GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 395-400

ON THE DIRICHLET PROBLEM FOR A SECOND ORDER

ELLIPTIC SYSTEM WITH DEGENERATION ON THE ENTIRE DOMAIN BOUNDARY

M. USANETASHVILI

Abstract. The solvability of the first boundary value problem is investigated for a second order elliptic system with degeneration on the entire domain boundary.

Consider a system of the form

L(u) ≡ Au_{xx} + 2Bu_{xy} + Cu_{yy} + au_{x} + bu_{y} + cu = 0  (1)

in the bounded simply connected domain D of the plane of variables x, y, where A, B, C, a, b ∈ H^{α}(\overline{D}) ∩ C^{3}(D) are given scalar functions and c is a given negatively definite n×n-matrix from the class H^{α}(\overline{D})∩C^{3}(D), n > 1, i.e.,

(ξ, c(x, y)ξ) ≤ c_{0}(ξ, ξ), c_{0} = const < 0 ∀ξ ∈ R^{n}, ∀(x, y) ∈ \overline{D}, (2)

u = (u_{1}, u_{2}, …, u_{n}) is the desired n-dimensional vector function, and (·, ·) is a scalar product.

It will be assumed below that system (1) is elliptic in the domain D and degenerates on the boundary Γ = ∂D, i.e.,

(B^{2} − AC)|_{D} < 0,  (3)
(B^{2} − AC)|_{∂D} = 0.  (4)

Obviously, since system (1) is elliptic in D, it can be assumed without loss of generality that A|_{D} > 0.

Let us write the equation ∂D in the form H(x, y) = 0, where H|_{D} > 0, H ∈ C^{2}(\overline{D}), H|_{Γ} = 0, ▽H|_{Γ} ≠ 0.

A vector u of the class C^{2}(D) satisfying system (1) in the domain D is called a regular solution of this system.

395

1991 Mathematics Subject Classification. 35J70.

Key words and phrases. Degenerating elliptic system, Dirichlet problem, extremum principle, barrier function.

1072-947X/99/0700-0395$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

Let us pose the Dirichlet problem for the system (1). Find, in the domain D, a regular solution of equation (1) which is continuous in the closed domain \overline{D} and satisfies the boundary condition

u|_{Γ} = f,  (5)

where f = (f_{1}, f_{2}, …, f_{n}) is a given continuous vector function on Γ.

Note that the Dirichlet problem was investigated in [1], [5] for some specific classes of second order elliptic systems.

When investigating the Dirichlet problem, the following two cases are to be distinguished:

(AH_{x}^{2} + 2BH_{x}H_{y} + CH_{y}^{2})|_{Γ} ≠ 0,  (6)
(AH_{x}^{2} + 2BH_{x}H_{y} + CH_{y}^{2})|_{Γ} = 0.  (7)

Remark 1. Equality (4) together with condition (6) imply that equation (1) parabolically degenerates on Γ and, at every point of the boundary, the tangential direction does not coincide with the characteristic direction. Conditions (4) and (7) are equaivalent to the fact that at the points of the boundary either the equation order degenerates or parabolic degeneration takes place and the characteristic direction coincides with the tangential direction.

When condition (3) is fulfilled, the following extremum principle [1] takes place: the norm R(x, y) = \sqrt{(u, u)} of a regular solution u of system (1) in the domain D cannot attain a nonzero relative maximum at any point (x, y) ∈ D.

The uniqueness of the solution of the Dirichlet problem for system (1) follows from the above-mentioned extremum principle.

To construct the solution of the Dirichlet problem we shall use the Wiener method. Using arbitrary vector functions f of the class C(\overline{D}) ∩ C^{3}(D) that are continuously extendable onto D, let us construct a sequence of domains D_{h} increasing as h → 0 and having a smooth boundary of the class C^{2} such that \overline{D}_{h_{1}} ⊂ D_{h_{2}} for h_{1} > h_{2}

Let u_{h}(x, y) be the solution of the Dirichlet problem for system (1) in D_{h} that coincides with f on the boundary of D_{h}. As is known, the solution u_{h}(x, y) exists and is unique, since in \overline{D}_{h} system (1) does not degenerate [6]. By virtue of (2) in D_{h} we have the inequality ∥u_{h}∥ ≤ M, where M = max ∥f(x, y)∥ in \overline{D}. Let us show that the set of functions \{u_{h}(x, y)\} is compact within D. Indeed, let h_{0} be an arbitrary fixed value of h. For h ≤ h_{0} the set {u_{h}(x, y)} will be uniformy bounded in D_{h_{0}},

∥u_{h}(x, y)∥ ≤ M.  (8)

On the other hand, we have the representation [6]

u_{h}(x, y) = ∫_{∂D_{h_{0}}} K_{h_{0}}(x, y; s)u_{h}(s)ds, (x, y) ∈ D_{h_{0}},  (9)

where the kernel is itself a regular solution of system (1) with respect to the variables x, y.

This immediately implies the equicontinuity of the family {u_{h}(x, y)}. From (8) and (9) we obtain the equicontinuity of the set of function {u_{h}} in D_{h_{0}}. According to Arcela’s theorem [1] the latter set may contain a subsequence uniformly converging to some function u(x, y) which by virtue of (9) is the solution of system (1) in the domain D. It remains to find out whether the constructed solution coincides with f on Γ.

We shall show that in the case of (6) for each point Q ∈ Γ = ∂D there exists a function v(x, y) called a barrier and possessing the following properties: (a) being continuous in some neighborhood of this point ω_{Q} = \{P ∈ \overline{D} : |P − Q| < ϵ\}; (b) being equal to zero at the point Q; (c) v(x, y) > 0 in ω_{Q}\Q; (d) everywhere in this neighborhood satisfying the condition

L_{1}^{0}(u) ≡ Au_{xx} + 2Bu_{xy} + Cu_{yy} + au_{x} + bu_{y} < 0.

In the considered case of (6) the function

v(x, y) = (x − x_{0})^{2} + (y − y_{0})^{2} + H^{β}(x, y), 0 < β < 1,

can serve a barrier. Indeed, the function v(x, y) evidently satisfies the conditions (a), (b) and (c). Let us verify the condition (d). Substituting the expression for v(x, y) into L_{1}^{0}(v), we obtain

L_{1}^{0}(v) =β(β − 1)H^{β−2}(AH_{x}^{2} + 2BH_{x}H_{y} + CH_{y}^{2}) +
+ βH^{β−1}(AH_{xx} + 2BH_{xy} + CH_{yy} + aH_{x} + bH_{y}) +
+ 2A + 2C + 2a(x − x_{0}) + 2b(y − y_{0}).  (10)

Hence we immediately conclude that by virtue of 0 < β < 1 there exists a neigborhood σ_{Q} of the point Q such that L_{1}^{0}(v) < 0.

Since the function f(P) is continuous, for given positive ϵ we can find a semicircular neighborhood σ′_{Q} ⊂ σ_{Q} of the point Q such that the inequality

∥f(P) − f(Q)∥ < ϵ, P ∈ σ′_{Q}, 0 < ϵ < 1,  (11)

be fulfilled.

Consider two functions v_{1}(P) = ϵ + Kv(P), K > 0, u_{h}^{∗}(P) = u_{h}(P) − f(Q), where P ∈ σ′_{Q}. In the domain ω_{h} = σ′_{Q} ∩ D_{h}, where h is a sufficiently small positive number, we have

sup_{(x,y)∈ω_{h}} L_{1}^{0}(v) ≤ α_{0} = const < 0,  (12)
L_{1}^{0}(u_{h}^{∗}) = L_{1}^{0}(u_{h}), ∥u_{h}^{∗}∥_{C(ω_{h})} ≤ 2M, M = max_{\overline{D}} ∥f∥.  (13)

We set g(P) = (u_{h}^{∗}(P), u_{h}^{∗}(P)), P ∈ ω_{h}. By virtue of (3), (10), (12), (13) and sufficiently large K we obtain

L_{1}^{0}(v_{1} − g) = KL_{1}^{0}(v) − 2[A(u_{hx}^{∗}, u_{hx}^{∗}) + 2B(u_{hx}^{∗}, u_{hy}^{∗}) + C(u_{hy}^{∗}, u_{hy}^{∗})] − −2(u_{h}^{∗}, Au_{hxx}^{∗} + 2Bu_{hxy}^{∗} + Cu_{hyy}^{∗} + au_{hx}^{∗} + bu_{hy}^{∗}) ≤

≤KL_{1}^{0}(v)−2(u_{h}(P)−f(Q), L_{1}^{0}(u_{h}))=KL_{1}^{0}(v)−2(u_{h}(P)−f(Q), −cu_{h})≤

≤ Kα_{0} − 2(f(Q), cu_{h}) = KL_{1}^{0}(v) + 2(u_{h}, cu_{h}) − 2(f(Q), cu_{h}) ≤

≤ Kα_{0} + 2∥f(Q)∥ ∥c∥ ∥u_{h}∥ ≤ Kα_{0} + 2M^{2}∥c∥ < 0.  (14)

Let us now find out which sign of v_{1} − g takes place on the domain boundary ω_{h}; ∂ω_{h} = γ_{h} ∪ γ_{1h}, where γ_{1h} = ∂ω_{h} ∩ ∂D_{h}, γ_{h} = ∂ω_{h}\γ_{1h}. By virtue of (11) we have

\begin{matrix}g|_{γ_{1h}} =(u_{h}(P) − f(Q), u_{h}(P) − f(Q))|_{γ_{1h}} =∥(f(P) − f(Q)∥^{2} <ϵ, \\ v_{1}|_{γ_{1h}} =ϵ + Kv(P)|_{γ_{1h}} > ϵ+K min_{γ_{1h}} v(P), g|_{γ_{h}} ≤4M^{2}. \end{matrix} (15)

By (15), for sufficiently large K we have

(v_{1} − g)|_{γ_{1h}} = (ϵ + Kv − g)|_{γ_{1h}} > ϵ + K min_{(x,y)∈γ_{1h}} v − ϵ = K min_{(x,y)∈γ_{1h}} v > 0,

(v_{1} − g)|_{γ_{h}} = (ϵ + Kv − g)|_{γ_{h}} ≥ ϵ + K min_{(x,y)∈γ_{h}} v − 4M^{2} > 0.

By (14) and (15) and applying the extremum principle we have [1] (v_{1} − g)|_{ω_{h}} ≥ 0, i.e., g ≤ v_{1} throughout the domain ω_{h}, or

(u_{h}(P) − f(Q), u_{h}(P) − f(Q)) ≤ ϵ + Kv(P).  (16)

Due to the properties of the barrier v there exists δ = δ(ϵ) > 0 such that for ∥P − Q∥ < δ and P ∈ σ′_{Q} we have

Kv(P) < ϵ.  (17)

From (16) and (17) we find that ∥u(P) − f(Q)∥ ≤ \sqrt{2ϵ} for ∥P − Q∥ < δ, which was to be proved.

Thus the following assertion is true.

Theorem 1. The Dirichlet problem (1), (5) always has a unique solution.

When considering the case of (7) it will be assumed that in some neighborhood of the boundary Γ there is a representation

AH_{x}^{2} + 2BH_{x}H_{y} + CH_{y}^{2} = H^{p}G,  (18)

where p = const > 0, G is a positive, continuous and bounded function in this neigborhood.

Lemma 1. If there exists a function W (P), which is positive in \overline{D}, uniformly converging to infinity as ρ(P, ∂D) → 0 and satisfying the inequality L_{1}^{0}(W) < 0, then system (1) has only a trivial solution in the class of bounded vector functions in D.

Proof. Indeed, let u(x, y) be a bounded regular solution of system (1) in D. Consider the expression L_{1}^{0}(ϵW − (u, u)) = ϵL_{1}^{0}(W) − L_{1}^{0}((u, u)). By calculating L_{1}^{0}((u, u)) we obtain

L_{1}^{0}((u, u)) = 2[A(u_{x}, u_{x}) + 2B(u_{x}, u_{y}) + C(u_{y}, u_{y})] +
+2(u, Au_{xx} + 2Bu_{xy} + Cu_{yy} + au_{x} + bu_{y}) ≥ 0,  (19)

since by virtue of (3), A|_{D} > 0, and the Cauchy–Bunyakovskii inequality we have

A(u_{x}, u_{x}) + 2B(u_{x}, u_{y}) + C(u_{y}, u_{y}) ≥ A∥u_{x}∥^{2} − 2|B||(u_{x}, u_{y})| + C∥u_{y}∥^{2} ≥

≥ A∥u_{x}∥^{2} − 2\sqrt{A}\sqrt{C}∥u_{x}∥ · ∥u_{y}∥ + C∥u_{y}∥^{2} = (\sqrt{A}∥u_{x}∥ − \sqrt{C}∥u_{y}∥)^{2} ≥ 0.

The conditions of Lemma 1 and (19) lead to L_{1}^{0}(ϵW − (u, u)) < 0, which by virtue of the extremum principle gives that the function ϵW − (u, u) cannot have a negative minimum in D, and since its limit value on the boundary are positive, we have (u, u) ≤ ϵW throughout D. Hence, since ϵ > 0 is arbitrary, it folows that ∥u∥ = 0 in D.

Taking into account (18), expression (10) can be rewritten as

L_{1}^{0}(v) = β(β − 1)H^{β+p+2}G + βH^{β−1}L_{1}^{0}(H) + 2A + 2C +
+2a(x − x_{0}) + 2b(y − y_{0}).

Theorem 2. If (2) holds and one of the conditions (i) 0 < p < 1; (ii) p = 1, (1−IG^{−1})|_{Γ} > 0, I = L_{1}^{0}(H); (iii) 1 < p < 2, I|_{Γ} ≤ 0; (iv) p ≥ 2, I|_{Γ} < 0 is fulfilled, then the Dirichlet problem has a solution.

Proof. The existence of a solution of the Dirichlet problem follows from the existence of a barrier function. For 0 < p < 1 the sign of L_{1}^{0}(v) coincides with the sign of β(β − 1)H^{p+β−2}, i.e., L_{1}^{0}(v) < 0. If the condition (ii) is fulfilled, then the sign of L_{1}^{0}(v) coincides with the sign of β[(β−1)G+I]H^{β−1} and, assuming that β < (1 − IG^{−1})|_{Γ}, we shall have L_{1}^{0}(v) < 0. Under the condition (iv) the sign of L_{1}^{0}(v) coincides with the sign of βH^{β−1}I, i.e., L_{1}^{0}(v) < 0.

Lemma 2. Let any one of the conditions (i) p = 1, IG^{−1}|_{Γ} ≥ 1; (ii) 1 < p < 2, I|_{Γ} > 0; (iii) p ≥ 2, I|_{Γ} ≥ 0 be fulfilled and, at each point P_{0} ∈ D, either the function Φ(P_{0}) = (Ag_{x}^{2} + 2Bg_{x}g_{y} + Cg_{y}^{2})(P_{0}) ≠ 0, where g ∈ C^{2}(\overline{D}), g > 1, or Φ(P_{0}) = 0 and c(P_{0}) < 0. Then there exists a function W (x, y) possessing the following properties: (a) W (P) > 0, P ∈ \overline{D}, (b)

lim_{ρ(P,∂D)→0} W (P) = +∞; (c) L_{1}^{0}(W) < 0, P ∈ D [7].

We can take as W, for instance, an expression W (x, y) = − log H(x, y) − g^{n}(x, y) + K_{0} for some natural n and positive constant K_{0}.

Lemmas 1 and 2 give rise to the following assertion.

Theorem 3. When the conditions of Lemma 2 are fulfilled, system (1) has only a trivial solution in the class of bounded vector functions in D.

References

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2. V. P. Didenko, The first boundary value problem for some elliptic systems of differential equations with degeneration on the boundary. (Russian) Sibirsk. Mat. Zh. 6(1965), No. 4, 814–831.

3. V. P. Didenko, On some elliptic systems degenerating on the domain boundary. (Russian) Dokl. Akad. Nauk SSSR 143(1962), No. 6, 1250– 1253.

4. E. A. Baderko, To the Dirichlet problem for degenerating elliptic systems. (Russian) Differentsial’nye Uravneniya 5(1969), No. 1, 131–140.

5. O. M. Jokhadze, Extremum principle for some classes of second order elliptic and parabolic systems. (Russian) Nonlocal boundary problems and related problems of mathematical biology, informatics and physics. (Abstr. intern. conf., 1996), 30–31, Kabard.–Balkarian Center of Russian Academy of Sciences, Nalchik, 1996.

6. A. V. Bitsadze, Boundary value problems for elliptic equations of second order. (Russian) Nauka, Moscow, 1966.

7. M. A. Usanetashvili, On the solvability of the Dirichlet problem for second order elliptic equations with degeneration on the entire boundary. (Russian) Differentsial’nye Uravneniya 21(1985), No. 1, 145–148.

(Received 18.12.1996)

Author’s address:

Department of Mathematics (99)

Georgian Technical University

77, M. Kostava St., Tbilisi 380075, Georgia