On the solution of a class of partial differential equations

In the paper we study the solution and smoothness of the solution of one class of partial differential equations of higher order in bounded domain G ⊂ Rn satisfying the flexible λ-horn condition.


Introduction
In the paper [7], Besov generalized spaces of the form n i=0 1 ≤ p i ≤ ∞, 1 ≤ θ i ≤ ∞, (i = 0, 1, . . ., n), l i = (l i 1 , . . ., l i n ), l 0 j ≥ 0, l i i > 0, l i j ≥ 0, (j = i = 1, 2, . . ., n), are introduced and studied.In the paper [13], the generalized spaces of Besov-Morrey type n i=0 with the finite norm where 1 1 , . . ., l i n ), l 0 j ≥ 0, l i i > 0, l i j ≥ 0, (j = i = 1, 2, . . ., n) are introduced, differential and difference-differential properties of functions from there spaces, determined in n-dimensional domains and satisfying the flexible horn condition are studied.In the case l 0 = (0, . . ., 0), l i = (0, . . ., l i , . . ., 0), p i = p, θ i = θ space (1.1) coincides with the space B l p,θ (G) studied in [2], the space (1.2) coincides with the space B l p,θ,a,κ,τ (G) studied in [9], while in the case a = 0, τ = ∞ it coincides with space (1.1).Note that consideration of such a space enables to study higher order differential equations of general form.In other words, the obtained imbedding theorems in the form of Sobolev type inequality in spaces (1.1) and (1.2) enable to estimate higher order generalized derivatives than in the case of spaces B l p,θ (G) and B l p,θ,a,κ,τ (G).Example 1.1.Let us consider an equation of the form u (5) in our case the solution of this equation is sought in the space . One can look for the solution of equation (1.6) in the space W ), but then this solution will require additional derivatives, in other words, in our case the solution belongs to a wider class.
In this paper we study the existence, uniqueness and smoothness of one class of higher order partial differential equations.Earlier, a problem of smoothness of another kind equations was studied in [1, 3-6, 8, 10-12].

Main results
At first we give two theorems proved in the paper [13]. and (G, λ) in the domain G there exists the generalized derivative D ν f , for which the the following inequalities are valid: and In particular, µ i,0 = ∑ n j=1 l i j λ j − ν j λ j − λ j − κ j a j where T is an arbitrary number from (0, min (1,

any numbers and satisfy the conditions
but with κ replaced by κ, C 1 and C 2 are constants independent of f , moreover C 1 is independent also of T.
Let γ be an n-dimensional vector.

Theorem 2.2 ([13]
).Let the conditions of Theorem 2.1 be fulfilled.Then for µ i > 0 (i = 1, 2, . . ., n) the derivative D ν f satisfies on G the Hölder condition in the metrics L p with the exponent σ, more exactly, here σ is any number satisfying the inequalities: where σ 0 satisfies the same conditions that σ satisfies, but with µ i replaced by µ i,0 .
Let us consider the Dirichlet problem for a higher order partial differential equation, i.e. consider a problem of the form where it is assumed that G is a bounded n-dimensional domain with piecewise-smooth boundary ∂G, ν = (ν 1 , . . ., ν n ), where |ν| We assume that the coefficients a αβ (x) are bounded measurable functions in the domain G, a αβ (x) ≡ a βα (x) and for ξ ∈ R n we also assume that (2.11) Prove that there exists a unique generalized solution of problems (2.8) and (2.9).Consider for φ, ψ ∈ .
The variational problem is stated as follows: it is required to find the function φ . ., n are entire, that gives the least value to the integral F(φ) and is unique.Equation (2.8) is the Euler equation for the considered variational problem.
Then means that F(φ) is lower bounded on n i=0 L <l i > 2 (G).Show that there exists (G) such that lim m→∞ F(φ m ) = k and let ε > 0. Choose m ε so that for m ≥ m ε and µ = 0, 1, 2, . . .would hold F(φ m+µ ) < k + ε.Then noting ≥ k.Further, by direct calculations we show that I φ m+µ −φ m 2 < 4ε.From the ellipticity condition (2.10) it follows that , and hence it follows that k = lim m→∞ F(φ m ) = F(φ 0 ).Show that the function delivering minimum to the functions F(φ) in the space n i=0 L <l i > 2 (G) is unique and satisfies then again by the ellipticity condition (2.10), φ coincides with φ 0 as an element of n i=0 L <l i > 2 (G).By Theorem 1 in [7] we have: , minimizing the integral F(φ), satisfies the following equation: Now prove that the function φ 0 ∈ n i=0 L <l i > 2 (G), minimizing the integral F(φ) is the solution (generalized) of problem (2.8)-(2.9).For that we suppose that a α,β (x) are bounded in absolute value in the domain G together with its derivatives and the function f α has derivatives belonging to the space L 2 (G).Denote by Θ(t) some monotonically decreasing function on the interval 1  2 ≤ t ≤ 1, and possessing the following properties: The function infinitely differentiable and finite over the whole axis.Let η > 0 and G η = {y : ρ(y, R n \G) > η}, x be an arbitrary point of domain G and r = ρ(x, y).Following S. L. Sobolev [16] we introduce the function , and D (s) ψ| ∂G = 0 for any s > 0. Then from the expression (2.12) by definition of the generalized derivative it follows that, where The function ω r h i possesses all the properties of a kernel.Then for the function φ 0 (the solution of the variational problem) we can construct the Sobolev averaging φ 0,h i (x), i = 1, 2 over the ball h i , (i = 1, 2) centered at the point x: Then we can rewrite equality (2.12) in the form φ 0,h 1 (x) = φ 0,h 2 (x).Consequently, for h < η φ 0,h (x) = φ 0 (x).
As the average function φ 0,h (x) is continuous and has any order continuous derivatives, then φ 0 (x) also possesses these properties.Making integration in parts in the equality Hence, by arbitrariness of the function ψ(x) it follows Thus the solution of the variational problem from the class n i=0 L <l i > 2 (G) is also the solution of problem (2.8)-(2.9)and this solution is unique.Proof.Let at first all a αβ (x) ≡ 0, except the ones for which |α| = |β| = |l i | (i = 1, 2, . . ., n) and the left hand side equals zero.For any Θ(x) ∈ Π b (x 0 ), such that Θ ≡ 1 in the vicinity of ∂Π b (x 0 ) any polynomial

Assume also that f
and for arbitrary solution u(x) from the variational principle it follows that , and the coefficients P (x) are chosen so that By means of (2.1) and (2.2) we get

.14)
As A (u(x) − p(x), G) = A (u(x), G), then in view of (2.14) by induction we get for any δ ≤ b, and consequently, L<l i > 2 (Π b (x 0 )).Existence of such a solution is proved by the functional method.Put in (2.11) ϑ ≡ u b,x 0 then from (2.10) we get As u = u − u b,x 0 is the solution of homogeneous equation (2.8) then the following inequality is valid for it: (2.17) From inequalities (2.16) and (2.17 < ∞. Here using Theorems 2.1 and 2.2 we get that u(x) is continuous and satisfies the Hölder condition on G d .Finally, we consider equations (2.8) where there are nonzero coefficients at minor derivatives of the solution.Then we take these terms to the right hand side of the equation and in this case we get the desired result.
and from the condition of Theorem 2.3 it follows that µ i > 0, µ i,0 > 0 (i = 1, 2, . .., n), i.e. the conditions of Theorems 2.1 and 2.2 are fulfilled.Thus, by Theorem 2.1, u(x) is continuous, by Theorem 2.2, u(x) satisfies the Hölder condition on G d .Let a αβ = 0, except a αβ , for which |α| = |β| = |l i |, and the right hand sides of equation (2.8) be nonzero.Let u b,x 0 be a generalized solution of this equation in Π b (x 0 ) from