We construct Green's function for second order elliptic operators of the form Lu = −∇ · (A∇u + bu) + c · ∇u + du in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients A is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition d − ∇ · b ≥ 0. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green's function in the case when the mean oscillations of the coefficients A and b satisfy the Dini conditions and the domain is C 1,Dini .2010 Mathematics Subject Classification. 35A08, 35J08.

We use the method of layer potentials to study the R 2 Regularity problem and the D 2 Dirichlet problem for second order elliptic equations of the form Lu = 0, with lower order coefficients, in bounded Lipschitz domains. For R 2 we establish existence and uniqueness assuming that L is of the form Lu = −div(A∇u + bu) + c∇u + du, where the matrix A is uniformly elliptic and Hölder continuous, b is Hölder continuous, and c, d belong to Lebesgue classes and they satisfy either the condition d ≥ div b, or d ≥ div c in the sense of distributions. In particular, A is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for D 2 for the adjoint equations L t u = 0.

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We consider the “thin one‐phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in ℝ+n+1 plus the area of the positivity set of that function in ℝn. We establish full regularity of the free boundary for dimensions n≤2, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced by Alt and Caffarelli in 1981. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments that are less reliant on the underlying PDE. © 2021 Wiley Periodicals LLC.

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation − div(A∇u + bu) + c∇u + du = − div f + g, assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions b, fand c ∈ L n,q for q ≤ ∞, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on b, d or c, d, we show a maximum principle and Moser's estimate for subsolutions with "good" constants. We also show the reverse Moser estimate for nonnegative supersolutions with "good" constants, under no smallness assumptions when q < ∞, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in R n+1 + plus the area of the positivity set of that function in R n . We establish full regularity of the free boundary for dimensions n ≤ 2, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight.While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in [AC81]. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.

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