GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 2, 1997, 109-116

ON THE SOLUTION OF THE DIRICHLET PROBLEM FOR

AN ELLIPTIC EQUATION DEGENERATING ON THE BOUNDARY

A. JVARSHEISHVILI

Abstract. The Dirichlet problem for the equation

yΔW + αW_{y} = −F is studied in the semi-circle x^{2} + y^{2} < 1, y > 0. The restrictions on F are established under which the problem is uniquely solvable in the definite generalized sense.

Let K = {(x, y) : (x, y) ∈ ℝ^{2}, x^{2} + y^{2} < 1, y > 0}, and let ∂K be the boundary of K. Denote ΔW = W_{xx} + W_{yy} and consider the equations

yΔW + αW_{y} = −F (x, y), (x, y) ∈ K,  (1)
yΔW + αW_{y} = 0, (x, y) ∈ K.  (2)

The classical Dirichlet problem for these equations is formulated as follows: Find in the domain K a regular solution u = u(x, y) of the given equation which is continuous in \overline{K} = K ∪ ∂K and satisfies the boundary condition u(x, y) = f(x, y) for (x, y) ∈ ∂K, where f(x, y) is the known continuous function on ∂K. Bitsadze [1] investigated the existence and uniqueness of the solution of the Dirichlet problem for equations of more general type than (1), while the solutions of boundary value problems for equations of more general type were studied by Keldysh [2]. In [3] Nikol’sky and Lizorkin considered some boundary value problems for degenerating elliptic equations. Weinstein [4], [5] constructed fundamental solutions for equation (2). As is well known (see, e.g., [6]), using fundamental solutions one can construct a Green function of the Dirichlet problem for equation

109 (1) which has the form

1991 Mathematics Subject Classification. 35J70.

Key words and phrases. Elliptic partial differential equation, type degeneration, Dirichlet problem, Green function, generalized solution.

1072-947X/97/0300-0109$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

G(x, y; ξ, η; γ)y^{1−α} = [Z_{0}(x, y; ξ, η; γ) −

−ρ^{−γ}Z_{0}(x, y; ξ/ρ^{2}, η/ρ^{2}; γ)]y^{1−α} = [Z_{0} − Z_{1}]y^{1−α},

Z_{0} = \frac{1}{2π} ∫_{0}^{π} sin^{γ−1} θ[r^{2} + 4yη sin^{2} θ/2]^{−γ/2}dθ,

ζ, z ∈K with ζ = ξ + iη, z = x + iy, ρ=|ξ|, r =|z − ζ|, γ =2−α.

(3)

A unique solution for the classical Dirichlet problem of equation (1) can be represented by the following

Theorem A. Let 0 < α < 1, and let the function F satisfy the conditions

|F (x_{1}, y_{1}) − F (x_{2}, y_{2})| ≤ c{|x_{1} − x_{2}|^{δ} + |y_{1} − y_{2}|^{δ}},
0 < δ ≤ 1, (x_{k}, y_{k}) ∈ K, k = 1, 2.

Moreover, F (x, y) = 0 when 0 ≤ y ≤ ω, where ω > 0 depends on F. Then for any function f continuous on ∂K there exists a unique function W = W (x, y) which is continuous on \overline{K}, W (x, y) = f(x, y), (x, y) ∈ ∂K, and satisfies equation (1) in K. For all (x, y) ∈ K we have [7]

W (x, y) = y^{1−α} ∫_{K} F (ξ, η)G(x, y; ξ, η; 2 − α) dξ dη +

+ (1 − α)y^{1−α} ∫_{−1}^{1} f(t, 0)G(x, y; t, 0; 2 − α) dt −

− y^{1−α} ∫_{0}^{π} f(φ) sin φ \frac{∂G(x, y; cos φ, sin φ; 2 − α)}{∂n} dφ =

= J_{1}(F, x, y) + J_{2}(f, x, y) + J_{3}(f, x, y),  (4)

where \frac{∂G}{∂n} is the derivative with respect to the outer normal to ∂K and f[φ] = f(cos φ, sin φ), 0 ≤ φ ≤ π.

The aim of this paper is to show that expression (4) is a solution of the Dirichlet problem of equation (1) for a wider class of given functions F and f. In the sequel we shall use the following propositions.

Proposition 1. Let γ < 0. Then for all (x, y) ∈ K, (ξ, η) ∈ K we have

Z_{0}(x, y; ξ, η; γ) ≥ C(y · η)^{−γ/2} ∫_{0}^{π/4} θ^{−1}dθ = ∞,

where C ∈ (0, ∞) is a constant independent of x, y, ξ, η.

Proposition 2. For each (a, b) ∈ (0, 1]×(0, 1] and a fixed p ∈ (0, ∞) the inequality ℓ ≤ \frac{(a+b)^{p}}{a^{p}+b^{p}} ≤ L holds with ℓ = min[1, 2^{p−1}], L = max[1, 2^{p−1}].

Proposition 3. For all (x, y) ∈ K, (ξ, η) ∈ K, 0 < α < 2, there exists a constant C ∈ (0, ∞) such that

|G(x, y; ξ, η; 2 − α)| ≤ \frac{C}{r^{2−α}} .

Proposition 4. If z = x + iy ∈ K, ζ = ξ + iη ∈ K, ζ = ρe^{iφ}, ζ^{∗} = ξ^{∗} + iη^{∗} = \frac{1}{ρ}e^{iφ}, then |z − ζ| ≤ |z − ζ^{∗}|,

lim |ζ| |z − ζ^{∗}| = 1, lim |ζ|^{2} · y · η^{∗} = 0 as |ζ| → 0.

Below we assume that F is summable on K, F (x, y) = 0 if (x, y) ∉ \overline{K}, and for any point (x, y) ∈ \overline{K} we have

∫_{0}^{2π} |F (x + ρ cos φ, y + ρ sin φ)| dφ ≤ C[1 + | ln ρ^{−1}|] = C ln^{+} ρ, (L)

where C ∈ (0, ∞) is a constant depending on F only.

Theorem 1. Let 0 < α < 2, and let F satisfy the condition (L). Then for all (x, y) ∈ \overline{K} there exists, in the Lebesgue sense, an integral

∫_{E} |F | |G(x, y; ξ, η; 2 − α)| dξ dη, E ⊆ K,

which is absolutely continuous with respect to the measurable set E uniformly for all (x, y) ∈ \overline{K}. If, however, y = 0, 0 < α < 1, then for an arbitrary summable function F (x, y) we have J_{1}(F, x, 0) = 0, −1 < x ≤ 1.

Theorem 2. Let 0 < α < 1 and let F satisfy the condition (L). Then for all (x_{0}, y_{0}) ∈ ∂K, x_{0} ≠ ±1 we have

lim y^{1−α} ∫_{K} F (ξ, η)G(x, y; ξ, η; 2 − α) dξ dη = 0 as x → x_{0}, y → y_{0}. (5)

Thus for the function F satisfying on K condition (L) there exists an expression J_{1}(F, x, y) for all (x, y) ∈ K, and at every point (x_{0}, y_{0}) ∈ ∂K, x_{0} ≠ ±1, equality (5) is fulfilled. As for the function G(x, y; ξ, η; γ), we are aware of the following inequalities ([4], [5]).

Proposition 5. Let us introduce the functions h and N defined by the equalities

G(x, y; ξ, η; γ) = \frac{1}{2π} (y·η)^{−γ/2} ln r^{−1}+N(z, ζ, γ) = h(y, η) ln r^{−1}+N(z, ζ, γ)

with z, ζ ∈ K. For all 0 < ω ≤ y, η ≥ ω, 0 < r ≤ ω we have

\begin{matrix}|N| ≤ C_{ω}, |N_{y}| ≤ C_{ω}, |N_{x}| ≤ C_{ω}r| ln r^{−1}|, \\ |N_{ξ}| ≤ C_{ω}r| ln r^{−1}|, |N_{η}| ≤ C_{ω}r| ln r^{−1}|, \end{matrix}  (6)

where C_{ω} ∈ (0, ∞) is a constant depending on ω.

To study boundary properties of the integral

J_{3}(f, x, y) = −y^{1−α} ∫_{0}^{π} f[φ] \frac{∂G(x, y; cos φ, sin φ; 2 − α)}{∂n} sin φ dφ

let us introduce an angular neighborhood of the point e^{iφ_{0}} ∈Γ=∂K\(−1, 1), 0 < φ_{0} < π. We say that the point z belongs to Δ_{1}(φ_{0}, δ) if

| arg z − φ_{0}| < δ(1 − |z|), δ > 0,  (7)

where δ is an arbitrarily fixed positive number. From (7) we obtain that for z ∈ Δ_{1}(φ_{0}, δ),

(a) |z − e^{iφ_{0}}| ≤ C_{δ}(1 − |z|) with a constant C_{δ} > 0; (8)

(b) there exist constants m > 0, n > 0 depending on δ such that for all ψ ∈ (φ_{0} − δ, φ_{0} + δ)

m|z − e^{iψ}| ≤ |z − e^{iφ_{0}}| ≤ n|z − e^{iψ}|;  (9)

(c) m|x − cos ψ| ≤ |x − cos φ_{0}| ≤ n|y − sin ψ|, ψ ∈ (φ_{0} − δ, φ_{0} + δ). (10)

Let f be a function which is summable on [0, π] and assume that

f̃(φ_{0}) = sup [|( ∫_{0}^{φ_{0}−ε}+ ∫_{φ_{0}+ε}^{π} )\frac{f[φ]}{φ − φ_{0}} dφ|, ε > 0],

M(f, φ_{0}) = sup [|\frac{1}{h + k} ∫_{φ_{0}−h}^{φ_{0}+k}|f[φ]| dφ|, h > 0, k > 0],

\overline{J}_{3}(f, φ_{0}) = sup [|J_{3}(f, z)|, z ∈ Δ_{1}(φ_{0}, δ)].

Suppose that the arbitrarily fixed numbers introduced in (7) and in Proposition 5 are equal, i.e., δ = ω = λ > 0. Introduce the set I(λ) = (λ, \frac{π}{2} − λ) ∪ (\frac{π}{2} + λ, π − λ), |I(λ)| = π − 4λ > 0, λ < \frac{π}{8} . Let φ_{0} ∈ I(2λ), in particular, φ_{0} ∈ (2λ, \frac{π}{2} − 2λ). Let 0 < α < 1, and let f be summable on [0, π]. Using Proposition 5 and relations (7)–(10), we can estimate the expression

| ∫_{φ_{0}−λ}^{φ_{0}+λ}y^{1−α}f[φ] \frac{∂}{∂n}(h · ln r^{−1}) sin φ dφ| ≤

≤ C_{λ}[ ∫_{φ_{0}−λ}^{φ_{0}+λ}|f[φ]| | ln r^{−1}| dφ + | ∫_{φ_{0}−λ}^{φ_{0}+λ}f · h \frac{∂ ln r^{−1}}{∂n} dφ| ].

For z ∈ Δ_{1}(φ_{0}, λ), z = |z|e^{iψ} = x + iy, r = |z − e^{iφ}| we have

\frac{∂h ln r^{−1}}{∂n} = \frac{1}{φ − φ_{0}} + R(x, y, φ) ln |φ − φ_{0}|,

where R(x, y, φ) is the bounded measurable function of x, y, φ, z = x + iy ∈ Δ_{1}(φ_{0}, λ). Thus from the latter relations we get

| ∫_{φ_{0}−λ}^{φ_{0}+λ}y^{1−α}f[φ] \frac{∂}{∂n} (h · ln r^{−1}) sin φ dφ| ≤

≤ C_{λ}[| ∫_{φ_{0}−λ}^{φ_{0}+λ}\frac{f_{1}[φ]}{φ − φ_{0}} dφ| + M(f, φ_{0})] ≤

≤ C_{λ}[f̃_{1}(φ_{0}) + M(f, φ_{0})], |f_{1}[φ]| ≤ |f[φ]|, φ ∈ (φ_{0} − λ, φ_{0} + λ), (11)

where C_{λ} ∈ (0, ∞) is a constant depending on λ only. Further, using inequalities (6)–(10), we can easily get the following inequalities:

| ∫_{0}^{φ_{0}−λ}y^{1−α}f[φ] \frac{∂}{∂n} (h · ln r^{−1}) sin φ dφ| ≤ C_{λ}M(f, φ_{0}),

| ∫_{φ_{0}+λ}^{π} y^{1−α}f[φ] \frac{∂}{∂n} (h · ln r^{−1}) sin φ dφ| ≤ C_{λ}M(f, φ_{0}),

| ∫_{0}^{π} y^{1−α}f[φ] \frac{∂N}{∂n} sin φ dφ| ≤ C_{λ}M(f, φ_{0}), z = |z|e^{iψ}, r = |z − e^{iφ}|.

(12)

Analogous inequalities are also valid for the rest of the summands.

Let f be summable on [−1, 1], and let Δ_{1}(x_{0}, λ) be an angular neighborhood of the point x_{0} ∈ [−1, 1], i.e., z = x + iy ∈ Δ_{1}(x_{0}, λ) if

|x − x_{0}|/y ≤ λ.  (13)

Using the previous arguments and all the inequalities provided by (13), we obtain the following inequality: let f be summable on [−1, 1], 0 < α < 1, −1 < x_{0} < 1. Then

|\overline{J}_{2}(f, x_{0})| ≤ C_{λ}M(f, x_{0}),  (14)

where C_{λ} ∈ (0, ∞) is a constant depending on λ > 0 only, while

\overline{J}_{2}(f, x_{0}) = sup [|J_{2}(f, z)|, z ∈ Δ_{1}(x_{0}, λ)].

On the basis of (14), (12) and (11) we obtain

Proposition 6. Let 0 < α < 1. Let f be summable on ∂K = Γ ∪ (−1, 1) and let λ > 0 be an arbitrarily fixed number, Γ_{λ} = {e^{iφ}, φ ∈ I(λ)}. Then for any point t ∈ Γ_{λ} ∪ (−1, 1) we have

\overline{J}_{3}(f, t) + \overline{J}_{2}(f, t) ≤ C_{λ}[M(f, t) + f̃_{1}(t)], |f_{1}(t)| ≤ |f(t)|. (15)

Theorem 3. Let 0 < α < 1, and let f be summable on ∂K. Then for almost all t ∈ Γ ∪ (−1, 1) we have

lim[J_{2}(f, z) + J_{3}(f, z)] = f(t), z → t, z ∈ Δ_{1}(t, λ),  (16)

where λ > 0 is an arbitrarily fixed number.

Proof. By virtue of Theorem A and Theorem 2 for all t_{0} = (x_{0}, y_{0}) ∈ ∂K, x_{0} ≠ ±1 and any function g continuous on ∂K we have

lim[J_{2}(g, z) + J_{3}(g, z)] = g(t_{0}), z → t_{0}, z ∈ Δ_{1}(t_{0}, λ).  (17)

Let f be a function summable on Γ ∪ (−1, 1) and suppose that (16) is not fulfilled. Then there exist sufficiently small λ_{0} > 0 and ε_{0} > 0 and a set E, where either |E ∩ Γ_{λ_{0}}| ≥ 16ε_{0} or |(−1, 1) ∩ E| ≥ 16ε_{0}, such that for every point t_{0} ∈ E there exists a sequence z_{n} → t_{0}, z_{n} ∈ Δ_{1}(t_{0}, λ_{0}), n ≥ 1, and

|[J_{2}(f, z_{n}) + J_{3}(f, z_{n})] − f(t_{0})| ≥ 3ε_{0}, n ≥ 1.  (18)

On the other hand, for ε_{0} > 0 and for the function f there exists a function g continuous on Γ ∪ (−1, 1) such that

∫_{∂K} |f(t) − g(t)| dt ≤ \frac{ε_{0}^{2}}{C_{λ_{0}}} , |f(t) − g(t)| < ε_{0}, t ∈ E.  (19)

Next, on the basis of the well-known Kolmogoroff inequality ([8], [9]) we have

|{t ∈ Γ ∪ (−1, 1); f̃_{1}(t) + M(f, t) > μ}| ≤
≤ \frac{4}{μ} ∫_{∂K}(|f_{1}(t)| + |f(t)|) dt = \frac{8}{μ} ∫_{∂K} |f| dt, |f_{1}| ≤ |f|.  (20)

Using relations (15), (17)–(20), we get

16ε_{0} ≤ |E| ≤ |{t_{0} ∈ E; |J_{2}(f, z_{n}) + J_{3}(f, z_{n}) − f(t_{0})| ≥ 3ε_{0}}| ≤ ≤|{t ∈ E; |J_{2}(f − g, z_{n})+J_{3}(f −g, z_{n})|>ε_{0}}|+|{t ∈ E; |f −g|>ε_{0}}|+ + |{t ∈ E; |J_{2}(g, z_{n}) + J_{3}(g, z_{n}) − g(t_{0})| > ε_{0}}| < 8ε_{0}.

This contradiction proves the theorem.

For our further investigation let us consider a maximal function for the function of two variables (see [8]).

Let f be the function summable on K. Assume that for (x_{0}, y_{0}) ∈ K

M_{1}(f, x_{0}, y_{0}) = sup \{\frac{1}{|σ|} ∫_{σ} |f| dx dy : |σ| > 0\},

σ = {(x − x_{0})^{2} + (y − y_{0})^{2} ≤ δ^{2}}.

Using (6) and the relation Δ[u, v] = v · Δu + 2[u_{x} · v_{x} + u_{y} · v_{y}] + u · Δv, we can prove the following

Proposition 7. Let 0 < α < 1, and let the function Ψ satisfy the condition (L). Then for all

(x_{0}, y_{0}) ∈ K_{δ} = {(x, y) ∈ K; x^{2} + y^{2} ≤ (1 − δ)^{2}, y ≥ δ},

where δ > 0 is an arbitrarily fixed number, we have

\begin{matrix}|Δ[J_{1}(Ψ, x_{0}, y_{0})]| ≤ C_{δ}M_{1}(Ψ, x_{0}, y_{0}), \\ |\frac{∂}{∂y} J_{1}(Ψ, x_{0}, y_{0})| ≤ C_{δ}M_{1}(Ψ, x_{0}, y_{0}). \end{matrix}  (21)

Here the constant C_{δ} ∈ (0, ∞) depends on δ only.

According to Kolmogoroff [9] the inequality

|\{(x, y) ∈ K; M_{1}(Ψ, x, y) > μ\}| ≤ \frac{θ}{μ} ∫_{K} |Ψ| dx dy  (22)

holds, where the constant θ ∈ (0, ∞) does not depend on μ and Ψ.

Proposition 8. Let 0 < α < 1, and let f be summable on ∂K. Then for all (x, y) ∈ K_{δ} we have

|Δ[J_{2}(f, x, y) + J_{3}(f, x, y)]| ≤ C_{δ}M_{1}(f, x, y),  (23)
|\frac{∂}{∂y} [J_{2}(f, x, y) + J_{3}(f, x, y)]| ≤ C_{δ}M_{1}(f, x, y).  (24)

Repeating now the arguments from the proof of Theorem 3 and taking into account relations (24), (23), (22), (21) and (20), we obtain the main result.

Theorem 4. Let 0 < α < 1. Let F satisfy the condition (L), and let f be a function summable on ∂K. Then we have

yΔ[J_{1}(F, x, y) + J_{2}(f, x, y) + J_{3}(f, x, y)] +

+α\frac{∂}{∂y} [J_{1}(F, x, y) + J_{2}(f, x, y) + J_{3}(f, x, y)] = −F (x, y)

for almost all (x, y) ∈ K and

lim W (x, y) = lim[J_{1} + J_{2} + J_{3}] = f(x_{0}, y_{0}) as z = x + iy → x_{0} + iy_{0},

z ∈ Δ_{1}(φ_{0}, λ), e^{iφ_{0}} = x_{0} + iy_{0},

for almost all (x_{0}, y_{0}) ∈ ∂K, where λ > 0 is an arbitrarily fixed number.

References

1. A. V. Bitsadze, Equations of mixed type. (Russian) Izdat. Akad. Nauk SSSR, 1950.

2. M. V. Keldysh, On some cases of degeneration of equations of elliptical type on the boundary of a domain. (Russian) Dokl. Akad. Nauk SSSR 77(1951), No. 2, 131–133.

3. S. M. Nikol’skii and P. I. Lizorkin, Some inequalities for functions of weighted classes and boundary value problems with strong degeneracy on the boundary. Dokl. Akad. Nauk SSSR 159(1964), No. 3, 512–515.

4. A. Weinstein, Discontinuous integrals and generalized potential theory. Trans. Amer. Math. Soc. 63(1948), No. 2, 342–354.

5. A. Weinstein, On Tricomi’s equation and generalized axially simmetric potential theory. Acad. Roy. Belg. Bull. 37(1951), 346–356.

6. M. M. Smirnov, Degenerating elliptical and hyperbolic equations. (Russian) Nauka, Moscow, 1966.

7. G. V. Jaiani, On the deflections of thin wedge-shaped shells. Bull. Acad. Sci. Georgian SSR 65(1972), No. 3, 543–546.

8. A. N. Kolmogoroff, Sur les fonctions conjugées et les series de Fourier. Fund. Math. 7(1925), 24–29.

9. I. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series. Fund. Math. 32(1939), 112–132.

(Received 29.03.1993; revised version 13.07.1995)

Author’s address: Dept. of Differential Equations S. Orjonikidze Moscow Aviation Institute 4, Volokalamsk Highway, Moscow 125871 GSP Russia