Abstract. We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the C ∞ regularity of the free boundary in the Signorini problem near non-degenerate points.
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functionalamong all functions u ≥ 0 which are fixed on ∂Ω.We prove that the free boundary F (u) = ∂ R n {u > 0} of a minimizer u has locally finite H n−1 measure and is a C 2,α surface except on a small singular set of Hausdorff dimension n − 3. We also obtain C 2,α regularity of Lipschitz free boundaries of viscosity solutions associated to this problem.
The initial value problem for the L 2 critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well posed in H s (T d ), for s > 4/9 and s > 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [3]. We use the "I-method". This method allows one to introduce a modification of the energy functional that is well defined for initial data below the H 1 (T d ) threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, T d λ = R d /λZ d , d = 1, 2.
Abstract. We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are C 1,γ . In particular, viscosity solutions are indeed classical.
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