V. P. GLUSHKO

ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION OF CERTAIN BOUNDARY-VALUE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

(Presented by Academician A. Yu. Ishlinskii, 28 XII 1964)

In the paper [1], M. Schechter proposed a sufficiently general method for proving the existence and uniqueness of the solution of the Dirichlet problem for degenerate elliptic equations of second order in the case when the boundary conditions on the “boundary of degeneration” are retained. In the present paper this method is generalized to the case when the boundary conditions on the boundary of degeneration are removed, and also to certain other boundary-value problems. As a special case, this yields the existence and uniqueness of the solution for the equations and boundary-value problems considered in [2–5].

1. Let \(D\) be a connected bounded domain in \(E_n\), lying in the half-space \(x_n \ge 0\). The boundary of the domain is \(\dot D=\dot D' \cup \dot D''\), where \(\dot D'=\dot D\cap(x_n>0)\), \(\dot D''=\dot D\cap(x_n=0)\); \(\dot D'\) belongs to the class \(C^{2+\alpha}\). In the domain \(D\) consider the differential equation of second order

\[ Lu \equiv \sum_{i,j=1}^{n} a_{ij}\frac{\partial^2 u}{\partial x_i\partial x_j} +\sum_{i=1}^{n} a_i\frac{\partial u}{\partial x_j} +au=f, \tag{1} \]

where \(L\) is an operator elliptic at every point of the domain \(D\cup \dot D'\). We shall assume that the coefficients of \(L\) belong to the class \(C^\alpha\) in any domain \(\overline{D}_\lambda=\overline D\cap(x_n\ge \lambda)\) for \(\lambda>0\).

On the boundary \(\dot D'\) an operator is given,

\[ \mathcal Ru=-\mathcal A\frac{\partial u}{\partial \gamma}+\mathcal Bu, \]

where the functions \(\mathcal A\ge 0\), \(\mathcal B>0\) are defined and belong to the class \(C^{1+\alpha}\) on \(\dot D'\), and \(\partial u/\partial\gamma\) is the derivative of \(u\) in some direction \(\gamma\), forming an acute angle with the inward normal \(n\) to \(\dot D'\).

Domains in which the operator \(L\) is uniformly elliptic will be denoted by \(G\). In particular, we shall consider domains \(G_m\) (\(m\) a positive integer) with the following properties: the boundary \(\dot G_m\) of the domain \(G_m\) belongs to the class \(C^{2+\alpha}\), \(\overline D_{1/m}\subseteq \overline G_m\subseteq \overline G_{m+1}\subset \overline D\), and \(\dot G_m\) do not intersect the hyperplane \(x_n=0\). We shall denote \(\dot G_m'=\dot G_m\cap\dot D'\), \(\dot G_m''=\dot G_m\setminus \dot G_m'\). We shall assume that the operator \(\mathcal R\) and the boundary \(\dot D'\) are such that there exists a continuous extension \(R_\gamma\) of this operator and of the field of directions \(\gamma\) in the domain \(D\), possessing the following properties:

a) \(R_\gamma=-A\,\partial/\partial\gamma+B\), where \(A\ge 0\) and \(B>0\) for \(x_n>0\) belong to the class \(C^{1+\alpha}\);

b) on the boundary \(\dot G_m\), \(\cos(\gamma,n)>0\) for sufficiently large \(m\ge m_0>0\);

c) for every \(m\ge m_0\) in the domain \(G_m\) the boundary-value problem is solvable:

\[ Lu=f \text{ in } G_m;\quad R_\gamma u=\Phi \text{ on } \dot G_m \]

for arbitrary \(f\in C^\alpha(G_m)\) and \(\Phi\in C^{1+\alpha}(\dot G_m)\) and

for the solution the Schauder estimate is valid

\[ \|u\|_{2+\alpha}^{D_\lambda}\leqslant c\{\|f\|_{\alpha}^{G_m} +\|\Phi\|_{1+\alpha}^{\dot G_m}+\|u\|_{0}^{G_m}\}, \]

where \(\|u\|_{1+\alpha}^{M}\) is the norm of \(u\) in the space \(C^{1+\alpha}(M)\), and \(\overline D_\lambda\subset \overline G_m\) \((\lambda>1/m)\). The constant \(c\) does not depend on \(u\), \(f\), and \(\Phi\).

Without loss of generality, we shall henceforth assume that the domain \(D\) lies in the strip \(0\leq x_n\leq 1\).

A function \(\psi(x_n)\), twice continuously differentiable in \(D\setminus \dot D'\), positive in \(D\), will be called a majorant of equation (1) in \(D\) if, for any domain \(G\subset D\), from the fact that for a solution of equation (1) on the boundary of \(G\) the conditions \(|R_\gamma u|\leq R_\gamma\psi\) on \(\dot G^{(1)}\) and \(|u|\leq \psi\) on \(\dot G^{(2)}\) \((\dot G=\dot G^{(1)}\cup \dot G^{(2)})\) are fulfilled, it follows that \(|u|\leq \psi\) in \(G\). If \(f\equiv 0\), then the corresponding majorant will be called homogeneous; otherwise, nonhomogeneous.

Let \(p(t)\) and \(q(t)\geq 0\) be functions continuous on the semi-interval \((0,1]\), and let \(h(t,\mu,\nu)\) be the general solution of the equation \(h''+ph'+q=0\), where \(\mu,\nu\) are arbitrary constants. The following is obvious.

Lemma 1. In order that \(h(t,\mu,\nu)\) be nonnegative on \([0,1]\) for some \(\mu,\nu\), it is necessary and sufficient that one of the following two pairs of conditions be fulfilled:

\[ \int_0^1 e^{P(r)}q(r)\,dr<\infty,\qquad \int_0^1 e^{P(s)}\,ds=\infty; \tag{2} \]

\[ \int_0^1 e^{P(s)}\int_s^1 e^{-P(r)}q(r)\,dr\,ds<\infty,\qquad \int_0^1 e^{P(s)}\,ds<\infty, \tag{3} \]

where

\[ P(s)=\int_s^1 p(r)\,dr. \]

Corollary. If condition (2) is fulfilled, the function

\[ h_1(t)=\int_t^1 e^{P(s)}\int_0^s e^{-P(r)}q(r)\,dr\,ds+\nu_1 \]

for any \(\nu_1>0\) is a positive and monotonically decreasing function on \((0,1)\).

If condition (3) is fulfilled, the function

\[ h_2(t)=\int_0^t e^{P(s)} \left[\int_s^1 e^{-P(r)}q(r)\,dr+\mu_2\right]\,ds \]

for any \(\mu_2>0\) is positive and monotonically increasing on \((0,1)\).

If the operator \(R_\gamma\) on the boundary \(\dot G^{(1)}\) of the domain \(G\subset D\) is such that \(A\geq 0\), \(B>0\), and \(\cos(\gamma,n)>0\) on \(\dot G^{(1)}\), then the following is valid.

Lemma 2. Let \(|Lu|\leq \lambda q(x_n)\) \((\lambda\geq 0)\), \(a_{nn}\equiv 1\), \(a\leq 0\), \(a_n\geq p(x_n)\) \([a_n\leq p(x_n)]\) in \(G\), and suppose condition (2) [(3)] is fulfilled. Then from

\[ |R_\gamma u|\leq \lambda R_\gamma h_1 \quad [\,|R_\gamma u|\leq \lambda R_\gamma h_2\,] \quad \text{on } \dot G^{(1)}, \]

\[ |u|\leq \lambda h_1 \quad [\,|u|\leq \lambda h_2\,] \quad \text{on } \dot G^{(2)}=\dot G\setminus \dot G^{(1)} \]

it follows that

\[ |u|\leq \lambda h_1 \quad [\,|u|\leq \lambda h_2\,] \quad \text{in } \overline G. \]

Corollary. It follows from Lemma 2 that the function \(\psi_1(x_n)=h_1(x_n)\) \([\psi_2(x_n)=h_2(x_n)]\) is a nonhomogeneous majorant of the equation \(Lu=f\)

in \(D\), if \(|f|\leq q(x_n)\) and the remaining conditions of Lemma 2 are fulfilled. The function \(\Psi_1(x_n)\equiv \lambda h_1(x_n)\) \([\Psi_2(x_n)=\lambda h_2(x_n)]\) is a homogeneous majorant of the equation \(Lu=0\) in \(D\) for any \(\lambda>0\), \(q\equiv 0\), if the conditions of Lemma 2 are fulfilled in \(D\).

We shall call the problem of finding a solution of equation (1) satisfying the condition \(\mathcal R u=0\) on \(\dot D'\) the problem \(K_0\).

Lemma 3 (existence). Suppose that in \(D\) there exists a continuous extension \(R_\gamma\) of the boundary operator \(\mathcal R\), possessing properties a), b), c), and suppose that there exists an inhomogeneous majorant \(\psi(x_n)\) of the equation \(Lu=f\) in \(D\) such that \(R_\gamma\psi\geq 0\) in \(D\). Then there exists a solution \(u\) of problem \(K_0\) in \(D\), satisfying the estimate \(|u|\leq \psi(x_n)\) in \(D\).

Lemma 4 (uniqueness). The solution of problem \(K_0\) is unique in the class of functions satisfying the condition \(|u|\leq \psi(x_n)\) in \(D\), if in \(D\) there exists a homogeneous majorant \(\Psi(x_n)\) such that \(R_\gamma\Psi\geq 0\) in \(D\) and \(\psi(x_n)\{\Psi(x_n)\}^{-1}\to 0\) as \(x_n\to 0\).

Choosing \(\nu_1\) \([\mu_2]\) sufficiently large and taking into account that \(B>0\) for \(x_n>0\), we can ensure that for \(x_n\geq \varepsilon>0\), \(R_\gamma\psi_1>0\) \([R_\gamma\psi_2>0]\). If, moreover, we require that

\[ \frac{A}{B}\leq \frac12 e^{-P(x_n)}\int_{x_n}^{1} e^{P(s)}\,ds \quad \text{for } 0<x_n<\varepsilon \text{ and } \cos(\gamma,x_n)<0, \tag{4} \]

then \(R_\gamma\psi_1>0\) on the boundary of any domain \(G_m\).

If instead we require that

\[ \frac{A}{B}\leq \frac12 e^{-P(x_n)}\int_{0}^{x_n} e^{P(s)}\,ds \quad \text{for } 0<x_n<\varepsilon \text{ and } \cos(\gamma,x_n)>0, \tag{5} \]

then \(R_\gamma\psi_2>0\) on the boundary of any domain \(G_m\). The functions \(\Psi_1(x_n)\) \([\Psi_2(x_n)]\) possess analogous properties.

With the aid of Lemmas 2, 3 and 4 one obtains

Theorem 1. Suppose \(|f|\leq q(x_n)\); \(a_{nn}\equiv 1\); \(a\leq 0\); \(a_n\geq p(x_n)\) \([a_n\leq p(x_n)]\), and condition (2) \([(3)]\) is fulfilled. Suppose that in \(D\) there exists a continuous extension \(R_\gamma\) of the boundary operator \(R\), possessing properties a), b) and c), and also satisfying condition (4) \([(5)]\). Then there exists a solution of problem \(K_0\) in \(D\), satisfying the estimate

\[ |u|\leq \psi_1(x_n)\quad [\,|u|\leq \psi_2(x_n)\,]\quad \text{in } D. \tag{6} \]

This solution is unique in the class of functions satisfying estimate (6).

Corollary. Since \(\psi_2(x_n)\to 0\) as \(x_n\to 0\), when conditions (3) and (5) are fulfilled it follows from estimate (6) that the solution \(u=0\) on \(D''\).

2. Consider boundary-value problems for equation (1) with nonhomogeneous boundary conditions. By the problem \(K_1\) \([K_2]\) we shall mean the problem of finding a solution of equation (1) satisfying the condition \(\mathcal R u=\varphi_1\) on \(\dot D'\) \([\mathcal R u=\varphi_1\) on \(\dot D'\), \(u=\varphi_2\) on \(\dot D''\)]. The problem \(K_1\) in essence does not differ from the problem \(K_0\), if there exists an extension in \(D\), \(\Phi_1\in C^{1+\alpha}(D)\), of the function \(\varphi_1\), satisfying the estimate

\[ |\Phi_1|\leq c_1\left\{\int_{x_n}^{1} e^{P(s)}\int_{0}^{s} e^{-P(r)}q(r)\,dr\,ds+\nu\right\}, \tag{7} \]

where \(c_1,\nu\) are positive constants. Condition (7) is fulfilled, for example, if \(\varphi_1\in C^{1+\alpha}(\dot D')\) and is bounded in the closure of \(\dot D'\).

When considering problem \(K_2\), suppose that \(\varphi_2 \in C^{2+\alpha}(\dot D'')\). Then, putting \(\Phi_2(x_1,x_2,\ldots,x_n)\equiv \varphi_2(x_n)\), by means of the substitution of the unknown function \(u=w+\Phi_2\) we can reduce problem \(K_2\) to an analogous problem, but with the condition \(w=0\) on \(\dot D''\) and with the condition \(\mathcal R w=\varphi_1-\mathcal R\Phi_2\) on \(\dot D'\). If there exists an extension to \(D\), \(\Phi\in C^{1+\alpha}(D)\), of the function \(\varphi=\varphi_1-\mathcal R\Phi_2\), satisfying the estimate

\[ |\Phi|\le c_2\left\{\int_0^{x_n} e^{P(s)}\int_s^1 e^{-P(r)}q(r)\,dr\,ds+\mu\int_0^{x_n} e^{P(s)}\,ds\right\}, \tag{8} \]

where \(c_2,\mu\) are positive constants, then the construction of a solution of problem \(K_2\) will be analogous to the construction of a solution of problem \(K_0\).

Thus we arrive at the following theorem:

Theorem 2. Suppose that the conditions of Theorem 1 are fulfilled and that there exists an extension to \(D\), \(\Phi_1\in C^{1+\alpha}(D)\) \([\Phi\in C^{1+\alpha}(D)]\), of the function \(\varphi_1[\varphi=\varphi_1-\mathcal R\Phi_2]\) given on \(\dot D'\), satisfying estimate (7) \([(8)]\). Suppose, moreover, that in the case of problem \(K_2\) the estimates \(|a_{ik}|, |a_i|, |a|\le q(x_n)\) hold for \(i,k\ne n\). Then in \(D\) there exists a solution of problem \(K_1[K_2]\), satisfying the estimate

\[ |u|\le \psi_1 \quad [|u-\varphi_2|\le \psi_2], \quad \text{in } D. \]

This solution is unique in the class of functions satisfying estimate (9).

Voronezh State University

Received
20 XII 1964

CITED LITERATURE

\(^{1}\) M. Schechter, Comm. Pure and Appl. Math., 13, No. 2, 321 (1960).
\(^{2}\) M. V. Keldysh, DAN, 77, No. 2, 181 (1951).
\(^{3}\) O. A. Oleinik, DAN, 87, No. 6, 885 (1952).
\(^{4}\) N. D. Vvedenskaya, DAN, 91, No. 4, 711 (1953).
\(^{5}\) S. A. Tersenov, DAN, 115, No. 4, 630 (1957).