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We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in L q with q > n. Our approach is based on the scaling and comparison arguments of [13] , and we show that all our assumptions are sharp. As a consequence of our results, we deduce the C 1 , α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem. [ABSTRACT FROM AUTHOR]

We establish sharp regularity estimates for solutions to L u = f in Ω ⊂ R n , L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on S n − 1 , called the spectral measure. First, we study the interior regularity of solutions to L u = f in B 1 . We prove that if f is C α then u belong to C α + 2 s whenever α + 2 s is not an integer. In case f ∈ L ∞ , we show that the solution u is C 2 s when s ≠ 1 / 2 , and C 2 s − ϵ for all ϵ > 0 when s = 1 / 2 . Then, we study the boundary regularity of solutions to L u = f in Ω, u = 0 in R n ∖ Ω , in C 1 , 1 domains Ω. We show that solutions u satisfy u / d s ∈ C s − ϵ ( Ω ‾ ) for all ϵ > 0 , where d is the distance to ∂Ω. Finally, we show that our results are sharp by constructing two counterexamples. [ABSTRACT FROM AUTHOR]

Abstract: We consider the monomial weight in , where is a real number for each , and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by , and they contain the best or critical exponent (which depends on ). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space (with no weight) when written for axially symmetric functions and domains in . [Copyright &y& Elsevier]

We study the regularity of the free boundary in the obstacle for the p -Laplacian, min ⁡ { − Δ p u , u − φ } = 0 in Ω ⊂ R n . Here, Δ p u = div ( | ∇ u | p − 2 ∇ u ) , and p ∈ ( 1 , 2 ) ∪ ( 2 , ∞ ) . Near those free boundary points where ∇ φ ≠ 0 , the operator Δ p is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when ∇ φ = 0 then Δ p is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where ∇ φ = 0 . On the one hand, for every p ≠ 2 we construct explicit global 2-homogeneous solutions to the p -Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not C 1 at points where ∇ φ = 0 . On the other hand, under the “concavity” assumption | ∇ φ | 2 − p Δ p φ < 0 , we show the free boundary is countably ( n − 1 ) -rectifiable and we prove a nondegeneracy property for u at all free boundary points. [ABSTRACT FROM AUTHOR]