The necessary and sufficient conditions for the solutions of elliptic problems

In engineering applications many problems in R are characterized by partial differential equations with piecewise analytic data such as nonsmooth domains, abruptly changes of types of boundary conditions, piecewise analytic coefficients and boudary conditions, etc., for instance, the physical domains of structral mechanical problems often have edges and vertices, interfaces between di?erent materials and material cracks [13-15]. The solutions of these problems have strong singularities at the edges and vertices and around the cracks, which make the conventional numerical approximation extremely difficult and inefficient. Hence comprehensive study on the regularity of the solutions of elliptic problems in R with piecewise analytic data is of great significance not only for theoretical reasons but also for the desigh of effective computations and the optimal convergence of numerical method for these problems [16-20].

In engineering applications many problems in R 3 are characterized by partial differential equations with piecewise analytic data such as nonsmooth domains, abruptly changes of types of boundary conditions, piecewise analytic coefficients and boudary conditions, etc., for instance, the physical domains of structral mechanical problems often have edges and vertices, interfaces between di?erent materials and material cracks [13][14][15]. The solutions of these problems have strong sin-gularities at the edges and vertices and around the cracks, which make the conventional numerical approximation extremely difficult and inefficient. Hence comprehensive study on the regularity of the solutions of elliptic problems in R 3 with piecewise analytic data is of great significance not only for theoretical reasons but also for the desigh of effective computations and the optimal convergence of numerical method for these problems [16][17][18][19][20]. E-mail: Gharibmusa@gmail.com These regularity results are important and useful for the regularity theory for elliptic problems on nonsmooth domains and for solving these problems by conventional numerical approaches. But these results do not characterize sufficiently the class of solutions of the problems in applications.
The conformal mapping and boundary value problems for harmonic functions; see, Lubuma [21], Maz'ya [22] or Maz'ya [23] was the earliest impetus. And the physical applications; examples can be found in [24], [25] and other standard monographs see also [26] and [27]. Also those problems play a role in numerical analysis, particulary in the study of the accuracy of finite element and finite difference approximation, acceleration of convergence, general convergence analysis, subtraction of singularities and other numerical techniques [28].

Theorem
Consider the initial Dirichlet problem u(x, y, 0) = K(x, y), The necessary and sufficient conditions for the solutions of problem (1) -(2) to belong to C m+2+α (Ω) are: The necessity of conditions (i) and (ii) is obvious. To prove the necessity of condition (iii), we notice that ( γq p ) cos γq−p ω sin p ω u (p,γq−p) (x, y, t), and form equation (5) we obtain

Proof
The sufficiency of the conditions depends on a constructed function (c.f. lemma 2). This function is constructed to remove the discontinuities at the boundary Γ 0 . These discontinuities are appeared where we continue the solution by symmetry across the boundary Γ 0 . The new boundary functions and right hand side of the equation are then shown to satisfy the compatibility condition (7 ). By repeating this process we can extend the domain until the angle is π, with the boundary function belonging to C m+2+α .
We first prove By the function f (x, y, t) ∈ C m+α (Ω), we can construct functions f p (x, y, t), P = 0, 1, · · · , m, defined on the whole plane and having the properties. IV. x Proof For simplicity we prove the lemma only for p = 0, the same proof can be used for p = 1, 2, · · · , m.
Changing s to −s in (3), and noting that K(s) is even, we obtain property II.
It is clear that f (x, y, t) ∈ C α (Ω), for where 1 −1 K(s)ds = 1, and for any K ≤ m, and this may be shown as before to belong to C α (Ω. This proves property III. Equation (5) may be written as Then which, after integrating by parts k times, gives where As before this may be shown to belong to C m+α (Ω) i.e. x k ∂ k f (x,y,t) ∂x k 1 ∂y k 2 ∂t k 3 ∈ C m+α (Ω). Property IV follows since where s 0 ∈ (−1, 1) and, α k = 1 −1 [K(s)(s) 2k ]ds > 0. Letting x tends to zero we obtain

From this, property V follows with
Then the lemma is proved.
Then, we get Noting that Then, the proof of lemma is complete.
Then R ∈ C m+α (Ω), ξ(0, t) = ψ(0, t), and form the compatibility condition (III), we get Then the functions g (x, y, t), ψ and φ 1 satisfy the compatibility conditions Thus to prove the theorem, it is sufficient to consider the problem where f and ψ satisfy the compatibility condition consider the function g(x, y, t) ∈ C m+2+α (Ω) given by g(x, y, t) = m p=0 g p (x, y, t), where g p (x, y, t) are the function constructed in lemma 2. Now Consider the function V (x, y, t) = U (x, y, t)−g(x, y, t). This function satisfies in Ω the initial Dirichlet problem ∆V − V t = h(x, y, t), where h(x, y, t) = f (x, y, t) − R (x, y, t), V = ψ (x, t) = ψ(x, t) − g(x, 0, t) on Γ 1 .