On the Cauchy problem for operators with injective symbols in the spaces of distributions

Abstract

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.

The Cauchy problem for operators with injective symbol in the Lebesgue space L 2 in a domain

“…)), u j y ds(y) ( 2 2 ) for all ν ∈ N. Moreover (see the proof of Corollary 2), we know that F is determined by (19) with the coefficients (22). The partial sums of this series converge to F in L 2 (Ω, E) and their restrictions to D − converge to F − in L 2 (D, E), i.e., [16]). It is appropriate to note that we actually obtain the same Carleman kernels as those for f = 0 (see [9,Theorem 12.6]).…”

“…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.…”

“…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.…”

On a mixed problem for the parabolic Lamé type operator

“…Actually, the approach was invented for the investigation of the famous ill-posed Cauchy problem for elliptic equations (see, for instance, [8] for the Cauchy-Riemann operator, [9] for the Laplace equation, [10] for the elliptic Lamé operator and [11], [12], [15], for general systems with injective principal symbols).…”

Conditions for the % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaGafyOaIyRbaebaaaa!3AE5! $$ \bar \partial $$ -closedness of differential forms

“…Other conditions are examined in [2] for extension from a part of the boundary (i.e., solution of the Cauchy problem).…”