Let D be a bounded domain in the n-dimensional Euclidian space (n 2) having smooth boundary @D. We indicate appropriate Sobolev spaces with negative smoothness in D in order to consider the non-homogeneous ill-posed Cauchy problem for an overdetermined operator A with injective symbol. We prove that elements of the indicated Sobolev spaces have traces on the boundary. This easily leads to a weak formulation of the Cauchy problem and to the corresponding uniqueness theorem. We also describe solvability conditions of the problem and construct its exact and approximate solutions. Namely, we obtain the Carleman formula recovering a vector-function u from the indicated negative Sobolev class via its Cauchy data on an open connected set @D and values of Au on the domain D. Some instructive examples are considered.
β‐Amino acids and their derivatives are important building blocks for the preparation of various bioactive compounds and materials. We developed a highly efficient method for the synthesis of β3‐tryptophan derivatives based on enantioselective Friedel‐Crafts alkylation of indoles with phthaloyl‐protected aminomethylenemalonate in the presence of chiral Cu()OTf2/iPrBox complex as a catalyst. A wide range of indoles with electron‐donating and electron‐withdrawing substituents gave the desired products in high yields (up to 99 %) and excellent enantioselectivities (up to 99 % ee). In the case of pyrrole the Friedel‐Crafts product was obtained in up to 90 % yield and up to 82 % ee.
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Let D be a bounded domain in R n (n ≥ 2) with infinitely smooth boundary ∂D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L 2 (D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman's formula that restores a (vector-)function in L 2 (D) from the Cauchy data given on a relatively open connected set Γ ⊂ ∂D and the values Au in D whenever the data belong to L 2 (Γ) and L 2 (D) respectively.As is known, the Cauchy problem for an elliptic system A is ill-posed in general (for instance, see [1]). However, this problem arises rather naturally in applications as the Cauchy problem for holomorphic functions in hydrodynamics, for the Laplace operator in geophysics, for the Lame system in elasticity, and so on (for instance, see [2] and the bibliography therein). The problem was extensively studied during the twentieth century (for instance, see [3-10], etc.); in particular, it provided a stimulus to the development of the theory of conditionally well-posed problems.In the present article, we follow the approach that was developed in [9] for the homogeneous Cauchy problem in the case of overdetermined elliptic systems (cf. [11][12][13]). We examine the inhomogeneous Cauchy problem. Of course, it is easy to see that for systems with invertible principal symbol these problems are equivalent (at least, locally). However, the equivalence for overdetermined systems holds provided that information is available about solvability of the equation Au = f in the solution domain. For example, these Cauchy problems are not equivalent for the operators with constant coefficients if this domain does not possess appropriate convexity properties with respect to A (for instance, see [14]). Furthermore, if the coefficients of A are C ∞ -smooth (and not analytic) then at present there are no general results even on local solvability of the equation Au = f (for instance, see [15, the Introduction and Section 3]).The above approach is easily implemented when the Cauchy problem is solved in the Sobolev spaces with sufficiently large smoothness exponents (see [16]). However, since inner products in these spaces are cumbersome, in our opinion, the Lebesgue space is the most appropriate for the constructive determination of exact and approximate solutions to the problem. Moreover, this space allows us to essentially broaden the classes of boundary data and solutions to the Cauchy problem. In this article, we consider a generalized statement of the Cauchy problem without any conditions on the "convexity" of the solution domain. Basic NotationLet X be a C ∞ -smooth manifold of dimension n ≥ 2 with smooth boundary ∂X; we assume that X is embedded into a (closed) smooth manifold X of the same dimension.
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