V. K. Zakharov

THE FIRST BOUNDARY VALUE PROBLEM FOR AN EQUATION OF ELLIPTIC TYPE OF FOURTH ORDER DEGENERATING ON THE BOUNDARY OF THE DOMAIN

(Presented by Academician S. L. Sobolev on 16 XI 1956)

In the present note the first boundary value problem is considered for the general fourth-order equation

\[ \begin{aligned} Lu \equiv -\Bigg\{& \frac{\partial^2}{\partial x^2}\left(a_{1111}\frac{\partial^2 u}{\partial x^2}\right) +\frac{\partial^2}{\partial x\partial y}\left(a_{1212}\frac{\partial^2 u}{\partial x\partial y}\right) +\frac{\partial^2}{\partial y^2}\left(a_{2222}\frac{\partial^2 u}{\partial y^2}\right)\\ &+\frac{1}{2}\frac{\partial^2}{\partial x\partial y}\left(a_{1112}\frac{\partial^2 u}{\partial x^2}\right) +\frac{1}{2}\frac{\partial^2}{\partial x^2}\left(a_{1112}\frac{\partial^2 u}{\partial x\partial y}\right) +\frac{1}{2}\frac{\partial^2}{\partial y^2}\left(a_{1222}\frac{\partial^2 u}{\partial x\partial y}\right)\\ &+\frac{1}{2}\frac{\partial^2}{\partial x\partial y}\left(a_{1222}\frac{\partial^2 u}{\partial y^2}\right)\Bigg\} +a_{111}\frac{\partial^3 u}{\partial x^3} +a_{122}\frac{\partial^3 u}{\partial x\partial y^2} +a_{112}\frac{\partial^3 u}{\partial x^2\partial y}\\ &+a_{222}\frac{\partial^3 u}{\partial y^3} +a_{11}\frac{\partial^2 u}{\partial x^2} +2a_{12}\frac{\partial^2 u}{\partial x\partial y} +a_{22}\frac{\partial^2 u}{\partial y^2}\\ &+a_1\frac{\partial u}{\partial x} +a_2\frac{\partial u}{\partial y} +a_0u=h \end{aligned} \tag{1} \]

of elliptic type with two independent variables, degenerating on a part of the boundary of the domain adjacent to the axis \(Ox\). Equation (1) is considered in a finite domain \(D\), situated in the upper half-plane and having a part of its boundary \(\Gamma_0\) on the axis \(Ox\). The remaining part of the boundary we denote by \(\Gamma_1\). We assume that the boundary \(\Gamma_1\) is such that the embedding theorems of S. L. Sobolev \((^1)\) hold for it.

We suppose that the coefficients of the highest derivatives in equation (1) are continuous in the closed domain \(\overline D \equiv D\cup \Gamma\), \(\Gamma=\Gamma_1\cup\Gamma_0\), and twice continuously differentiable in \(D^\delta=D\cap(y>\delta)\), where \(\delta>0\) is arbitrary; \(a_{111}\), \(a_{112}\), \(a_{122}\), and \(a_{222}\) are three times continuously differentiable in \(D^\delta\); \(a_{11}\), \(a_{12}\), and \(a_{22}\) are twice continuously differentiable in \(D^\delta\); \(a_1\) and \(a_2\) are continuously differentiable in \(D^\delta\); \(a_0\) is continuous in \(D^\delta\). We also assume that

\[ c^2y^\alpha \leq a_{2222} \leq C^2y^\alpha, \]

where \(\alpha \geq 0\). If \(\alpha=0\), then we shall assume that \(a_{1111}\to0\) as \(y\to0\). For definiteness we shall suppose that at every point of the domain \(\overline D\) the form

\[ A^4(\xi_1,\xi_2;x,y)\equiv a_{1111}\xi_1^4+a_{1212}\xi_1^2\xi_2^2+a_{2222}\xi_2^4 +a_{1112}\xi_1^3\xi_2+a_{1222}\xi_1\xi_2^3 \geq 0 \]

for \(\xi_1^2+\xi_2^2>0\), with equality attained only at the points of \(\Gamma_0\).

Theorem 1. Let, for any real numbers \(\xi_1,\xi_2\) \((\xi_1^2+\xi_2^2>0)\), the form \(A^4(\xi_1,\xi_2;x,y)\geq0\), where equality is attained only at the points \(\Gamma_0\). Make the substitution: \(\xi_1^2=\xi_{11}\), \(\xi_1\xi_2=\xi_{12}\), \(\xi_2^2=\xi_{22}\). Then there exists a sufficiently smooth function \(\lambda=\lambda(x,y)\) such that the form

\[ B(\xi_{11},\xi_{12},\xi_{22};x,y) \equiv A^2(\xi_{11},\xi_{12},\xi_{22};x,y)+\lambda(\xi_{12}^2-\xi_{11}\xi_{12})\equiv \]

\[ \equiv a_{1111}\xi_{11}^2+(a_{1212}+\lambda)\xi_{12}^2+a_{2222}\xi_{22}^2 +a_{1112}\xi_{11}\xi_{12}-a_{1222}\xi_{12}\xi_{22} -\lambda\xi_{11}\xi_{22}\geq0 \]

provided that \(\xi_{11}^{2}+\xi_{12}^{2}+\xi_{22}^{2}>0\); \(\xi_{11},\xi_{12},\xi_{22}\) are arbitrary real numbers, and the equality sign can occur only at points of \(\Gamma_0\).

As in the paper \((^3)\), the space \(\dot{\Omega}\) and the space \(\dot{R}\) are introduced, with metric defined by the scalar product

\[ \begin{aligned} \{Gu,Gv\}=\iint_D \bigg[& a_{1111}\frac{\partial^2 u}{\partial x^2}\frac{\partial^2 v}{\partial x^2} +(a_{1212}+\lambda)\frac{\partial^2 u}{\partial x\,\partial y}\frac{\partial^2 v}{\partial x\,\partial y} +a_{2222}\frac{\partial^2 u}{\partial y^2}\frac{\partial^2 v}{\partial y^2} \\ &+\frac12 a_{1112}\frac{\partial^2 u}{\partial x^2}\frac{\partial^2 v}{\partial x\,\partial y} +\frac12 a_{1112}\frac{\partial^2 u}{\partial x\,\partial y}\frac{\partial^2 v}{\partial x^2} +\frac12 a_{1222}\frac{\partial^2 u}{\partial y^2}\frac{\partial^2 v}{\partial x\,\partial y} \\ &+\frac12 a_{1222}\frac{\partial^2 u}{\partial x\,\partial y}\frac{\partial^2 v}{\partial y^2} +\frac{\lambda}{2}\frac{\partial^2 u}{\partial x^2}\frac{\partial^2 v}{\partial y^2} +\frac{\lambda}{2}\frac{\partial^2 u}{\partial y^2}\frac{\partial^2 v}{\partial x^2} \bigg]\,dx\,dy, \end{aligned} \tag{2} \]

which is possible by virtue of Theorem 1.

We assume that, for \(\alpha \geq 0,\ \beta \geq 0\), the following conditions are satisfied:

\[ c_1^2 y^\beta \leq a_{1212}+\lambda \leq C_1^2 y^\beta,\qquad 0 \leq y^\alpha \xi_{22}^{2}\leq C_2^2 B(\xi_{11},\xi_{12},\xi_{22};x,y), \]

\[ 0\leq y^\beta \xi_{12}^{2}\leq C_3^2 B(\xi_{11},\xi_{12},\xi_{22};x,y),\qquad 0\leq a_{1111}\xi_{11}^{2}\leq C_4^2 B(\xi_{11},\xi_{12},\xi_{22};x,y) \]

for arbitrary real \(\xi_{11},\xi_{12},\xi_{22}\).

We introduce the notation:

\[ N_1=a_2-2\frac{\partial \bar a_{12}}{\partial x} -\frac{\partial \bar a_{22}}{\partial y} +\frac{\partial^2 a_{122}}{\partial x\,\partial y} +\frac{\partial^2 a_{222}}{\partial y^2}, \qquad N_2=2\frac{\partial a_{222}}{\partial y} +\frac{\partial a_{122}}{\partial x} -\bar a_{22}, \]

\[ \iint_D uv\,dx\,dy=[u,v], \]

\[ \bar a_{11}=a_{11}-\frac12\frac{\partial^2\lambda}{\partial y^2}, \qquad \bar a_{12}=a_{12}+\frac12\frac{\partial^2\lambda}{\partial x\,\partial y}, \qquad \bar a_{22}=a_{22}-\frac12\frac{\partial^2\lambda}{\partial x^2}. \]

By \(\mathscr{L}_D^2(\sigma)\), \(\sigma=\sigma(x,y)\), \((x,y)\in D\), we shall denote the space of functions square-summable with weight \(\sigma(x,y)\) over the domain \(D\): \([\sigma u,u]<+\infty\).

Case a) For \(0\leq\alpha<1,\ \beta\geq 0\), the first boundary-value problem for equation (1) with homogeneous boundary conditions is posed as follows: find a generalized solution of equation (1) which vanishes on \(\Gamma=\Gamma_1\cup\Gamma_0\) together with its normal derivative.

Suppose that in some neighborhood of \(\Gamma_0\) the following conditions are satisfied:

Case b) \(1\leq\alpha<3,\ \beta\geq1;\ 0\leq\beta<1,\ \alpha\geq1:\)

\[ \begin{array}{rll} 1)\quad N_1\geq -c_1^2 y^{-2}|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } \alpha=1,\\[2mm] N_1\geq -c_1^2 y^{\alpha-3}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } 1<\alpha<2,\\[2mm] N_1\geq -c_1^2 y^{\beta-1}\ \text{or}\ N_1\geq -c_1^2 y^{\alpha-3}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } 2\leq\alpha<3,\\[2mm] N_1\geq -c_1^2 y^{\beta-1} & \text{for } \alpha\geq3; \end{array} \]

\[ \begin{array}{rll} 2)\quad N_2\leq c_2^2 y^{-1}|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } \alpha=1,\\[2mm] N_2\leq c_2^2 y^{\alpha-2}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } 1<\alpha<2,\\[2mm] N_2\leq c_2^2 y^\beta\ \text{or}\ N_2\leq c_2^2 y^{\alpha-2}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } 2\leq\alpha<3,\\[2mm] N_2\leq c_2^2 y^\beta & \text{for } \alpha\geq3; \end{array} \]

\[ \begin{array}{rll} 3)\quad \text{either}\quad |a_{222}|\leq c^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } \alpha=1,\\[2mm] |a_{222}|\leq c^2 y^{\alpha-1}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } 1<\alpha<3,\ \alpha>3,\\[2mm] |a_{222}|\leq c^2 y^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} & \text{for } \alpha=3; \end{array} \tag{3} \]

\[ \begin{gathered} |a_{122}| \leq c_3^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{for } \alpha=1,\ \beta\leq 1;\ \alpha\leq 1,\ \beta=1,\\ |a_{122}| \leq c_3^2 y^{\gamma-1}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{for } \alpha\geq 1,\ \beta>1;\ \alpha>1,\ \beta\geq 1; \end{gathered} \tag{4} \]

\[ \begin{gathered} a_{112}\leq c_4^2 y^{\beta-1} \quad \text{for } 0\leq \beta<1,\\ a_{122}\leq c_4^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{for } \beta=1,\\ a_{112}\leq c_4^2 y^{\beta-1}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{for } \beta>1; \end{gathered} \tag{5} \]

or

\[ \begin{gathered} -\tilde c^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1}\leq a_{222}\leq 0 \quad \text{for } \alpha=1,\\ -\tilde c^2 y^{\alpha-1}(\ln|\ln y|)^{-\varepsilon_1}\leq a_{222}\leq 0 \quad \text{for } 1<\alpha<3,\ \alpha>3,\\ -\tilde c^2 y^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1}\leq a_{222}\leq 0 \quad \text{for } \alpha=3;\\ a_{112}\leq 0;\qquad a_{122}^2-3a_{112}a_{222}\leq 0. \end{gathered} \tag{6} \]

Case c) \(\alpha\geq 3,\ \beta\geq 1\):

1)
\[ \begin{gathered} N_1\geq -c_1^2|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{for } \alpha=3,\ \beta\geq 1;\ \alpha\geq 3,\ \beta=1,\\ N_1\geq -c_1^2y^{\alpha-3}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{or}\quad N_1\geq -c_1^2y^{\beta-1}(\ln|\ln y|)^{-\varepsilon_1}\\ \text{for } \alpha>3,\ \beta>1; \end{gathered} \]

2)
\[ \begin{gathered} N_2\leq c_2^2 y|\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{for } \alpha\geq 3,\ \beta=1;\ \alpha=3,\ \beta\geq 1,\\ N_2\leq c_2^2y^{\alpha-2}(\ln|\ln y|)^{-\varepsilon_1} \quad \text{or}\quad N_2\leq c_2^2y^\beta(\ln|\ln y|)^{-\varepsilon_1}\\ \text{for } \alpha>3,\ \beta>1; \end{gathered} \]

3) either (3), (4), and (5) for \(\alpha\geq 3,\ \beta\geq 1\), or (6) for \(\alpha\geq 3\), where \(\varepsilon_1\) is an arbitrary positive number, \(c\) with subscripted values are certain constants, \(\gamma=\max(\alpha,\beta)\).

The conditions just listed will be called “conditions \(S\).”

When conditions \(S\) are satisfied, the first boundary-value problem with homogeneous boundary conditions consists in finding a generalized solution of equation (1): in case b), one that vanishes on \(\Gamma\), whose first derivatives vanish on \(\Gamma_1\); on \(\Gamma_0\) no conditions are imposed on the first derivatives of the sought solution; in case c), one that vanishes on \(\Gamma_1\) together with its first derivatives; on \(\Gamma_0\) no boundary conditions at all are imposed.

We note that the required boundary conditions are satisfied, on the basis of the embedding theorems proved in note \((^3)\), by functions from \(\dot\Omega\). The vanishing of a function and of derivatives on the boundary is understood in the mean.

Under the conditions formulated above, by a generalized solution of the first boundary-value problem for equation (1) in cases a), b), and c) we mean such a function \(u\in\dot\Omega\) that, for any function \(v\in\dot\Omega\) that vanishes in some neighborhood of \(\Gamma_0\), the following integral relation is fulfilled:

\[ \begin{aligned} [h,v]=&-\{Gu,Gv\} +\left[\frac{\partial u}{\partial x},\, a_{111}\frac{\partial^2 v}{\partial x^2} +a_{112}\frac{\partial^2 v}{\partial x\,\partial y}\right]\\ &+\left[\frac{\partial u}{\partial x},\, \left(2\frac{\partial a_{111}}{\partial x} +\frac{\partial a_{112}}{\partial y} -\bar a_{11}\right)\frac{\partial v}{\partial x} +\left(\frac{\partial a_{112}}{\partial x} -\bar a_{12}\right)\frac{\partial v}{\partial y}\right]\\ &+\left[\frac{\partial u}{\partial y},\, \left(2\frac{\partial a_{222}}{\partial y} +\frac{\partial a_{122}}{\partial x} -\bar a_{22}\right)\frac{\partial v}{\partial y} +\left(\frac{\partial a_{122}}{\partial y} -\bar a_{12}\right)\frac{\partial v}{\partial x}\right]\\ &+\left[\frac{\partial u}{\partial x},\, \left(\frac{\partial^2 a_{111}}{\partial x^2} +\frac{\partial^2 a_{112}}{\partial x\,\partial y} -\frac{\partial a_{11}}{\partial x} -\frac{\partial a_{12}}{\partial y}\right)v\right]\\ &+\left[\frac{\partial u}{\partial y},\, a_{222}\frac{\partial^2 v}{\partial y^2} +a_{122}\frac{\partial^2 v}{\partial x\,\partial y}\right]\\ &+\left[\frac{\partial u}{\partial y},\, \left(\frac{\partial^2 a_{222}}{\partial y^2} +\frac{\partial^2 a_{122}}{\partial x\,\partial y} -\frac{\partial a_{22}}{\partial y} -\frac{\partial a_{12}}{\partial x}\right)v\right]\\ &+\left[u,\left(a_0-\frac{\partial a_1}{\partial x} -\frac{\partial a_2}{\partial y}\right)v\right] -\left[u,\, a_1\frac{\partial v}{\partial x} +a_2\frac{\partial v}{\partial y}\right]. \end{aligned} \tag{7} \]

Theorem 2. Suppose that in cases b) and c) the conditions \(S\) are satisfied. Suppose also that in case a), in some neighborhood of \(\Gamma_0\), the inequalities hold:

\[ N_1 \geqslant c_1^2 y^{\alpha-3}; \qquad N_2 \leqslant c_2^2 y^{\alpha-2}; \]

or (5) and

\[ |a_{222}| \leqslant c^2 y^{\alpha-1}, \]

\[ |a_{122}| \leqslant c_3^2 y^{\gamma-1}(|\ln|(\ln y|)|)^{-\varepsilon_1} \qquad \text{for } 0 \leqslant \beta < 1, \]

\[ |a_{122}| \leqslant c_3^2 y^{\beta-1}(\ln|\ln y|)^{-\varepsilon_1} \qquad \text{for } \beta > 1, \]

\[ |a_{122}| \leqslant c_3^2 |\ln y|^{-1}(\ln|\ln y|)^{-\varepsilon_1} \qquad \text{for } \beta = 1; \]

or

\[ -\widetilde c^{\,2} y^{\alpha-1} \leqslant a_{222} \leqslant 0,\quad a_{112} \leqslant 0,\quad a_{122}^2 - 3a_{112}a_{222} \leqslant 0, \]

where \(\gamma=\max(\alpha,\beta)\), \(\varepsilon_1>0\) and is arbitrary. If, in addition to the conditions listed above, in each case, for any point of the domain \(D\), the following is satisfied:

\[ a_0-\frac12\frac{\partial a_1}{\partial x} -\frac12\frac{\partial a_2}{\partial y} +\frac12\frac{\partial^2 a_{11}}{\partial x^2} +\frac{\partial^2 a_{12}}{\partial x\,\partial y} +\frac12\frac{\partial^2 a_{22}}{\partial y^2} - \]

\[ -\frac12\frac{\partial^3 a_{111}}{\partial x^3} -\frac12\frac{\partial^3 a_{112}}{\partial x^2\partial y} -\frac12\frac{\partial^3 a_{122}}{\partial x\,\partial y^2} -\frac12\frac{\partial^3 a_{222}}{\partial y^3} \leqslant 0; \]

\[ \frac32\frac{\partial a_{111}}{\partial x} +\frac12\frac{\partial a_{112}}{\partial y} -\bar a_{11}\leqslant 0,\qquad \frac32\frac{\partial a_{222}}{\partial y} +\frac12\frac{\partial a_{122}}{\partial x} -\bar a_{22}\leqslant 0, \]

\[ \left( \frac32\frac{\partial a_{111}}{\partial x} +\frac12\frac{\partial a_{112}}{\partial y} -\bar a_{11} \right) \left( \frac32\frac{\partial a_{222}}{\partial y} +\frac12\frac{\partial a_{122}}{\partial x} -\bar a_{22} \right) - \]

\[ -\left( \frac12\frac{\partial a_{112}}{\partial x} +\frac12\frac{\partial a_{122}}{\partial y} -\bar a_{12} \right)^2 \geqslant 0, \]

then the first boundary-value problem in the formulation indicated above has a unique solution for any right-hand side \(h\in \mathscr L_2^2(\sigma_0^{-1})\), where \(\sigma_0\) is the weight function defined in the embedding theorems \((^3)\).

The proof is carried out by the functional method, the same as in the work of M. I. Vishik \((^2)\). The lower-order terms of equation (1) are realized in the form of an operator acting in \(\mathring R\). It is proved that the operator corresponding to the operator \(\bar L\) has everywhere a dense domain of definition and a bounded inverse operator.

In conclusion I take this opportunity to express my sincere gratitude to Academician S. L. Sobolev, under whose supervision the present work was carried out.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
13 XI 1956

CITED LITERATURE

\(^1\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
\(^2\) M. I. Vishik, Mat. sbornik, 35 (77), 513 (1954).
\(^3\) V. K. Zakharov, DAN, 114, No. 3 (1957).