We study the nonlinear elliptic problemis nondecreasing, sublinear and f u is continuous. For every λ > 0, we obtain a maximal solution u λ 0 and prove its global regularity C 1,γ (Ω). There is a constant λ * such that u λ vanishes on a set of positive measure for 0 < λ < λ * , and u λ > 0 for λ > λ * . If f is concave, for λ > λ * we characterize u λ by its stability.

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, J γ → min, ruled by nonlinear, p-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl-Batchelor type; singular degenerate elliptic equations; and obstacle type systems. The Euler-Lagrange equation associated to J γ becomes singular along the free interface {u = 0}. The degree of singularity is, in turn, dimed by the parameter γ ∈ [0, 1]. For 0 < γ < 1 we show local minima is locally of class C 1,α for a sharp α that depends on dimension, p and γ. For γ = 0 we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.KEYWORDS: free boundary problems, degenerate elliptic operators, regularity theory. MSC2000: 35R35, 35J70, 35J75, 35J20.

We study local behavior of positive solutions to the fractional Yamabe equation with a singular set of fractional capacity zero.

Let (M, g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev exponent on the boundary. 4 n−2 g with scalar * Supported by CNPq/Brazil grant 309007/2016-0 and CAPES/Brazil grant 88881.169802/2018-01. † Supported by CNPq/Brazil grant 310983/2017-7. arXiv:1807.08406v2 [math.DG] 21 Apr 2019 In the case of manifolds without boundary, the question of compactness of the full set of smooth solutions to the Yamabe equation was first raised by R.Schoen in a topics course at Stanford University in 1988. A necessary condition is that the manifold M n is not conformally equivalent to the sphere S n . This problem was studied in [12,13,21,22,24,25,28,30] and was completely solved in [6,8,20]. In [6], Brendle discovered the first smooth counterexamples for dimensions n ≥ 52 (nonsmooth examples were obtained by Ambrosetti and Malchiodi in [5]). In [20], Khuri, Marques and Schoen proved compactness for dimensions n ≤ 24. Their proof contains both a local and a global aspect. The local aspect involves the vanishing of the Weyl tensor up to order [ n−6 2 ] at any blow-up point and the global aspect involves the positive mass theorem. Finally, in [8], Brendle and Marques extended the counterexamples of [6] to the remaining dimensions 25 ≤ n ≤ 51. In the case of nonempty umbilical boundary, the same compactness and noncompactness results were obtained by Disconzi and Khuri in [11] for the boundary condition B g u = 0.Despite its additional technical difficulties, the question of compactness of the solutions of (1.1) turns out to have great similarity with the one above for the classical Yamabe equation. In [17] Felli and Ould Ahmedou prove compactness for locally conformally flat manifolds with umbilic boundary, a result previously obtained by Schoen [28] for the classical Yamabe equation. In [1] the first author proves the vanishing of the trace-free boundary second fundamental form in dimensions n ≥ 7 at any blow-up point, a result inspired by the vanishing of the Weyl tensor in dimensions n ≥ 6 obtained by Li-Zhang and Marques independently in [21,22,25]. On the other hand, the noncompactness results of Brendle and Marques inspired the first author's paper [2] which provides counterexamples in dimensions n ≥ 25 to compactness in (1.1). So Theorem 1.1 ensures that there is a critical dimension 3 < n 0 ≤ 25 such that compactness for the set of positive smooth solutions of (1.1) holds for n < n 0 and fails for n ≥ n 0 .Although the corresponding result for the classical Yamabe equation in dimension 3 was obtained by Li and Zhu in [24], our approach to Theorem 1.1 makes use of some further techniques of the later works [20,25]. This is because the canonical bubble, coming from the Euclidean metric on B 3 , fails to provide a good approximation for the blowing up solutions of (1.1).

We study regularity results for almost minimizers of the functionalwhere is a matrix with Hölder continuous coefficients. In the case 0 < ≤ 1 we show that an almost minimizer belongs to 1, , where the exponent is related with the competition between the Hölder continuity of the matrix , the parameter of almost minimization and . In some sense, this regularity is optimal. As far as the case = 0 is concerned, our results show that an almost minimizer is 1, locally in each phase { > 0} and { < 0}, improving in some sense a recent result of David & Toro. K E Y W O R D S almost minimizers, free boundary problems, regularity theory M S C ( 2 0 1 0 ) 35J20, 35J61, 35R35, 49J10 1486