R. I. Anishchenko

On a Boundary-Value Problem for the Thomas–Fermi and Thomas–Fermi–Dirac Equations

(Presented by Academician M. A. Lavrentiev, 9 III 1962)

Nonnegative solutions of the boundary-value problem are considered:

\[ \varphi''(x)=f(x,\varphi(x))\psi(x); \tag{1} \]

\[ \varphi(0)=y_0,\qquad R\varphi'(R)-\varphi(R)=-q, \tag{2} \]

where \(y_0>0,\ q<y_0,\ R>0\). It is assumed that the functions \(f(x,y)\) and \(\psi(x)\) satisfy the conditions: 1) \(f(x,y)\) is defined and continuous for \(x\ge 0,\ y\ge 0\); 2) \(f(x,y)\) is an increasing function of \(y\) in the domain \(x\ge 0,\ y\ge 0\); 3) \(\psi(x)\) is continuous for \(x>0\), positive for \(x>0\), and integrable on every interval \([0,X]\), where \(X>0\). Let, in addition, either conditions 4)—6) be fulfilled: 4) \(f(x,0)>0\) for \(x>0\); 5) \(f(x,y)\) satisfies a Lipschitz condition with respect to \(y\) in the domain \(Q\ge y\ge a>0,\ P\ge x\ge 0\), for arbitrary positive \(P,Q,a\); 6)

\[ \int_0^{+\infty}\psi(x)f(x,0)\,dx=+\infty; \]

or conditions \(4'\)—\(6'\): \(4'\) \(f(x,0)\equiv 0\) for \(x\ge 0\); \(5'\) \(f(x,y)\) satisfies a Lipschitz condition with respect to \(y\) in the domain \(Q\ge y\ge 0,\ P\ge x\ge 0\), for arbitrary \(P>0,\ Q>0\);

\[ 6')\quad \int_0^{+\infty}\psi(x)\,dx=+\infty. \]

Problems of this type occur in the statistical theory of the atom for the Thomas–Fermi and Thomas–Fermi–Dirac equations \((^1)\).

Problem (1), (2) for \(q=0\) and under conditions 1)—3), \(4'\)—\(6'\) was considered in the note \((^2)\). Under assumptions 1)—6) and for \(y_0>q\ge 0\) \((^3)\) it was proved:

Theorem 1. There exists an \(\bar x(q)\) such that, for all \(R\) satisfying the inequalities

\[ 0<R\le \bar x(q), \tag{3} \]

and only for them, problem (1), (2) has a solution, and it is unique for each \(R\).

Here results are obtained for the case \(q<0\).

Theorem 2. Under conditions 1)—3), \(4'\)—\(6'\), and \(q<0\), for every \(R>0\) there exists a unique solution of problem (1), (2).

Theorem 3. Under conditions 1)—6) and \(q<0\), there exists an \(\bar x(q)\) such that, for every \(R\) satisfying the inequalities \(0<R<\bar x(q)\), problem (1), (2) has a unique solution. For \(R\ge \bar x(q)\), all solutions of problem (1), (2) are obtained as solutions of equation (1) under the conditions

\[ \varphi(0)=y_0,\qquad \varphi(\xi)=0,\qquad \varphi'(\xi)=0, \tag{4} \]

where \(\xi=\bar x(0)<\bar x(q)\).

If, in addition to conditions 1)—6), the following condition is fulfilled: 7) \(f(x,y)\) has a continuous derivative \(\partial f/\partial y\) for \(\delta\ge y>0\), for some \(\delta>0\), such that \(\partial f/\partial y=O(y^{-\mu})\), where \(\mu<1\), for small \(y\), or condition \(5'\), then problem (1), (4) has a unique solution, and for \(x\ge \xi\) (uniqueness for \(0\le x\le \xi\)

follows from the maximum conditions), and the solution of problem (1), (2) exists for those \(R\) satisfying (3), and only for them.

To prove Theorems 1–3, one considers the problem equivalent to the given one: to find \(y(x)\) and \(C\) such that the equation

\[ y''(x)=f(x,y(x)+Cx)\psi(x) \tag{5} \]

and the conditions

\[ y(0)=y_0, \tag{6} \]

\[ y(R)=q,\qquad y'(R)=0,\qquad y(x)\geq -Cx\quad (0\leq x\leq R). \tag{7} \]

are satisfied.

Obviously, \(\varphi(x)=y(x)+Cx\), \(\varphi'(R)=C\). Under conditions 1)–6) it is proved that there exists an interval of values of \(C\), \((\overline C,+\infty)\), such that for every \(C\in[\overline C,+\infty)\) there exists a solution \(y(x)\) of equation (5) satisfying the initial condition (6) and, at some point \(x=\overline x(q,C)\), the conditions

\[ y(\overline x)=q,\qquad y'(\overline x)=0, \tag{8} \]

with \(y(x)>-Cx\) for \(C>\overline C\), \(\overline x\geq x\geq 0\). For fixed \(q\), the function \(\overline x=\overline x(q,C)\) depends continuously on \(C\) and decreases monotonically as \(C\) increases; moreover, for every \(\eta>0\) there is a sufficiently large \(C\) such that \(\overline x(q,C)\leq \eta\). Denote \(\overline x(q)=\overline x(q,\overline C)\), and let \(Y(x)\) be the corresponding solution of (5), (6), (8), where \(C=\overline C\). In the case \(q>0\) we have \(\overline x=q/|\overline C|\), \(\overline C<0\), \(Y(x)>-\overline Cx\) for \(0\leq x<\overline x\). In the case \(q<0\), \(y=Y(x)\) is tangent to the straight line \(y=-\overline Cx\) at the point \(x=\xi=\overline x(0)\), \(\overline C>0\). The uniqueness of the solution of problem (1), (4) under the additional condition 7) for \(f(x,y)\) can be proved with the aid of the corresponding integral equation and inequalities of Chaplygin type \({}^{(5)}\). For the Thomas–Fermi–Dirac equation \(y''=(y^{1/2}+\beta x^{1/2})^3x^{-1/2}\), \(\beta>0\), condition 7) is fulfilled and the solution exists only for \(R\leq \overline x\).

No two integral curves of problem (1), (2) intersect for \(x>0\). Under conditions 1)–6) and fixed \(q<0\), all integral curves of problem (1), (2) for \(R<\overline x\) are bounded below by the curve \(\varphi=\Phi(x)\), the greatest solution of problem (1), (4) for \(x\geq 0\), with
\[ \overline x\,\Phi'(\overline x)-\Phi(\overline x)=-q. \]
If \(\varphi=\varphi(x)\) is the least solution of (1), (4), then there exists an \(x_0\) such that
\[ x_0\varphi'(x_0)-\varphi(x_0)=-q. \]
For \(R>x_0\), problem (1), (2) has no solution. From the continuity and monotonicity of the function \(D(x)=x\varphi'(x)-\varphi(x)\) it follows that any solution \(\varphi(x)\) of problem (1), (2) for \(R=R_1\), \(q=q_1<y_0\), is the analytic continuation of the solution for each \(q\in(q_1,y_0)\) and some \(R=R_q\). If \(q_1<q_2<y_0\), then \(\overline x(q_1)>\overline x(q_2)\); in particular, if \(q_1>0\), \(q_2<0\), then \(\overline x(q_1)<\overline x(0)<\overline x(q_2)\). Under conditions 1)–3), 4′)–6′), the domain of admissible values of \(C\) (i.e., values for which there exist \(y(x)\geq -Cx\), tangent to the straight line \(y=q\) and satisfying condition (6)) is the interval \((0,+\infty)\), and the solution exists for every \(R>0\) (\(q<0\)).

The desired solution (5), (6), (7) satisfies the equation

\[ y(x)=q+\int_x^R (s-x)\psi(s)f(s,y(s)+Cs)\,ds,\qquad 0\leq x\leq R. \tag{9} \]

Conversely, \(y(x)\) and \(C\) satisfying (9), (6) with \(y(x)\ge -Cx\) satisfy (5), (6), (7). Let \(0<R<\bar{x}\). Consider the case \(q<0\) (the case \(q\ge 0\) was considered in (3)). Set

\[ y^{(0)}(x)= \begin{cases} q, & x\ge \dfrac{|q|}{C},\\[4pt] -Cx, & x<\dfrac{|q|}{C}; \end{cases} \qquad y^{(k)}(x)= \begin{cases} Y_k(x), & Y_k(x)\ge -Cx,\\ -Cx, & Y_k(x)<-Cx, \end{cases} \tag{10} \]

where

\[ Y_k(x)=q+\int_x^R (s-x)\psi(s) f\bigl(s,y^{(k-1)}(s)+Cs\bigr)\,ds,\qquad k=1,2,\ldots \]

Theorem 4. If there exists a solution of equation (9), \(y(x)>-Cx\) for \(0\le x\le R\), for some \(C\), then the sequence (10) converges uniformly to the solution.

Theorem 5. If for some \(C=C_1\) there exists a solution \(y_1(x)\) of equation (9), with \(y_1(x)>-C_1x\) for \(0\le x\le R\), then there exists a \(C_2<C_1\) such that for all \(C_2\le C\le C_1\) there exists a solution of equation (9), and it depends continuously on the parameter \(C\).

Let \(C\in (C^{(1)},C^{(2)})\) (\(C^{(2)}\) can be found using (10), (6), \(C^{(1)}\ge |q|/R\)).

Set, for example, \(C^{(3)}=\dfrac{C^{(1)}+C^{(2)}}{2}\), and find \(y^{(k)}(x,C^{(3)})\). If there exist \(k_1\) and \(x_1\) such that \(y^{(k_1)}(x,C^{(3)})>-C^{(3)}x\) for all \(x_1\le x\le R\) and \(y^{(k_1)}(x_1,C^{(3)})>y_0\), then \(C<C^{(3)}\), \(C\in (C^{(1)},C^{(3)})\). Otherwise \(C\in [C^{(3)},C^{(2)}]\). Putting \(C^{(4)}=\dfrac{C^{(1)}+C^{(3)}}{2}\) or \(C^{(4)}=\dfrac{C^{(2)}+C^{(3)}}{2}\) in the latter case, and continuing this process, we obtain a sequence of intervals nested in one another and contracting to the point \(C=C(R)\). Starting from some \(C^{(n)}\), all solutions of (9) corresponding to the left endpoints of these intervals converge to the desired one according to Theorem 5. We shall now find \(\xi\), \(\varphi(x)\), \(\Phi(x)\), \(\bar{x}\), \(\bar{C}\). Put

\[ \varphi^{(0)}(x,\xi)=0,\qquad \varphi^{(k)}(x,\xi)=\int_x^\xi (s-x)\psi(s)f\bigl(s,\varphi^{(k-1)}(s,\xi)\bigr)\,ds \quad (k=1,2,\ldots). \tag{11} \]

If there exists a solution \(\varphi(x,\xi)\) of the integral equation

\[ \varphi(x,\xi)=\int_x^\xi (s-x)\psi(s)f\bigl(s,\varphi(s,\xi)\bigr)\,ds,\qquad x\ge 0, \tag{12} \]

for some \(\xi\), then the sequence (11) converges to the solution \(\varphi(x,\xi)\). With the aid of (11) and the condition \(y(0)=y_0\), as well as the inequality \(\varphi(x)\le y_0(1-x/\xi)\), \(0\le x\le \xi\), one can find an interval \((\xi_1,\xi_2)\) such that \(\xi\in(\xi_1,\xi_2)\). Next, proceeding in the same way as in the determination of \(C=C(R)\), we find \(\xi\) and then \(\varphi(x)\) for \(x\ge 0\), using (11)*. To determine \(\Phi(x)\) for \(x\ge \xi\) (\(\Phi(x)=\varphi(x)\) for \(0\le x\le \xi\)) and \(\bar{x}\), put

\[ \Phi_0(x)=u(x),\qquad \Phi^{(k)}(x)=\int_\xi^x (x-s)\psi(s)f\bigl(s,\Phi^{(k-1)}(s)\bigr)\,ds, \tag{13} \]

where

\[ u(x)=\frac12 A_n(x-\xi)^2,\qquad A_n=\max_{\substack{\xi\le x\le R_1\\ 0\le y\le y_0}}\{\psi(x)f(x,y)\}. \]

From the inequalities

* If equation (12) has more than one solution, then, in order to find \(\varphi(x)\) for \(0\le x\le \xi\), we obtain a two-point problem \(\bigl(\varphi(0)=y_0,\ \varphi(\xi)=0\bigr)\).

it follows from Chaplygin’s theorem \((^5)\) that any solution of (1), (4) is smaller than \(u(x)\) inside the rectangle \(\xi \leq x \leq R_1,\ 0 \leq y \leq n y_0\). Then the sequence (13) converges to \(\Phi(x)\), the greatest solution of problem (1), (4) for \(x \geq \xi\). One can choose \(n\) and \(R_1\) so that, for some \(x=\bar{x}<R_1\), the equality

\[ \int_\xi^{\bar{x}} s\psi(s) f(s,\Phi(s))\,ds=|q|, \quad \text{where } \Phi(x)=\lim_{k\to\infty}\Phi^{(k)}(x), \]

will hold, i.e. \(\Phi(x)\) satisfies the condition \(\bar{x}\Phi'(\bar{x})-\Phi(\bar{x})=-q\). In this case

\[ \bar{C}=\Phi'(\bar{x})=\int_\xi^{\bar{x}}\psi(s) f(s,\Phi(s))\,ds. \]

If

\[ \int_\xi^{x_0}\psi(s) f(s,\varphi(s))\,ds=|q|, \]

where \(\varphi(x)\) is the least solution of (1), (4) for \(x\geq \xi\), then for \(R>x_0\) problem (1), (2) has no solution.

For the Thomas–Fermi–Dirac equation in the case \(q=0\), an example was considered in the note \((^4)\).

Remark. For \(q\geq y_0\), problem (1), (2) has no solution.

Theorem 6. Under conditions 1)—3), 4′)—6′) and \(y_0>q>0\), there exists such a finite interval \((0,\bar{x}(q))\) that for all \(R\in(0,\bar{x}(q)]\), and only for them, there exists a solution of problem (1), (2), and it is unique for each \(R\).

Institute of Physics of Metals
Academy of Sciences of the USSR

Received
22 II 1962

CITED LITERATURE

1. P. Gombás, Statistical Theory of the Atom and Its Applications, IL, 1951.
2. R. I. Anishchenko, Izv. Vyssh. uchebn. zaved., Matematika, No. 6, 3 (1960).
3. R. I. Anishchenko, Sibirsk. matem. zhurn., 11, No. 2 (1961).
4. R. I. Anishchenko, Fiz. met. i metalloved., 12, issue 1 (1961).
5. S. A. Chaplygin, A New Method of Approximate Integration of Differential Equations, Moscow–Leningrad, 1950.