ON A VARIATIONAL METHOD FOR SOLVING A CLASS OF DEGENERATE ELLIPTIC EQUATIONS

I. A. Kipriyanov

(Presented by Academician S. L. Sobolev on 21 III 1963)

1. In the preceding note \((^1)\) we indicated a class of weighted embedding theorems that admit a complete converse. In this connection, the results of Theorems 8 and 9 \((^1)\) also carry over to bounded domains. In the present note we intend to supplement the preceding results and to indicate a class of well-posed boundary-value problems for a new type of degenerate elliptic equations.

For simplicity of exposition, we restrict ourselves to considering the half-space \(\overset{+}{R}_2\) in the two-dimensional case. Consider the set of functions \(f(x,y)\), finite in the half-space \(x>0\) with the manifold \(x=0\) removed (see the definition in \((^1)\)), and define on it the differential operator

\[ D_x f=\frac{1}{x}\frac{\partial f(x,y)}{\partial x} \tag{1} \]

and the powers of this operator

\[ D_x^l f=\frac{\partial}{x\partial x} \left( \frac{\partial^{\,l-1} f(x,y)}{(x\partial x)^{l-1}} \right) \quad (l=2,3,\ldots). \tag{2} \]

Define the functional space \(W_{x,2,\gamma}^{(l)}(\overset{+}{R}_2)\) as the closure of the set of functions \(f(x,y)\), finite in the half-space \(\overset{+}{R}_2\) with the manifold \(x=0\) removed, with respect to the norm

\[ \|f\|^2_{W_{x,2,\gamma}^{(l)}(\overset{+}{R}_2)} = \int_{\overset{+}{R}_2} |f(x,y)|^2 x^{2\gamma}\,dx\,dy + \sum_{k=1}^{l} \int_{\overset{+}{R}_2} \left| x^k \frac{\partial^k f(x,y)}{(x\partial x)^k} \right|^2 x^{2\gamma}\,dx\,dy, \tag{3} \]

where \(\gamma\) is a positive number.

We define the functional space \(W_{y,2,\gamma}^{(l_1)}(\overset{+}{R}_2)\) as the closure of the set of functions \(f(x,y)\), finite in the half-space \(\overset{+}{R}_2\) with the manifold \(x=0\) removed, with respect to the norm

\[ \|f\|^2_{W_{y,2,\gamma}^{(l_1)}(\overset{+}{R}_2)} = \int_{\overset{+}{R}_2} |f(x,y)|^2 x^{2\gamma}\,dx\,dy + \sum_{k=1}^{l_1'} \int_{\overset{+}{R}_2} \left| \frac{\partial^k f(x,y)}{\partial y^k} \right|^2 x^{2\gamma}\,dx\,dy + \sum_{k=1}^{l_1'} \mathscr{L}_{k,\gamma}^{(2)} \left( \frac{\partial^k f(x,y)}{\partial y^k} \right), \tag{4} \]

where by \(\mathscr{L}_{k,\gamma}^{(2)}(\partial^k f/\partial y^k)\) one should understand the expression

\[ \mathscr{L}_{k,\gamma}^{(2)} \left( \frac{\partial^k f}{\partial y^k} \right) = \int_{-\infty}^{\infty} \int_{\overset{+}{R}_2} \frac{ \left| \partial^k f(x,y+h)/\partial y^k - \partial^k f(x,y)/\partial y^k \right|^2 }{ |h|^{1+2\lambda} } x^{2\gamma}\,dx\,dy\,dh, \tag{5} \]

and here \(l_1=l_1' + \lambda\) and \(0<\lambda<1\).

We define the functional space \(W_{x,y,2,\gamma}^{(l,l_1)}(\overset{+}{R}_2)\) as the intersection of the functional classes \(W_{x,2,\gamma}^{(l)}(\overset{+}{R}_2)\) and \(W_{y,2,\gamma}^{(l_1)}(\overset{+}{R}_2)\), and introduce the norm, as usual, by the formula

\[ \|f\|^2_{W_{x,y,2,\gamma}^{(l,l_1)}(\overset{+}{R}_2)} = \|f\|^2_{W_{x,2,\gamma}^{(l)}(\overset{+}{R}_2)} + \|f\|^2_{W_{y,2,\gamma}^{(l_1)}(\overset{+}{R}_2)}. \tag{6} \]

The case \(\gamma=0\) is contained in the preceding note \((^1)\). As in the preceding case \((^1)\), instead of the indicated norms one may consider norms equivalent to them. Here the index \(l\) assumes only positive integer values.

Theorem 1. Let \(f\in W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)\), where \(\gamma>0\), and let, for nonnegative integers \(k\), the inequality
\[ \mu=\mu(k,\gamma)=1-2k/l-(2\gamma+1)/2l>0 \tag{7} \]
hold. Then the derivatives \(\partial^k f(x,y)/(x\partial x)^k\), for all those \(y\) for which they are quadratically summable over \(R_1\) with respect to \(x\), belong to the space \(W^{(\bar l_1)}_{y,2}(R_1)\) with \(\bar l_1=\mu l_1\). Moreover, the inequality
\[ \left\| \left. \frac{\partial^k f}{(x\partial x)^k} \right|_{x=0} \right\|_{W^{(\bar l_1)}_{y,2}(R_1)} \leq c\|f\|_{W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)} \tag{8} \]
holds, where the constant \(c\) does not depend on \(f\).

Remark 1. \(W^{(\bar l_1)}_{y,2}(R_1)\) denotes the usual Aronszajn–Slobodeckii space \((^2)\). The converse assertion is also true.

Theorem 2. Let nonnegative integers \(k\) be given, for which the inequality
\[ \mu=\mu(k,\gamma)=1-2k/l-(2\gamma+1)/2l>0 \tag{9} \]
holds. For such \(k\), let functions \(\varphi^{(k)}(y)\in W^{(\bar l_1)}_{y,2}(R_1)\) with \(\bar l_1=\mu l_1\) be prescribed on \(R_1\). There exists a function \(\bar f(x,y)\in W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)\) such that
\[ \lim_{x\to +0} \left\| \frac{\partial^k\bar f(x,y)}{(x\partial x)^k} -\varphi^{(k)}(y) \right\|_{W^{(\bar l_1)}_{y,2}(R_1)} =0 \tag{10} \]
for all admissible \(k\). Moreover, the inequality
\[ \|\bar f\|_{W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)} \leq c\sum_k \|\varphi^{(k)}(y)\|_{W^{(\bar l_1)}_{y,2}(R_1)} \tag{11} \]
holds with a constant \(c\) independent of \(\varphi^{(k)}\).

Let us note that the theorems stated here remain valid if, instead of nonnegative integers \(k\) satisfying the inequality
\[ \mu=\mu(k,\gamma)=1-2k/l-(2\gamma+1)/2l>0, \tag{12} \]
one simply takes nonnegative numbers \(k\) satisfying the same inequality. What is meant by this is clear from the theorems given below.

Theorem 3. Let \(f\in W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)\) \((\gamma>0)\), and let, for nonnegative numbers \(k\), the inequality
\[ \mu=\mu(k,\gamma)=1-2k/l-(2\gamma+1)/2l>0 \tag{13} \]
hold. Then the derivatives \(\widetilde{\mathscr D}^{\,k}_{x} f\), for all those \(y\) for which they are quadratically summable over \(R_1\) with respect to \(x\), belong to the space \(W^{(\bar l_1)}_{y,2}(R_1)\) with \(\bar l_1=\mu l_1\). Moreover, the inequality
\[ \left\| \left.\widetilde{\mathscr D}^{\,k}_{x}f\right|_{x=0} \right\|_{W^{(\bar l_1)}_{y,2}(R_1)} \leq c\|f\|_{W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)} \tag{14} \]
holds with a constant \(c\) independent of \(f\).

Here and in the following theorem, in the case of an integer \(k\), \(\widetilde{\mathscr D}^{\,k}_{x}f\) should be understood as the derivative \(\partial^k f/(x\partial x)^k\), while in the case of fractional \(k\), \(\widetilde{\mathscr D}^{\,k}_{x}f\) should be understood as the integro-differential operator \((k=\bar k+\beta/2,\ \bar k\) a nonnegative integer and \(0<\beta/2<1\), with \(\beta<\gamma)\)
\[ \frac{\partial^{\bar k+\beta/2}f}{x^{\bar k-\beta/2}\partial x^{\bar k+\beta/2}} = \]
\[ = \frac{1}{x^{2\bar k-\beta+2\gamma}} \frac{\partial}{\partial x} \left[ \frac{1}{\Gamma(1-\beta/2)} \int_0^x (x^2-\tau^2)^{-\beta/2}\tau^{\bar k} \left( \tau^{\bar k}\frac{\partial^k f(\tau,y)}{(\tau\partial\tau)^k} \right) \tau^{2\gamma}\,d\tau \right]. \tag{15} \]
For the definition of the integro-differential operator (15), see \((^1)\). The converse theorem is also true.

Theorem 4. Let nonnegative numbers \(k\) be given, for which inequality (13) is satisfied. For such numbers \(k\) let us prescribe on \(R_1\) functions
\(\varphi^{(k)}(y)\in W^{(\bar l_1)}_{y,2}(R)\), where \(\bar l_1=\mu l_1\). There exists an \(\bar f\in W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)\) such that

\[ \lim_{x\to +0}\left\|\widetilde{\mathscr D}^{\,k}_x\bar f(x,y)-\varphi^{(k)}(y)\right\|_{W^{(\bar l_1)}_{y,2}(R)}=0. \tag{16} \]

for all admissible \(k\). Moreover, the inequality

\[ \|f\|_{W^{(l,l_1)}_{x,y,2,\gamma}(\overset{+}{R}_2)} \le c\sum_k \left\|\varphi^{(k)}(y)\right\|_{W^{(\bar l_1)}_{y,2}(R_1)} \tag{17} \]

holds, with a constant \(c\) independent of \(\varphi^{(k)}\).

It is interesting to note that many properties of the fractional differentiation operator previously studied by the author \((3–7)\) carry over to integro-differential operators of the type of operator (15), which here plays the role of a boundary operator.

1. We shall consider the equation

\[ \mathscr L(f)=\Delta^2 f+ \frac{c_1+2(3+2\gamma)}{x}\frac{\partial^3 f}{\partial x^3} +\frac{4\gamma}{x}\frac{\partial^3 f}{\partial x\,\partial y^2} + \]

\[ +\frac{c_2+\widetilde c_1\,2(3+2\gamma)+(3+2\gamma)(1+2\gamma)}{x^2} \frac{\partial^2 f}{\partial x^2} + \]

\[ +\frac{c_3+2\widetilde c_2(3+2\gamma)-(3+2\gamma)(1+2\gamma)}{x^3} \frac{\partial f}{\partial x} +f=0, \tag{18} \]

which is the Euler equation for the functional

\[ \widetilde{\mathscr D}^{(2)}_\gamma(f) = \int_{\overset{+}{R}_2}|f|^2x^{2\gamma}\,dx\,dy + \]

\[ + \int_{\overset{+}{R}_2} \left\{ \left[x^2\frac{\partial^2 f}{(x\,\partial x)^2}\right]^2 + 2\left[x\frac{\partial^2 f}{x\,\partial x\,\partial y}\right]^2 + \left[\frac{\partial^2 f}{\partial y^2}\right]^2 \right\} x^{2\gamma}\,dx\,dy. \tag{19} \]

(the constants \(c_1,c_2,c_3,\widetilde c_1,\widetilde c_2\) have quite definite values). The functional (19), as is not hard to see, is a norm in the space \(W^{(2,2)}_{x,y,2,\gamma}(\overset{+}{R}_2)\). The principal part of equation (18) can be written in self-adjoint form, and it will have the form

\[ \mathscr L(f)= \frac{\partial^2}{\partial x^2}\left(x^3\frac{\partial^2 f}{\partial x^2}\right) + 2\frac{\partial^2}{\partial x\,\partial y}\left(x^3\frac{\partial^2 f}{\partial x\,\partial y}\right) + \frac{\partial^2}{\partial y^2}\left(x^3\frac{\partial^2 f}{\partial y^2}\right) + \]

\[ + x^2 c^{(1)}_\gamma\frac{\partial^3 f}{\partial x\,\partial y^2} + x^2 c^{(2)}_\gamma\frac{\partial^3 f}{\partial x^3} + x c^{(3)}_\gamma\frac{\partial^2 f}{\partial x^2} + c^{(4)}_\gamma\frac{\partial f}{\partial x} + x^3 f=0. \tag{20} \]

Statement of the problem. It is required to find a generalized solution of equation (20), \(f(x,y)\), taking on the boundary of the domain \(x=0\) the prescribed value \(\varphi(y)\).

We shall agree to call a function \(\varphi(y)\), prescribed on the boundary of the domain \(\overset{+}{R}_2\), admissible if there exists a function
\(f\in W^{(2,2)}_{x,y,2,\gamma}(\overset{+}{R}_2)\) for which \(\varphi(y)\) serves as the boundary value. Denote by
\(W^{(2,2)}_{x,y,2,\gamma}(\varphi)\) the set of functions
\(f\in W^{(2,2)}_{x,y,2,\gamma}(\overset{+}{R}_2)\) such that
\(\left.f\right|_{x=0}=\varphi\). From item 1 it follows that
\(W^{(2,2)}_{x,y,2,\gamma}(\varphi)\) is a nonempty set. For every
\(f\in W^{(2,2)}_{x,y,2,\gamma}(\varphi)\) we have
\(0\le \widetilde{\mathscr D}^{(2)}_\gamma(f)<\infty\). Therefore there exists an exact lower bound of the values \(\widetilde{\mathscr D}^{(2)}_\gamma(f)\):

\[ d=\inf_{f\in W^{(2,2)}_{x,y,2,\gamma}(\varphi)} \widetilde{\mathscr D}^{(2)}_\gamma(f). \tag{21} \]

Obviously, from the set \(W^{(2,2)}_{x,y,2,\gamma}(\varphi)\) one can extract a sequence \(\{f_k\}\) for which

\[ \lim_{k\to\infty}\widetilde{\mathscr D}^{(2)}_\gamma(f_k)=d. \tag{22} \]

The sequence \(\{f_k\}\) is called minimizing.

Theorem 5. The minimizing sequence of functions \(\{f_k\}\) converges in \(W_{x,y,2,\gamma}^{(2,2)}(\overset{+}{R}_2)\) to a certain function \(f_0(x,y)\). This function belongs to \(W_{x,y,2,\gamma}^{(2,2)}(\varphi)\) and gives the functional \(\widetilde{\mathfrak D}^{(2)}_\gamma(f)\) its least value.

Theorem 6. The function \(f_0\), giving the minimum to the functional \(\widetilde{\mathfrak D}^{(2)}_\gamma(f)\) in \(W_{x,y,2,\gamma}^{(2,2)}(\varphi)\), is a generalized solution of equation (20) under the boundary condition

\[ f\big|_{x=0}=\varphi . \tag{23} \]

Remark 2. We call the function \(f_0\) a generalized solution of equation (20) under the boundary condition (23) if, for every function \(z\), finite in the half-space \(\overset{+}{R}_2\) with the manifold \(x=0\) removed, the relation

\[ \widetilde{\mathfrak D}^{(2)}_\gamma(f_0,z)= \int_{\overset{+}{R}_2} f_0 z x^{2\gamma}\,dx\,dy+ \]

\[ +\int_{\overset{+}{R}_2} \left\{ \left[ x^2\frac{\partial^2 f_0}{(x\partial x)^2} x^2\frac{\partial^2 z}{(x\partial x)^2} \right] + 2\left[ x\frac{\partial^2 f_0}{x\partial x\,\partial y} x\frac{\partial^2 z}{x\partial x\,\partial y} \right] + \frac{\partial^2 f_0}{\partial y^2} \frac{\partial^2 z}{\partial y^2} \right\}x^{2\gamma}\,dx\,dy=0 . \tag{24} \]

By the very definition of a generalized solution, it is unique. From the preceding theorems of Sections 1 and 2 it follows:

Theorem 7. In order that the problem of finding a generalized solution of equation (20) under the boundary condition (23) be solvable in \(W_{x,y,2,\gamma}^{(2,2)}(\overset{+}{R}_2)\), it is necessary and sufficient that \(\varphi\in W_{y,2}^{(\bar l_1)}(R_1)\) with \(\bar l_1=3/2-\gamma\).

For the solution \(f_0\) the two-sided estimate is valid

\[ c_2\|\varphi\|_{W_{y,2}^{(\bar l_1)}(R_1)} \le \|f_0\|_{W_{x,y,2,\gamma}^{(2,2)}(\overset{+}{R}_2)} \le c_1\|\varphi\|_{W_{y,2}^{(\bar l_1)}(R_1)} . \tag{25} \]

If in equation (20) the number \(\gamma\) is different from zero, then the necessary and sufficient condition for solvability of problem (20)—(23) can also be formulated in terms of the integro-differential operator (15). Everywhere in Theorems 5, 6, and 7, the boundary condition \(f|_{x=0}=\varphi\), on the basis of Theorems 3 and 4 of Section 1, may be replaced by the boundary condition

\[ x^{\beta/2}\partial^{\beta/2}f/\partial x^{\beta/2}\big|_{x=0}=\varphi_1 \qquad(\beta<\gamma). \tag{26} \]

Then Theorem 7 takes the following form:

Theorem 8. In order that the problem of finding a generalized solution of equation (20) under the boundary condition (26) be solvable in \(W_{x,y,2,\gamma}^{(2,2)}(\overset{+}{R}_2)\), it is necessary and sufficient that \(\varphi_1\in W_{y,2}^{(\bar l_1)}(R_1)\) with \(\bar l_1=3/2-\gamma-\beta\). For the solution \(f_0\) the two-sided estimate holds

\[ \bar c_2\|\varphi_1\|_{W_{y,2}^{(\bar l_1)}(R_1)} \le \|f_0\|_{W_{x,y,2,\gamma}^{(2,2)}(\overset{+}{R}^2)} \le \bar c_1\|\varphi_1\|_{W_{y,2}^{(\bar l_1)}(R_1)} . \tag{27} \]

In a completely analogous way one may consider the case of equations of order \(2m\), in which the order of degeneration will be correspondingly equal to \(2m-1\). Similar results are also valid for bounded domains in \(R_n\).

The author expresses gratitude to S. G. Krein for his attention to the work.

Note added in proof. At the present time the author has obtained a priori estimates for solutions of certain classes of degenerating equations with variable coefficients.

Voronezh Technological Institute

Received
14 III 1963

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