Duality by reproducing kernels

Тарханов

2016

Sib. Adv. Math.

Abstract. We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain D of R n for a second order elliptic differential operator A(x, ∂). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ∂) is of Robin type on ∂D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ⊂ ∂D and control the growth of solutions near Y . We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y . Moreover, we prove the completeness of root functions related to L.

Section: Proof If F ∈ Hmentioning

confidence: 88%

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

Тарханов

2016

Sib. Adv. Math.

“…This is the reason we use slightly different spaces; we follow [18] (cf. [20], [2,Chapters 1,9], [24]). More exactly, denote by…”

Section: Sobolev Spacesmentioning

confidence: 99%

“…It follows from [18, theorems 2.1 and 2.2] (see also [20], [24] for systems of equations) that Uniqueness Theorem and Existence Theorem are valid for problem (7) on the Sobolev scale H s (D, E), s ∈ Z for data w ∈ H(D, E, | · | s−2 ) and u 0 ∈ H s−1/2 (∂D, E). Denote by P (D) the operator mapping u 0 and w = 0 to the unique solution to the Dirichlet problem (7).…”

Section: Strong Traces On the Boundarymentioning

confidence: 99%

“…Putting the solution u(f ) into (23) and using formula (24) and Definition 5.1, we obtain for all v ∈ C ∞ comp (Ω, E i ): Also we would like to note that Theorem 5.2 gives not only the solvability conditions to Problem 4.1 but the solution itself, of course, if it exists (see (33)). It is clear that we can use the theory of functional series (Taylor series, Laurent series, etc.)…”

Section: A Homotopy Formulamentioning

confidence: 99%

See 1 more Smart Citation

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

Fedchenko

Complex Variables and Elliptic Equations

2013

Self Cite

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.

“…In addition to the standard scale Q H s .D; E/, s 2 N (see [1]), we consider also the following two scales of spaces adopted for studying the Dirichlet problem for strongly elliptic operators (cf. [14,15,19] and [22, Chapters 1 and 9]). We denote by C 1 m 1 .D; E/ the subspace in C 1 .D; E/ consisting of sections vanishing up to order m 1 on @D. For s 2 N and u 2 C 1 .D; E/ we define two types of negative norms as kuk s D sup However there are no reasons that Au 2 H s m .D; F / for an element u 2 H s .D; E/ and there are no reasons for elements of H s .D; E/ to have traces on @D. Thus, we introduce two more types of negative norms.…”

Section: Differential Operators and Sobolev Spacesmentioning

confidence: 99%

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

Shestakov

2011

Jiip

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.