On the Cauchy problem for the elliptic complexes in spaces of distributions

Abstract

Let D be a bounded domain in R n with a smooth boundary ∂D. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A i } of first order operators. In particular, we describe traces on ∂D of tangential part τ i (u) and normal part ν i (u) of a (vector)-function u from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue spaces L 2 (D) we construct the approximate and exact solutions to the Cauchy problem with maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman's formulae for a (vector-) function u from the Sobolev space H 1 (D) by its Cauchy data τ i (u) on a subset Γ ⊂ ∂D and the values of A i u in D modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.Key words: Elliptic differential complexes, ill-posed Cauchy problem, Carleman's formula.It is well-known that the Cauchy problem for an elliptic system A is ill-posed (see, for instance, [1]). Apparently, the serious investigation of the problem was stimulated by practical needs. Namely, it naturally appears in applications: in hydrodynamics (as the Cauchy problem for holomorphic functions), in geophysics (as the Cauchy problem for the Laplace operator), in elasticity theory (as the Cauchy problem for the Lamé system) etc., see, for instance, the book [2] and its bibliography. The problem was actively studied through the XX century (see, for instance, [3] Differential complexes appeared as compatibility conditions for overdetermined operators (see, for instance, [13]). Thus, the Cauchy problem for them is of special interest. One of the first problems of this kind was the Cauchy problem for the Dolbeault complex (the compatibility complex for the multidimensional Cauchy-Riemann system), see [14]. The interest to it was great because of the famous example by H. Lewy of differential equation without solutions, constructed with the use of the tangential Cauchy-Riemann operator, see [15]. Recently new approaches to the problem were found in spaces of smooth functions (see [16], [17]).We consider the Cauchy problem in spaces of distributions with some restrictions on growth in order to correctly define its traces on boundaries of domains (see, for instance, [2], [18], [19], [20]). In this paper we develop the approach presented in [9] to study the homogeneous Cauchy problem for overdetermined elliptic partial differential operators. Instead we consider the nonhomogeneous Cauchy problem for elliptic complexes.

On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

“…Proof. Follows from [27, Theorems 2.8 and 5.2] for the case f = 0 and [6] for f = 0 because both the set S = ∂Ω where the boundary Cauchy data are defined and its complement ∂Ω b \ S = ∂Ω m are non empty and open in the relative topology.…”

“…Moreover, since the operator ∆ (b) is elliptic and its coefficients are real analytic, it admits a bilateral fundamental solutions to the operator, say ϕ b , in a neighbourhood of the compact Ω b . Let us indicate a solvability criterion and formulas for its solutions based on results from [27], [6] and [36]. With this purpose, we set…”

On Carleman-type Formulas for Solutions to the Heat Equation

“…equality the first equation in (4) is valid for U − . Formula (9) means that that the normal derivative of the volume potential G + Ω,0 (f ) is continuous if the point (x, t) passes over the surface Γ T . Therefore…”

“…He found principal ingredients, leading to the construction of integral formulas for its solution (Carleman formulas): a proper integral formula recovering the function via the data on the whole boundary, the uniqueness theorem and an effective tool, providing the analytic continuation from a domain to a larger one. This method was successfully used in the framework of Hilbert space methods to investigate the Cauchy problem for general elliptic systems of partial differential equations, see [6][7][8], and even to elliptic complexes of differential operators, see [9]. It provided both a solvability criterion and formulas for exact and approximate solutions.…”

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

“…[1], [2], [3], è äð.). Íà ñàìîì äåëå îíà ÿâëÿåòñÿ òèïè÷íûì ïðèìåðîì íåêîððåêòíîé çà-äà÷è äëÿ áîëåå îáùåãî êëàññà ýëëèïòè÷åñêèõ ñèñòåì (ñì., íàïðèìåð, [4], [5], [3]) èëè äàaeå ýëëèïòè÷åñêèõ äèôôåðåíöèàëüíûõ êîìïëåêñîâ ( [6], [7]). …”