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UDC 517.946.4
MATHEMATICS
M. E. LERNER, S. P. PULKIN
ON A SINGULAR PROBLEM WITH CONDITIONS OF F. I. FRANKL AND F. TRICOMI
(Presented by Academician I. N. Vekua, 21 VI 1966)
1. Consider the equation
\[ S(u)\equiv u_{xx}+\operatorname{sgn} y\,u_{yy}+\frac{2q}{x}u_x=0,\qquad q>1 \tag{S} \]
in the domain \(D\), bounded by: 1) the arc \(AB\) of the circle \(x^2+y^2=1\), \(A(1;0)\), \(B(0;1)\); 2) the segment \(OB\) of the axis \(OY\); 3) the segment \(OC\) of the characteristic \(x+y=0\), \(C(c;-c)\), \(c\geq 1/2\); 4) the chordal arc \(AC\), which either coincides completely with the segment \(CI\) \((I(2c;0))\) of the characteristic \(x-y=2c\), or is situated between this characteristic and the segment \(AE\) \((E(1/2;-1/2))\) of the characteristic \(x-y=1\). The arc \(AC\) is such that any characteristic can intersect it in no more than one point. Let \(D_1\) and \(D_2\) be, respectively, the subdomains of ellipticity and hyperbolicity of equation (S). The magnitude of the arc \(s\) along the boundary of the domain \(D\) will be measured from the point \(A\) counterclockwise. Let \(l\) be the length of \(AC\), and \(TR\) a segment of the arc \(AB\), whose length is equal to the length of \(AC\) and whose endpoints \(T\) and \(R\) have respectively the arc coordinates \(\varepsilon>0\), \(\varepsilon+l<\pi/2\).
Singular problem \(\widetilde A_1\). In the domain \(D\), find a function \(u(x,y)\) possessing the properties: \(S(u)=0\) in \(D_1\cup D_2\), \(u\in C(\overline D)\), \(u\in C^1(D\setminus OA)\),
\[ u_y(x;0-0)=u_y(x;0+0) \]
(the “gluing” condition), and satisfying the following boundary conditions:
\[ \begin{array}{ll} 1.\quad u\big|_{\overline{AT}}=\gamma_1(s). & 3.\quad u\big|_{\overline{OC}}=\psi(x).\\[4pt] 2.\quad u\big|_{\overline{RB}}=\gamma_2(s). & 4.\quad u(\varepsilon+\mu)-u(-\mu)=g(\mu),\quad 0\leq \mu\leq l. \end{array} \tag{I} \]
Here \(\gamma_1,\gamma_2,\psi,g\) are prescribed functions. Condition 4 defines a curvilinear jump of compactification \({}^{1-3}\).
A generalized solution of problem \(\widetilde A_1\) of class \(\widetilde{\mathscr G}_2\) (of class \(\widetilde{\mathscr G}_2^0\)) will mean a function \(u(x,y)\), defined in \(D\) and satisfying the conditions: 1) \(S(u)=0\) in \(D_1\); 2) \(u\in C^2(D_1)\); 3) \(u\in C(\overline D)\); 4) \(u\) is a generalized solution of equation (S) of class \(\widetilde{\mathscr G}_2\) (class \(\widetilde{\mathscr G}_2^0\)) in the domain \(D_2\) (such classes of solutions are introduced below); 5) for any \(x\), \(0<x<1\), there exists
\[ \lim_{y\to 0} u_y=v(x), \]
where \(v(x)\) is absolutely integrable on \(0\leq x\leq 1\) and satisfies the Hölder condition on \(0<x<1\); 6) \(u\) satisfies the boundary conditions (I).
2. Problem NE. In the domain \(D_1\), find a bounded solution of equation (S) from the data:
\[ \lim_{y\to 0}u_y=v(x),\quad 0<x<1;\qquad u\big|_{AB}=\varphi(s). \tag{II} \]
Without loss of generality one may assume
\[ \gamma_1(0)=\gamma_2(\pi/2)=\psi(0)=0. \]
In the domain \(D_1\), the generalized solution of problem \(\widetilde A_1\) of class \(\widetilde{\mathscr G}_2\) is determined by the formula \({}^{(4)}\)
\[ u(x_0,y_0) = -\int_0^1 G_0(t,0;x_0,y_0)v(t)\,dt - \int_0^{\pi/2}\frac{\partial G_0}{\partial n}\,\varphi(s)\,ds, \tag{1} \]
where \(G_0(x,y;x_0,y_0)\) is the Green function of problem NE.
Lemma 1. Let \(u(x,y)\) be a generalized solution of problem \(\widetilde A_1\) of class \(\widetilde{\mathscr G}_2\). Then \(u(x,y)\) cannot attain in \(\bar D_1\) a greatest positive (least negative) value on the interval \(x=0,\ 0<y<1\), greater (smaller) than the values of \(u(x,y)\) on \(\overline{AB}\cup\overline{AO}\).
3. Problem \(\widetilde{\mathscr G}_2\). In the domain \(D_2\), find a solution \(u(x,y)\) of equation \((S)\) from the data
\[ \lim_{y\to 0} u_y=\nu(x),\quad 0<x<1;\qquad u\big|_{OC}=\psi(x). \tag{III} \]
By the change of variables \(\xi=y+x,\ \eta=y-x\) in the domain \(D_2\), equation \((S)\) is transformed to the form
\[ S_0(u)\equiv u_{\xi\eta}-\frac{q}{\xi-\eta}(u_\xi-u_\eta)=0. \]
The domain \(D_2\) goes over into the triangle \(O'A'C'\) (the domain \(\Delta\)). The boundary conditions (III) take the form
\[ \lim_{(\xi,\eta)\to(x;-x)}(u_\xi+u_\eta)=\nu(x),\quad 0<x<1;\qquad u(0;\eta)=\psi(-\eta/2)=\psi_1(\eta). \]
We shall call the function \(u(\xi,\eta)\) a regular solution of equation \((S_0)\) in \(\Delta\) (of equation \((S)\) in \(D_2\)) if \(S(u)=0\) in \(\Delta\), \(u\in C^2(\Delta)\), \(u\in C(\bar\Delta)\), \(u\in C^1\bigl(\bar\Delta\setminus (O'\cup A')\bigr)\).
Lemma 2. If the function \(\nu(\xi)\) is continuously differentiable on \(0<\xi<1\) and absolutely integrable on \(0\le \xi\le 1\), the function \(\psi_1(\eta)\) is continuous on \(-2c\le \eta\le 0\), twice continuously differentiable on \((-2c,0)\), and \(\psi_1(\eta)=(-\eta)^\alpha\widetilde\psi(\eta)\) \((0<\alpha\le 1,\ \widetilde\psi(\eta)\) bounded), then the regular solution \(u(\xi_0,\eta_0)\) of equation \((S_0)\) is determined by the formula
\[ \begin{aligned} u(\xi_0,\eta_0)=& \int_0^{\xi_0} H(\xi,\xi_0,\eta_0)\nu(\xi)\,d\xi -\int_{\eta_0}^{0} \left[ \frac{\partial A(0,\eta;\xi_0,\eta_0)}{\partial\eta} -\frac{q}{\eta}A(0,\eta;\xi,\eta_0) \right]\times \\ &\times \psi_1(\eta)\,d\eta +\left(\frac{\xi_0}{\xi_0-\eta_0}\right)^q\psi_1(-\xi_0) +\left(\frac{-\eta_0}{\xi_0-\eta_0}\right)^q\psi_1(\eta_0). \end{aligned} \tag{2} \]
Here \(H(\xi,\xi_0,\eta_0)\) is the value for \(\xi=-\eta\) of the Riemann function; \(A(\xi,\eta;\xi_0,\eta_0)\) is the Riemann–Hadamard function of problem \(C\mathscr G\) \((^4)\). The proof is analogous to \((^5)\), Theorem 6.
By a generalized solution of class \(\widetilde{\mathscr G}_2\) of equation \((S)\) in the domain \(D_2\) we shall mean the function defined by formula (2), in which \(\nu(\xi)\) is continuous on \(0<\xi<1\), \(\psi_1(\eta)\) is continuous on \(-2c\le \eta\le 0\), and
\[ \psi_1(\eta)=(-\eta)^\alpha\widetilde\psi(\eta)\quad (0<\alpha<1,\ \widetilde\psi(\eta)\ \text{bounded}). \]
If, in addition, \(\psi'(0)=0\), \(\nu(\xi)\) and \(\psi_1'(\eta)\) satisfy a Hölder condition respectively on \(0<\xi<1\) and \(-2c\le \eta\le 0\), then the function \(u(\xi_0,\eta_0)\) will be regarded as belonging to the class \(\widetilde{\mathscr G}_2^{\,0}\) of generalized solutions of equation \((S)\) in \(D_2\).
Lemma 3. Every generalized solution of class \(\widetilde{\mathscr G}_2\) of equation \((S)\) in the domain \(D_2\) can be represented as the limit of a sequence, uniformly convergent in \(\bar D_2\), of regular solutions of equation \((S)\) in \(D_2\).
Lemma 4. Let \(u(\xi,\eta)\) be a regular solution of equation \((S_0)\), with \(u(0,\eta)=0\). Then the maximum of \(u(\xi,\eta)\) in \(\bar\Delta\), if it is positive, is attained only on the segment \(O'A'\).
The proof follows from \(q/(\xi-\eta)>0,\ (q-q^2)/(\xi-\eta)<0\) in \(\Delta\).
Lemma 5. Let \(u(\xi,\eta)\) be a generalized solution of class \(\widetilde{\mathscr G}_2\) of equation \((S)\) in \(D_2\), with \(u(0,\eta)=0\). Then the maximum of \(u(\xi,\eta)\) in \(\bar\Delta\), if it is positive, is attained on the segment \(O'A'\).
The proof follows from Lemmas 3 and 4.
4. Theorem 1. There can exist no more than one generalized solution of problem \(\widetilde A_1\) of class \(\widetilde{\mathscr G}_2\).
Suppose the contrary. Then there exists a generalized solution \(u(x,y)\) of problem \(\widetilde A_1\) of class \(\widetilde{\mathscr G}_2\), taking positive values, satisfying
satisfying homogeneous boundary conditions. Suppose the maximum of \(u(x,y)\) in \(\overline D\) is attained in \(\overline D_2\). Then, by Lemma 5, it is attained at an interior point of the segment \(OA\), which contradicts the gluing condition. Assuming that the maximum of \(u(x,y)\) in \(\overline D\) is attained in \(\overline D_1\), by virtue of the preceding, Lemma 1 and the density jump condition we obtain that it is attained in \(\overline D_2\), which is impossible.
5. Solution of problem \(\widetilde A_1\). Let \(u(x,y)\) be a generalized solution of problem \(\widetilde A_1\) of the class \(\mathscr G_2^0\). Taking into account (1), (2) and the gluing condition, we obtain an integral equation of the first kind with unknown function \(\nu(x)\). Differentiating and transforming this equation, we arrive at the singular integral equation
\[ \nu(x)+\frac1\pi\int_0^1\left[\frac1{t-x}-\frac1{1-tx}+\frac{\mathscr L(t,x)}{t+x}\right]\nu(t)\,dt = \int_0^1 R(t,x)\nu'(t)\,dt+f(x), \tag{3} \]
where \(\mathscr L(t,x)\) is a homogeneous function, not depending, like \(R(t,x)\), on the boundary conditions. The kernel \(R(t,x)\) is continuous everywhere in the square \(0\le t,x\le 1\), with the exception of the point \((1;1)\), at which it has a singularity of arbitrarily small order. The function \(f(x)\) depends on \(\gamma_1(s)\), \(\gamma_2(s)\), \(\psi(x)\), \(g(\mu)\), \(\nu(x)\).
Lemma 6. In equation (3) the function \(f(x)\) can be represented in the form
\[ f(x)=f_0(x)+\int_0^1 Z(t,x)\nu(t)\,dt, \tag{4} \]
where \(f_0(x)\) does not depend on \(\nu(x)\), and the kernel \(Z(t,x)\) is continuous in the square \(0\le t,x\le 1\).
Taking Lemma 6 into account, we write equation (3) in the form
\[ \nu(x)+\frac1\pi\int_0^1\left[\frac1{t-x}-\frac1{1-tx}+\frac{\mathscr L(t,x)}{t+x}\right]\nu(t)\,dt = \int_0^1 R_0(t,x)\nu(t)\,dt+f_0(x), \tag{5} \]
where \(R_0(t,x)=R(t,x)+Z(t,x)\).
Lemma 7. If the function \(\gamma_1(s)\) has a derivative of second order at least in a neighborhood of the point \(s=0\), then the function \(f_0(x)\) satisfies the Hölder condition on \(0\le x\le 1\).
Applying the method of work (6), we reduce equation (5) to the equivalent Fredholm equation
\[ \varphi(\xi)=\int_0^\infty R_{00}(\eta,\xi)\varphi(\eta)\,d\eta+f_{00}(\xi), \tag{6} \]
\[ \varphi(\xi)=e^{-\xi}\nu(e^{-2\xi}),\qquad x=e^{-2\xi},\qquad t=e^{-2\eta}. \tag{7} \]
The function \(f_{00}(\xi)\equiv 0\) if \(\gamma_1(s)=\gamma_2(s)=\psi(x)=g(\mu)\equiv 0\). From Theorem 1 follows the uniqueness and existence of a solution of equation (6) in the class of functions satisfying the Hölder condition and absolutely integrable with some power \(p>1\) on \(0<\xi<+\infty\). Having found \(\varphi(\xi)\) from (6), we find \(\nu(x)\) from (7). By formula (2) we find \(u(x,y)\) in \(\overline D_2\), and, consequently, also \(u|_{\overline{AC}}\). With the help of the density jump condition we find \(u|_{\overline{TR}}\), and then by formula (1) determine \(u(x,y)\) in \(\overline D_1\). Thus, the following is valid.
Theorem 2. If the function \(\gamma_1(s)\) has a derivative of second order at least in a neighborhood of the point \(s=0\), then there exists a unique generalized solution of problem \(\widetilde A_1\) of the class \(\mathscr G_2^0\).
Received
4 VI 1966
CITED LITERATURE
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- Linear Equations of Mathematical Physics, “Nauka,” 1965.
- S. P. Pul’kin, Uch. zap. Kuibyshev Ped. Inst., no. 21 (1958).
- M. E. Lerner, Volga Mathematical Collection, no. 3 (1965).
- S. P. Pul’kin, Proceedings of the Second Scientific Conference of Mathematics Departments of Pedagogical Institutes of the Volga Region, no. 1, 1962.