We consider the problem of finding, from the final data u(x, T ) = ϕ(x), the temperature function u(x, t), x ∈ (0, π), t ∈ [0, T ] satisfies the following nonlinear systemThe nonlinear problem is severely ill-posed. We shall improve the quasi-boundary value method to regularize the problem and to get some error estimates. The approximation solution is calculated by the contraction principle. A numerical experiment is given.
On a smooth domain Ω ⊂⊂ C n , we consider the Dirichlet problem for the complex Monge-Ampère equation ((dd c u) n = f dV, u| bΩ ≡ ϕ). We state the Hölder regularity of the solution u when the boundary value ϕ is Hölder continuous and the density f is only
This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in [Kh10] which refines former work by Nicoara [N06]. It has been proved by Raich [R10] that on a CR manifold of dimension 2n − 1 which is compact pseudoconvex of hypersurface type embedded in C n and orientable, the property named "(CR − P q )" for 1 ≤ q ≤ n−1 2 , a generalization of the one introduced by Catlin in [C84], implies compactness estimates for the Kohn-Laplacian b in degree k for any k satisfying q ≤ k ≤ n − 1 − q. The same result is stated by Straube in [S10] without the assumption of orientability. We regain these results by a simplified method and extend the conclusions in two directions. First, the CR manifold is no longer required to be embedded. Second, when (CR − P q ) holds for q = 1 (and, in case n = 1, under the additional hypothesis that∂ b has closed range on functions) we prove compactness also in the critical degrees k = 0 and k = n − 1. MSC: 32F10, 32F20, 32N15, 32T25
Abstract. Let Ω be a bounded, pseudoconvex domain of C n satisfying the "f -Property". The f -Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also occur for many relevant classes of domains of infinite type. In this paper, we prove the existence, uniqueness and "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equationThe idea of our proof goes back to Bedford and Taylor's [BT76]. However, the basic geometrical ingredient is based on a recent result by Khanh [Kha13].
We prove that for certain classes of pseudoconvex domains of finite type, the Bergman-Toeplitz operator T ψ with symbol ψ " K´α maps from L p to L q continuously with 1 ă p ď q ă 8 if and only if α ě 1 p´1 q , where K is the Bergman kernel on diagonal. This work generalises the results on strongly pseudoconvex domains byČučković and McNeal, and Abeta, Raissy and Saracco.
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