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UDC 517.946
MATHEMATICS
A. M. NAKHUSHEV
ON A PROBLEM OF MIXED TYPE FOR THE EQUATION \(y(y-1)u_{xx}+u_{yy}=0\)
(Presented by Academician M. A. Lavrent’ev on 11 V 1965)
The equation
\[ L(u)\equiv k(y)u_{xx}+u_{yy}=0, \tag{L} \]
where \(k(y)=y^2-y\), is an equation of mixed type; it is elliptic for \(y<0\) and \(y>1\), hyperbolic for \(0<y<1\), and along the straight lines \(y=0,\ y=1\) it degenerates parabolically. The families of curves
\(x\pm { }^1\!/\!_4 k'\sqrt{-k}\pm { }^1\!/\!_8\arcsin k'=\mathrm{const}\) are real characteristics of equation (L), while the straight lines \(y=0,\ y=1\) constitute the locus of points of return of this family.
- We introduce the following notation:
\[ \omega(y)= { }^1\!/\!_4 k'\sqrt{-k}+ { }^1\!/\!_8 \arcsin k', \qquad \tilde y(y)=-(3/2a)^{2/3}(\omega\mp \pi/16)^{2/3}, \]
\[ 0\leq y\leq 1, \tag{1} \]
\[ \omega(y)= { }^1\!/\!_4 k'\sqrt{\bar k}-{ }^1\!/\!_8 \ln |k'+2\sqrt{\bar k}|, \qquad \tilde y(y)=(3\omega/2a)^{2/3}, \]
\[ y\leq 0,\quad y\geq 1; \]
\[ \xi(x,y)=(x+\omega-\pi/16)/a,\qquad \eta(x,y)=(x-\omega+\pi/16)/a, \]
\[ a=\mathrm{const},\quad \tilde x=x/a; \tag{2} \]
\[ \pi/8<a<\pi/4,\qquad b(\tilde y)=[\tilde y'(y)]^{-2}\tilde y'''(y), \qquad \tilde\omega(y)=\exp\left({ }^1\!/\!_2\int_0^{\tilde y} b(s)\,ds\right). \]
In the plane of the variables \(x,y\), consider the points
\(A_1(0,0)\), \(A_0(\pi/8,0)\), \(A_2(a,0)\), \(B(a-\pi/16,{ }^1\!/\!_2)\), \(A_3(a,1)\), \(A(a-\pi/8,1)\), \(A_4(0,1)\), \(B_1(\pi/16,{ }^1\!/\!_2)\). Let \(D^*\) be a bounded simply connected domain bounded by: 1) the arc \(\sigma_0\):
\((x-a/2)^2+\omega^2=a^2/4,\ y\leq 0\); 2) the arc \(\sigma_1\):
\((x-a/2)^2+\omega^2=a^2/4,\ y\geq 1\); 3) the characteristics \(A_1B_1:\eta=\pi/8a\), \(B_1A_4:\xi=0\), \(A_2B:\xi=1-\pi/8a\), \(BA_3:\eta=1\) of equation (L). Let, further, \(B_0(D)\) be the point of intersection of the characteristics \(A_0B_0:\eta=\pi/4a\) and \(AB_0:\xi=1-\pi/8a\) \((AD:\eta=1-\pi/8a\) and \(B_1A_4)\); \(D\) be the subdomain of \(D^*\) lying above the curve \(DAB\); \(D^+\) the elliptic part of the domain \(D\); \(\Delta\) (\(\Delta^*\)) the hyperbolic part of \(D\) lying above the curve \(A_4DA\) \((ABA_3)\); \(D^-\) the domain of the characteristic quadrilateral \(AB_0A_0D\).
Following A. V. Bitsadze \((^1)\), by a solution of equation (L) that is regular in the domain \(D^*\) we shall mean a solution \(u(x,y)\) possessing the following properties: 1) \(u\) is continuous everywhere in the closed domain \(\overline{D^*}\), except at the points \(A,A_0,A_2\) and, possibly, at the points \(A_1,A_3,A_4\), where it tends to \(\infty\) of order \(\alpha<{}^1\!/\!_6\); 2) \(u_y\) (\(u_x\)) is continuous everywhere in \(\overline{D^*}\), except, possibly, at the points \(A,A_i,\ i=0,\ldots,4\), and at the characteristics issuing from these points, where it tends to \(\infty\) of order \(<{}^5\!/\!_6\) (\(<{}^7\!/\!_6\)) and \(<1\) (\(<1\)), respectively; 3) \(u\) is twice continuously differentiable everywhere in \(D^*\), except, possibly, at the boundary of the domain \(D^-\).
Problem 1. Find a regular solution in the domain \(D^*\) of equation (L), satisfying the conditions:
\[ (\widetilde{\omega}u)_{\sigma_0}=\varphi_0(\widetilde{x}),\quad (\widetilde{\omega}u)_{\sigma_1}=\varphi_1(\widetilde{x}),\quad 0<\widetilde{x}<1; \tag{3} \]
\[ (\widetilde{\omega}u)_{AB_0}=\psi_1(\eta),\quad x_0<\eta\leq \pi/4a;\quad (\widetilde{\omega}u)_{AD}=\psi_2(\xi), \]
\[ 0\leq \xi < x_0=1-\pi/8a, \]
where \(\widetilde{\omega}\) is given by formulas (1), (2), with in (1) the minus sign taken before \(\pi/16\), and \(\varphi_i\) and \(\psi_i\) are representable in the form
\[ \varphi_0(\widetilde{x})=\alpha_0(\widetilde{x})\widetilde{x}^{-\chi_0}(1-\widetilde{x})^{-\alpha/2},\quad \varphi_1(\widetilde{x})=\alpha_1(\widetilde{x})(\widetilde{x}-\widetilde{x}^2)^{-\chi_1}, \]
\[ \psi_1(\eta)=\beta_1(\eta)(\eta-x_0)^{-\alpha}(\pi/4a-\eta)^{1/6-\alpha},\quad \psi_2(\xi)=\beta_2(\xi)(x_0-\xi)^{-\alpha}, \]
where
\[ \alpha_i\in C^{(2)}\quad (0\leq \widetilde{x}\leq 1),\quad \alpha_0(1)\neq 0;\quad \beta_1\in C^{(4)}\quad (x_0\leq \eta\leq \pi/4a), \]
\[ \beta_i(x_0)\neq 0,\quad \beta_1(\pi/4a)\neq 0,\quad \beta_2\in C^{(4)}\quad (0\leq \xi\leq x_0),\quad 1/12>\chi_i=\mathrm{const}. \]
2. Uniqueness of the solution of Problem 1 follows directly from the uniqueness of the solution of the Tricomi problem for equation (L) in the domain \(D^{-}\) with data on the characteristics \(AB_0, AD\), and from the following two lemmas.
Lemma 1. Let \(u(x,y)\) be a regular solution in the domain \(\Delta\) \((\Delta^*)\) of equation (L), vanishing on \(AD\) \((AB)\). Then
\[ \int_0^{x_1} u(x,1)u_y(x,1)\,dx\geq 0 \quad \left( \int_{x_1}^{a} u(x,1)u_y(x,1)\,dx\geq 0 \right), \quad x_1=a-\pi/8. \]
Lemma 2. Let \(u(x,y)\) be a regular solution in the domain \(D^+\) of equation (L), vanishing on \(\sigma_1\). Then
\[ \int_0^a u(x,1)u_y(x,1)\,dx\leq 0. \]
The validity of Lemma 1 is established by the method of F. I. Frankl \((^2)\). Lemma 2 is proved in the same way as in the case of the Tricomi equation \((^2)\).
3. Proof of existence of the solution.
Lemma 3. Let: 1) \(u\in C(\overline{D^+})\); 2) \(u\in C^{(1)}(D^+-A_3-A_4)\); 3) \(u\in C^{(2)}(D^+)\), \(L(u)\equiv 0\) in \(D^+\); 4) \((u)_{\sigma_1}=0\), \(u_y(x,1)+{}^{1}/_{5}u(x,1)=0\) for \(0<x<a\). Then \(u\equiv 0\) in \(\overline{D^+}\).
Lemma 4. Let \(u\) satisfy the conditions of Lemma 1 and
\[ u_y(x,1)+{}^{1}/_{5}u(x,1)=0 \]
for \(0<x<x_1\). Then \(u\equiv 0\) in \(\Delta\).
Lemma 3 follows simply from the known Zaremba–Giraud extremum principle \((^{3,4})\), and Lemma 4 is a consequence of Lemma 1.
In equation (L) we pass to Frankl’s variables \((^5)\) \(\widetilde{x},\widetilde{y}\) and to the new function \(v=\widetilde{\omega}u\). Then \(v\) will satisfy the equation
\[ \widetilde{y}v_{\widetilde{x}\widetilde{x}}+v_{\widetilde{y}\widetilde{y}}+c(\widetilde{y})v=0, \tag{\(\widetilde{L}\)} \]
where
\[ c(\widetilde{y})=-{}^{1}/_{2}\left(b_{\widetilde{y}}+{}^{1}/_{2}b^2\right), \]
and the domain \(D\) \((D^+,\Delta,\Delta^*)\) passes into the domain \(\widetilde{D}\) \((\widetilde{D}^+,\widetilde{\Delta},\widetilde{\Delta}^*)\), bounded by the normal contour
\[ \widetilde{\sigma}_1:\quad (\widetilde{x}-{}^{1}/_{2})^2+{}^{4}/_{9}\widetilde{y}^3={}^1/_4 \]
and the characteristics \(\widetilde{A}_4\widetilde{D},\ \widetilde{D}\widetilde{A},\ \widetilde{A}\widetilde{B},\ \widetilde{B}\widetilde{A}_3\) of equation \((\widetilde{L})\).
It is easy to show that
\[ \widetilde{y}=(y-1)a^{-2/3}F^{2/3}\left(-{}^{1}/_{2},{}^{3}/_{2},{}^{5}/_{2},1-y\right) \]
in \(\overline{D}\); consequently, \(c(\widetilde{y})\) for
\[ \widetilde{y}>-(3\pi/16a)^{2/3}\quad (y>0) \]
by virtue of (2) admits derivatives of any order, and \(c(0)>0\).
Problem (1,1). Find a regular solution in the domain \(\widetilde{D}^{+}\) of equation \((\widetilde{L})\), satisfying the conditions:
\[ (v)_{\widetilde{\sigma}_1}=\varphi_1(\widetilde{x})\quad \text{for}\quad 0<\widetilde{x}<1, \]
\[ (v_{\widetilde{y}})_{\widetilde{y}=0}=\nu(\widetilde{x})\quad \text{for}\quad 0<\widetilde{x}<x_0,\quad x_0<\widetilde{x}<1. \]
Problem (1,2). Find a regular solution in the domain \(\widetilde{\Delta}\) \((\widetilde{\Delta}^*)\) of equation \((\widetilde{L})\), satisfying the conditions
\[ (v)_{\widetilde{A}\widetilde{D}}=\psi_2(\xi)\quad \text{for}\quad 0\leq \xi<x_0; \]
\((v\tilde y)_{\tilde y=0}=\nu(\tilde x)\) for \(0<\tilde x<x_0\) (\(v=\psi_1(\eta)\) on \(\widetilde{AB}\) for \(x_0<\eta\leqslant 1\); \(v_{\tilde y}^{\,2}=\nu(\tilde x)\) for \(\tilde y=0\) and \(x_0<\tilde x<1\)).
In both problems it is assumed that \(\nu(\tilde x)\in C^{(1)}\) \((0<\tilde x<x_0,\ x_0<\tilde x<1)\), but may tend to infinity of order \(<5/6\) for \(\tilde x=0,x_0,1\).
The uniqueness of the solution of problems (1,1) and (1,2) is established with the aid of Lemmas 3 and 4, while the proof of existence is carried out according to the scheme proposed by Gellerstedt in [6].
Lemma 5. Suppose: 1) there exist solutions of Problem 1 in the domain \(D\); 2) \(\tau(x)=(v)_{\tilde y=0}\), \(\nu(x)=(v_{\tilde y})_{\tilde y=0}\). Then
\[ \tau(x)=\gamma\int_0^1 \nu(t)\left[L(x,t)-|t-x|^{-1/3}+(t+x-2tx)^{-1/3}\right]\,dt+\Phi(x), \qquad 0<x<1; \tag{4} \]
\[ \tau(x)=\gamma\int_x^{x_0}\nu(t)(t-x)^{-1/3}P\bigl((t-x)^{4/3}\bigr)\,dt+\Psi_2(x), \qquad 0<x<x_0; \tag{5} \]
\[ \tau(x)=\gamma\int_{x_0}^{x}\nu(t)(x-t)^{-1/3}P\bigl((x-t)^{4/3}\bigr)\,dt+\Psi_1(x), \qquad x_0<x<1; \tag{6} \]
where: 1) \(\gamma\equiv\mathrm{const}\), \(L\in C\) \((0\leq x,t\leq 1)\), \(L\in C^{(2)}\) \((0<x,t<1)\); moreover for \(x=t\) \((x=t=0,1)\) the first derivatives may tend to infinity as \(\ln|x-t|\) \(\bigl((t+x-2tx)^{-1/3}\bigr)\); 2) \(\Phi\in C^{(3)}\) \((0<x<1)\) and for \(x=0,1\) tends to infinity of order \(2\chi_1\); 3) \(P\bigl((x-t)^{4/3}\bigr)\) admits derivatives of any order for \(x\ne t\), moreover \(P(0)=1\), and the first derivatives for \(x=t\) tend to zero of order not less than \(1/3\); 4) \(\Psi_2\in C^{(3)}\) \((0\leq x<x_0)\), \(\Psi_1\in C^{(3)}\) \((x_0<x\leq 1)\), and for \(x=x_0\) \(\Psi_i\) tends to infinity of order \(\alpha\).
Lemma 5 follows directly from the constructive properties of the solution of problems (1,1) and (1,2).
From (4), (5), and (6), after a series of transformations entirely analogous to the transformations used in the case of the Tricomi problem, for determining \(\nu(x)\) we obtain the singular integral equation
\[ \nu(x)=\lambda\int_0^{x_0}\nu(t)\left[ \left(\frac{t-x_0}{x-x_0}\right)^{2/3}\frac{1}{t-x} + \left(\frac{t+x_0-2tx_0}{x-x_0}\right)^{2/3}\frac{1}{t+x-2tx} \right]dt- \]
\[ -\lambda\int_{x_0}^{1}\nu(t)\left[ \left(\frac{t-x_0}{x-x_0}\right)^{2/3}\frac{1}{t-x} - \left(\frac{t+x_0-2tx_0}{x-x_0}\right)^{2/3}\frac{1}{t+x-2tx} \right]dt+ \tag{7} \]
\[ +\lambda\int_0^1 \nu(t)k(x,t)\,dt+h(x), \qquad \lambda=\frac{1}{\pi\sqrt{3}}, \]
where \(h(x)\in C^{(2)}\) \((0<x<x_0,\ x_0<x<1)\); \(h(x)\) for \(x=x_0\) tends to infinity of order \(\alpha+2/3\) and to infinity of the same order at the points \(x=0,\ x=1\), if \(2\chi_1=\alpha\); the kernel \(k(x,t)\) has the following properties: 1) it is regular for \(0<x,t<1\) and \(x\ne t\); for \(x=t\) the first derivatives may tend to infinity of order \(\leq \varepsilon+2/3\); 2) for \(0<x,t<1,\ x\ne x_0\), it satisfies the inequality
\[ |k(x,t)|\leq M_1(x-x_0)^{-2/3}\bigl[|L(x_0,t)|+ \]
\[ +|x-x_0|^{1-\varepsilon}|\ln(t+x-2tx)|\bigr]+M_2, \]
where \(0<M_i=\mathrm{const}\), and \(\varepsilon\) is an arbitrarily small positive number.
By the change of variable of integration \(t=(s+1)/2\), equation (7) can be rewritten in the form
\[ \tilde v(y)=\lambda\int_{-1}^{y_0}\tilde v(s)\left[\left(\frac{s-y_0}{y-y_0}\right)^{2/3}\frac1{s-y}+ \left(\frac{1-sy_0}{y-y_0}\right)^{2/3}\frac1{1-sy}\right]\,ds- \tag{8} \]
\[ -\lambda\int_{y_0}^{1}\tilde v(s)\left[\left(\frac{s-y_0}{y-y_0}\right)^{2/3}\frac1{s-y}- \left(\frac{1-sy_0}{y-y_0}\right)^{2/3}\frac1{1-sy}\right]\,ds+ \lambda\int_{-1}^{1}\tilde v(s)\tilde k(y,s)\,ds+\tilde h(y), \]
where \(y=2x-1,\ y_0=2x_0-1,\ \tilde v(y)=v((y+1)/2)\).
Equation (8) for \(\tilde k(y,s)\equiv 0\) was studied by Gellerstedt (7). Using his results and relying on the uniqueness of the solution of problems 1; 1.1; 1.2, it is not difficult to show that equation (7) is unconditionally solvable (in the class of sought solutions) and that its solution \(v(x)\in C^{(1)}\) \((0<x<x_0,\ x_0<x<1)\), moreover it becomes infinite of order \(\alpha+2/3\) for \(x=0,\ x_0,\ 1\).
Knowing \(v(x)\) from (7), with the help of problems 1.1 and 1.2 we prove the existence of a solution of problem 1 in the domain \(D\).
The construction of the solution \(u(x,y)\) of problem 1 in the domain \(D^{-}\) presents no essential difficulty, and therefore we restrict ourselves to giving the final result.
There exists a solution \(u(x,y)\) of problem 1 in the domain \(D^{-}\), and it has the following properties: 1) \(u\in C(\bar D^{-}-A-A_0)\), and at the points \(A,\ A_0\) it becomes infinite of order \(\alpha\); 2) \(\partial u/\partial \xi\) is continuous everywhere in the closed domain \(\bar D^{-}\), except for the characteristic \(AB_0\) and the point \(A_0\), where it becomes infinite of order \(\alpha+5/6\) for \(y\ne 1\) and infinite of order \(\alpha+1\) at the points \(A,\ A_0\); 3) \(\partial u/\partial \eta\) is continuous everywhere in \(\bar D^{-}\), except for the characteristics \(AD,\ A_0B_0\), where it becomes infinite of order \(\alpha+5/6\) for \(y\ne 1,\ y\ne 0\) and infinite of order \(\alpha+1\) at the points \(A,\ A_0\); 4) let \(\widetilde{(\omega u)}_{A_0B_1}=\tilde\psi_1(\eta),\ \pi/8a\le \eta<\pi/4a;\ \widetilde{(\omega u)}_{A_0B_0}=\tilde\psi_2(\xi),\ 0<\xi\le x_0\), while \(\omega\) is given by formulas (1), (2), where before \(\pi/16\) the sign \(+\) remains. Then
\[ \tilde\psi_1^{(i)}(\eta)=b_i(\eta)(\pi/4a-\eta)^{-\alpha-i},\qquad i=0,1,2,3,4; \]
\[ \tilde\psi_2^{(i)}(\xi)=a_i(\xi)\xi^{-\alpha-i}(x_0-\xi)^{1/6-\alpha-i},\qquad i=1,2,3,4; \]
\[ \tilde\psi_2^{(i)}(\xi)=a_0(\xi)\xi^{-\alpha},\qquad i=0, \]
where
\[ b_i\in C\left(\pi/8a\le \eta\le \pi/4a\right),\qquad a_i\in C(0\le \xi\le x_0),\qquad b_i(\pi/4a)\ne 0, \]
\[ a_i(0)\ne 0, \]
\((i)\) denotes the order of the derivative.
The construction of the solution \(u(x,y)\) of problem 1 in the domain \(D^*-(D+D^{-})\) is carried out in the same way as in the domain \(D\), proceeding only from the enumerated properties of the function \(u\) on the characteristics \(A_0B_1\) and \(A_0B_0\).
I express my deep gratitude to A. V. Bitsadze, who suggested studying boundary-value problems for equations of type (L) in domains containing both degeneration lines, and under whose supervision the present work was carried out, and to S. A. Tersenov for valuable advice and attention.
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
6 V 1965
CITED LITERATURE
- A. V. Bitsadze, DAN, 70, No. 4 (1950).
- F. I. Frankl, Izv. AN SSSR, ser. matem., 9, 121 (1945).
- A. V. Bitsadze, Equations of mixed type, Moscow, 1959.
- S. Agmon, L. Nirenberg, M. Protter, Com. Pure and Appl. Math., 6, 455 (1953).
- F. I. Frankl, Izv. AN SSSR, ser. matem., 9, 387 (1945).
- S. Gellerstedt, Sur un problème aux limites pour une équation linéaire aux dérivées partielles du second ordre de type mixte, Thèse, Uppsala, 1935.
- S. Gellerstedt, Ark. Mat., Astron., Fys., 26A, No. 3 (1937).