On the problem of determining the parameter of an elliptic equation in a Banach space

. The boundary value problem of determining the parameter of an elliptic equation −

c Vilnius University, 2014 Let us F (E) is the some space of smooth E-valued functions on [0, T ].We say that (u(t), p) is the solution of problem (1) in F (E) × E 1 if the following conditions are fulfilled: (i) u (t), Au(t) ∈ F (E), p ∈ E 1 ⊂ E; (ii) (u(t), p) is satisfies the equation and boundary conditions (1).
Problem (1) was considered in [13].The solvability and uniqueness under some condition for operator A were proved.The well-posedness of this problem in a Hilbert space with the self-adjoint operator A was studied in the paper [3].
Applying ( 2), ( 4), ( 5), ( 6) and ( 7), we can obtain the following presentation of the solution (u(t), p) of inverse problem (1): is the Banach space obtained by completion of the set of smooth E-valued functions ρ(t) on [0, T ] in the norm From positivity of operator A in Banach space E it follows that B = A 1/2 is strongly positive operator in E. Hence, the operator −B is a generator of an analytic semigroup exp{−tB} (t 0) with exponentially decreasing norm (see [29]) as t → ∞, i.e., for some M (B) ∈ [1, +∞), α(B) ∈ (0, +∞) and t > 0, the following estimates are valid: Let us give lemmas which we need in future.
In the present paper, the exact estimates for the solution of problem (1) in the Hölder norms are obtained.In applications, the exact estimates are established for the solution of the boundary value problems for multi-dimensional elliptic equations with the parameter.

C α,α
0T (E)-estimates for the solution of (1) Then the following estimates are satisfied for the solution where M is independent on α, u(0), u(λ), u(T ) and f (t).

So, we have proved
M Au(0) E .
In a similar manner, we can establish estimates for S 2 (t) and S 3 (t): M Au(T ) E .
We denote by C α,α 0T (E) = C α,α 0T ([0, T ], E), 0 < α < 1, the Banach space obtained by completion of the set of smooth E-valued functions ρ(t) on [0, T ] in the norm In exactly similar manner as Theorem 1, we can establish the following result.

Applications
In this section, we consider applications of abstract Theorems 1 and 2. First, we consider the boundary value problem on the range {0 t T, x ∈ R n } for 2m-order multidimensional elliptic equation where a r (x) and ϕ(x), ψ(x), ξ(x) are given sufficiently smooth functions and a r (x) > 0, δ > 0 is the sufficiently large number.Suppose that the symbol of the differential operator of the form acting on functions defined on the space R n , satisfies the inequalities for ζ = 0. So, we have boundary value problem in a Banach space E = C µ (R n ) of all continuous bounded functions defined on R n satisfying a Hölder condition with the indicator µ ∈ (0, 1) and with a strongly positive operator A x = B x + δI defined by (31).
The proof of Theorem 4 is based on the abstract Theorems 1, 2 and the positivity of the operator A x in C µ (R n ), the structure of the fractional spaces E α ((A x ) 1/2 , C(R n )) [29] and the coercivity inequality for an elliptic operator A x in C µ (R n ) [30].

Conclusion
In the present paper, the well-posedness of the boundary value abstract elliptic problem with the unknown parameter in Holder spaces with a weight is established.In practice, new Schauder type exact estimates in Holder norms for the solution of three boundary value problems for elliptic equations with the unknown parameter are obtained.Moreover, applying the result of the monograph [29] the high order of accuracy two-step difference schemes for the numerical solution of the boundary value elliptic problem with the unknown parameter can be presented.Of course, the coercive stability estimates for the solution of these difference schemes have been established without any assumptions about the grid steps.