈ V, I 2 : V → H is the natural embedding and I 1 : H → V * is an embedding given also by (5.11), we can write J = I 1 • I 2 . It is clear that J is compact as I 2 is compact and I 1 is continuous.
The interior data naturally induces a linear functional on V by so we wish to solve the equation Lu = F . This is is equivalent to L σ u − σJu = F , which in turn, can be written as σ J is compact as J is compact and L −1 σ is continuous. Thus, by the Fredholm alternative, (5.12) has a unique solution if and only if w = 0 is the unique function in V satisfying w − σL −1 σ Jw = 0 (or else Lw = 0). But this readily follows from the weak maximum principle in Theorem 5.1 and thus, a solution of ( 5.4) exists in bounded domains.
An important consequence of this theorem is the following: Proof.
If Ω is bounded, the proof is a consequence of Theorem 5.6 and can be found in [KrS,Theorem 6.6].
Let Ω be an unbounded open set and assume that u and v are supersolutions of Lw = F in Ω. Since they are supersolutions of the same equation in any bounded open set D ⊂ Ω, min(u, v) is a supersolution in any such D as well. Using a partition of unity, this yields that min(u, v) is a supersolution in Ω.
The proof of the following theorem can be found for instance in [KrS,Theorem 6.9].
Then there exists a non-negative Radon measure so that (5.20) In particular,

GREEN'S FUNCTIONS IN UNBOUNDED DOMAINS
Here we construct the Green's function associated with an elliptic operator given by (1.1) satisfying either negativity assumption as well as b + c ∈ K Dini,1/2 (Ω). We follow the approach of Hofmann and Kim [HK] along with its variation due to Kang and Kim [KK]. 6.1. Construction of Green's functions. Before we start, we should mention that the equation formal adjoint operator of L is given by with corresponding bilinear form Moreover, if L satisfies (1.6), then its adjoint satisfies (1.7) and vice versa.

Lemma 6.2.
Let Ω ⊂ R n be an open set and L be the operator given by (1.1) that satisfies either (1. 6) or (1.7). Let B s = B(x, s) be a ball of radrius s centered at x ∈ Ω such that 3B s ⊂ Ω and u ∈ Y 1,2 (Ω \ B s ) be a solution of Lu = 0 in Ω \ B s that vanishes on ∂Ω. Then for any r ≥ 4s we have (6.21) where the implicit constants depend only on λ, Λ, b + c L n (Ω; R n ) , and C s . Proof. The proof can be found in [KSa,Lemma 3.19] with the difference that we use Theorems 3.3 instead of [KSa,Lemma 3.18] that only holds for r ≤ 1.
We will now demonstrate that for a fixed y ∈ Ω, G(·, y) ≥ 0 a.e. in Ω \ {y}. Assume that σ n is the sequence converging to zero for which G σn (·, y) converge to G(·, y) in the sense of (6.26) and (6.33). If necessary, we can pass to a subsequence so that σ n < min(|x − y|, d y )/10. Fix x ∈ Ω so that x = y and let ρ m be a sequence converging to zero so that ρ m ≤ min(|x − y|, d x )/10. Therefore, since G σn (·, y) ≥ 0 in Ω, we have that G(·, y), as n → ∞, where we used (6.26) in the case B ρm (y) ⊂ Ω \ B r (x) for some r > 0 and ( 6.33) in the case B ρm (x) ∩ B σn (y) = ∅. By Lebesgue's differentiation theorem, if we let m → ∞, we infer that G(x, y) ≥ 0 for a.e. x ∈ Ω \ {y}.
To prove uniqueness of the Green's function, we assume that G(·, y) is another Green's function for the same operator. Then for f ∈ C ∞ c (Ω) and g = 0, we have that for fixed y ∈ Ω, Ω G(·, y) f = u(y) ∈ Y 1,2 0 (Ω) and L t u = f.
Notice that we can apply the previous considerations to construct the Green's function G t (·, y) associated with the operator L t with all the properties above. The only thing that remains to be shown is that G t (x, y) = G(y, x) for a.e. (x, y) ∈ Ω 2 \ {x = y}. We will first prove it in the case that solutions of Lu = 0 and L t u = 0 are locally Hölder continuous in Ω \ {x} and Ω \ {y} respectively. In this case, all the properties that hold a.e. in Ω \ {pole}, due to the continuity away from the pole, will hold everywhere (apart from the pole).
To this end, let σ n and ρ m be the sequences converging to zero for which G σn (·, x) and G t ρm (·, y) converge to G(·, x) and G t (·, y) in the sense of (6.26), (6.33), and (6.34). If necessary, we may further pass to subsequences so that σ n < min(|x − y|, d x )/10 and ρ m ≤ min(|x − y|, d y )/10.
For k ∈ N, set ψ k (x) = k n ψ(kx) and define b k = (b χ Ω k ) * ψ k , c k = (c χ Ω k ) * ψ k and d k = (d χ Ω k ) * ψ k . It is clear that b k → b in L n (Ω), c k → c in L n (Ω), and d k → d in L n/2 (Ω). Define L k u = −divA∇u − div(b k u) − c k ∇u − d k u and set which are open sets such that ∪ k≥1 Ω k = Ω. If ϕ ∈ C ∞ c (Ω), then for k sufficiently large, ϕ ∈ C ∞ c (Ω k ) and Remark 4.7 applies. Therefore, since, for such k, Theorem 4.5 holds for L k in Ω k with bounds independent of k, we can construct the Green's functions G k and G t k associated with L k and L t k in Ω k as above, with the additional property that G k (·, x) and G t k (·, y) are locally Hölder continuous away from x and y respectively. In the last part we used Theorem 4.12, which applies in this situation, since |b k | 2 , |c k | 2 , |d k | ∈ K Dini,1/2 (Ω k ). Extend both G k (·, x) and G t k (·, y) by zero outside Ω k and note that (6.5)-( 6.9) hold in Ω with constants independent of k (see Remark 4.7). Therefore, repeating essentially the arguments concerning the convergence of G ρ and the inheritance of the bounds from G ρ , we can find G(·, y) which is non-negative a.e. in Ω \ {y} and vanishes on ∂Ω. Additionally, it satisfies (6.5)-( 6.9), and, after passing to a subsequence, G k (·, y) ⇀ G(·, y) in Y 1,2 (Ω \ B r (y)) for all r > 0, (6.40) G k (·, y) ⇀ G(·, y) in W 1,p (B r (y)), for all r < d y , (6.41) G k (·, y) * ⇀ G(·, y) in L n n−2 ,∞ (Ω), (6.42) ∇G k (·, y) * ⇀ ∇G(·, y) in L n n−1 ,∞ (Ω). (6.43) Of course, the considerations above apply to G t k as well. To show (6.36) we let ϕ ∈ C ∞ c (Ω) and pick k large enough so that ϕ ∈ C ∞ c (Ω k ) for every such k. If L k,Ω k (resp. L k,Ω ) is the bilinear form associated with L k on Ω k (resp. Ω), and we extend G k by zero outside Ω k , the two forms coincide. Therefore, by (6.36) it holds that for k large enough, (6.44) L k,Ω (G k (·, y), ϕ) = L k,Ω k (G k (·, y), ϕ) = ϕ(y), for all ϕ ∈ C ∞ c (Ω), and by (6.42) and (6.43), we can pass to the limit and prove (6.36) for G(·, y). If x, y ∈ Ω, there exists k large enough so that x, y ∈ Ω k for every such k. If f ∈ C ∞ c (Ω) and g ∈ C ∞ c (Ω), by (6.4) and G k (x, y) = G k (y, x) for all x, y ∈ Ω k so that x = y, we have that (6.45) u k (y) = Ω G k (y, ·) f + Ω ∇G k (y, ·) g = Ω G t k (·, y) f + Ω ∇G t k (·, y) g.
Since u k ∈ Y 1,2 0 (Ω k ), we can extend it by 0 outside Ω k . Recall that u k satisfies L t k u k = f − divg in Ω k and also u k Y 1,2 (Ω) = u k Y 1,2 (Ω k ) f L 2 * (Ω k ) + g L 2 (Ω k ) ≤ f L 2 * (Ω) + g L 2 (Ω) , where the implicit constant is independent of k. If we take limits in (6.45) as k → ∞ and use (6.42) and (6.43) for G t k (·, y), we can show that for all y ∈ Ω, lim k→∞ u k (y) = lim k→∞ Ω G t k (x, y) f (x) dx + lim k→∞ Ω ∇G t k (x, y) g(x) dx ( 6.46) = Ω G t (x, y) f (x) dx + Ω to argue by density using ( 6.8) and ( 6.9) for the part around the pole and ( 6.5) for the part far away from the pole. Details are left to the reader. Remark 6.3. Notice that the construction of the Green's function associated with L and the proof of several of its properties can be accomplished in the absence of b+c ∈ K Dini,1/2 (Ω). Although, we prefer to keep the statement of the theorem more simple and we impose the condition |b + c| 2 ∈ K Dini,1/2 (Ω) from the very beginning.
Finally, we can prove that, under certain restrictions, the Green's function has pointwise lower bounds as well.