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.

We show the geometric and analytic consequences of a general estimate in the∂-Neumann problem: a "gain" in the estimate yields a bound in the "type" of the boundary, that is, in its order of contact with an analytic curve as well as in the rate of the Bergman metric. We also discuss the potential-theoretical consequence: a gain implies a lower bound for the Levi form of a bounded weight.

Abstract2015 Mathematica Josephina, Inc. Under a potential-theoretical hypothesis named f-property which holds for all pseudoconvex domains of finite type and many examples of infinite type, we give a new method for constructing a family of bumping functions and hence plurisubharmonic peak functions with good estimates. The rate of lower bounds on the Kobayashi metric follows by the estimates of peak functions. The application to the continuous extendibility of proper holomorphic maps is given.

Abstract. We discuss, both for systems of complex vector fields and for sums of squares, the phenomenon discovered by Kohn of hypoellipticity with loss of derivatives.MSC: 32W05, 32W25, 32T251. Estimates for vector fields and sums of squares in R 3A system of real vector fields {X j } in T R n is said to satisfy the bracket finite type condition if (1.1) commutators of order ≤ h − 1 of the X j 's span the whole T R n .Explicitly:This system enjoys δ-subelliptic estimates for δ = 1 h and therefore it is hypoelliptic according to Hörmander [6]. (See also [5] and [10] for elliptic regularization which yields regularity from estimates.) This remains true for systems of complex vector fields {L j } stable under conjugation (both in C ⊗ T R n or C ⊗ T C n ) once one applies Hörmander's result to {Re L j , Im L j }. Stability under conjugation can be artificially achieved by adding {ǫL j } in order to apply Hörmander's theorem u. (Precision about ǫ and c ǫ is not in the statement but transparent from the proof.) On the other hand, by integration by partsThus if the type is h = 2, and hence δ = -norm is abbsorbed in the left: {ǫL j } can be taken back and one has 1 2 -subelliptic estimates for {L j }. The restraint h = 2 is substantial and in fact Kohn discovered in [9] a pair of vector fields {L 1 , L 2 } in R 3 of finite type k + 1 (any fixed k) which are not subelliptic but, nonetheless, are hypoelliptic. Precisely, in the terminology of [9], they loose k−1 2 derivatives and the related sum of squaresL 1 L 1 +L 2 L 2 looses k − 1 derivatives. The vector fields in question are L 1 = ∂z + iz∂ t and L 2 =z k (∂ z −iz∂ t ) in C ×R. Writing t = Im w, they are identified tō L andz k L for the CR vector fieldL tangential to the strictly pseudoconvex hypersurface Re w = |z| 2 of C 2 . Consider a more general hypersurface M ⊂ C 2 defined by Re w = g(z) for g real, and use the notations g 1 = ∂ z g, g 11 = ∂ z ∂zg and g 111 = ∂ z ∂z∂zg. Suppose that M is pseudoconvex, that is, g 11 ≥ 0 and denote by 2m the vanishing order of g at 0, that 1