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UDC 517.946.9
M. I. ALIEV
THE PROBLEM E FOR ONE CLASS OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS
(Presented by Academician M. A. Lavrent’ev, 19 XI 1966)
Let \(D\) be a finite simply connected domain bounded by the segment \(AB\) of the \(x\)-axis and by a Jordan curve \(\sigma\), lying in the half-plane \(y>0\) and with endpoints at \(A\) and \(B\). The equation
\[ Gu \equiv y^m u_{yy}+u_{xx}-f(x,y,u,u_x,u_y)=0, \tag{1} \]
where \(m>0\), and \(f\) is a given function of its arguments, is elliptic for \(y>0\), while on the line \(y=0\) it degenerates parabolically.
In [1] the solvability of the Dirichlet problem for equation (1) was investigated. In the present paper the problem E (see [2]) for equation (1) is studied.
Problem E. Find a twice continuously differentiable in the domain \(D\) solution of equation (1), bounded in the domain \(D\), continuous up to \(\sigma\), and taking prescribed continuous values on \(\sigma\).
Without loss of generality, we shall assume that zero values are prescribed on \(\sigma\).
Suppose that the function \(f\) and the curve \(\sigma\) satisfy the following conditions:
Condition A. 1) \(f\) is continuously differentiable with respect to its arguments, and
\(-f_u\le -k<0\) for \((x,y)\in D\) and all values of \(u,u_x,u_y\); 2) for \((x,y)\in D\), \(|u|\le N\), \(-\infty<u_x,u_y<\infty\), the estimates \(|f_u|\le M\), \(|f_{u_x}|\le M\), \(|f_{u_y}|\le M\) hold, where \(M(N)\) is some positive number.
Condition B′. a) For \(m=1\),
\(-\partial f/\partial u_y\ge 1\) in the domain
\[
\mathfrak{R}_h=\{(x,y)\in D-\overline{D}_h,\ |u|\le N,\ -\infty<u_x,u_y<\infty\},
\]
where
\[
D_h=\{(x,y)\in D,\ y>h\},
\]
and \(h\) is a sufficiently small positive number; b) for \(1<m<2\),
\(-\partial f/\partial u_y>y^{m-1}\) in \(\mathfrak{R}_h\); c) for \(m\ge 2\),
\(-\partial f/\partial u_y>0\) in \(\mathfrak{R}_h\).
Condition C. Let the curve \(\sigma\) at the points \(A(a,0)\) and \(B(b,0)\) \((a<b)\) end in arbitrarily small arcs \(AA'\) and \(BB'\), whose equations may be given in the form \(y=l(x)\), where \(l'(x)\ge 0\) on \(AA'\) and \(l'(x)\le 0\) on \(BB'\). The remaining part \(\sigma'\) of the curve \(\sigma\) is such that, for the \(\sigma_0'\)-image of \(\sigma'\) under the mapping \(\xi=x,\ t=2\sqrt{y}\), at each point \(\zeta\in\sigma_0'\) there exists a certain circle \(S\) whose intersection with the closed domain \(\overline{D}_0\) consists of just one point \(\zeta\), \(S\cap\overline{D}_0=\zeta\), where \(D_0\) is the image of the domain \(D\).
Theorem. If the function \(f\) satisfies conditions A and B′, and the curve \(\sigma\) satisfies condition C, then the solution of problem E for equation (1) exists and is unique.
We note that, under conditions A and B′, for the solution \(u(x,y)\) of problem E the estimate
\[ |u|\le \frac{1}{k}\max_{\overline{D}} |f_0(x,y)|\equiv \mu, \tag{2} \]
holds, where \(k\) is the number from condition A, and \(f_0(x,y)=f(x,y,0,0,0)\).
The uniqueness of the solution of problem E follows from the following lemma:
Lemma. Let the linear equation
\[ y^m u_{yy}+u_{xx}+a u_x+b u_y+cu=F(x,y) \tag{3} \]
with coefficients bounded in \(D\), where \(c\ge 0\) and the conditions are fulfilled: a) for \(m=1\), \(b\ge 0\) in \(D-D_h\); b) for \(1<m<2\), \(b>y^{m-1}\) in \(D-D_h\); c) for \(m\ge 2\), \(b>0\) in \(D-D_h\).
Then the solution of problem E for equation (3) is unique.
This lemma is proved analogously to the uniqueness lemma in [2]. The proof of the existence of a solution is carried out according to the scheme of [1], which is a certain modification of Perron’s method [3].
Let a continuous function \(\varphi(x,y)\) be given in the closed domain \(D+\partial D\), and let a disk \(K\subset D\) be given. By \(\mathfrak M_K(\varphi)\) we denote the function which, outside the disk \(K\) and on its boundary \(\partial K\), coincides with \(\varphi\), while inside \(K\) it is the solution of equation (1) constructed from the boundary values \(\varphi\) on \(\partial K\).
The function \(\varphi\) is called a supersolution (subsolution) if for every disk \(K\subset D\) one has \(\varphi\ge \mathfrak M_K(\varphi)\) \([\varphi\le \mathfrak M_K(\varphi)]\). A supersolution (subsolution) \(\varphi\) is called an upper (lower) function if \(\varphi\ge 0\) \((\varphi\le 0)\) on \(\partial D\).
From condition C it follows that for each point \((x_0,y_0)\in\sigma\) (we consider \(\sigma\) closed) there exists some point \((\bar x,\bar y)\) \((\bar y\ge 0)\) such that for any point \((x,y)\in D\)
\[ r^2=(x-\bar x)^2+4(\sqrt y-\sqrt{\bar y})^2 >R^2=(x_0-\bar x)^2+4(\sqrt{y_0}-\sqrt{\bar y})^2 . \]
The function
\[
w(x,y;x_0,y_0)=\mu\left[1-(1+R^2)^p(1+r^2)^{-p}\right],
\]
where \(\mu\) is the number from formula (2), for a suitable choice of the parameter \(p>0\), satisfies the conditions: 1) \(w(x,y;x_0,y_0)>0\) in \(D\), 2) \(w(x_0,y_0;x_0,y_0)=0,\ (x_0,y_0)\in\sigma\), 3) \(Gw\le 0\) in \(D\).
Using \(w(x,y;x_0,y_0)\), we form new functions
\[
\varphi_0(x,y)=\varepsilon/2+w(x,y;x_0,y_0),\qquad
\psi_0(x,y)=-\varepsilon/2-w(x,y;x_0,y_0),
\]
which will be, respectively, upper and lower functions for equation (1), and \(\varphi_0(x,y)=\varepsilon/2,\ \psi_0(x,y)=-\varepsilon/2\).
Since \(\varphi_0\) and \(\psi_0\) are continuous functions, there is a certain neighborhood \(\mathfrak A_0\subset\sigma\) of the point \(P_0(x_0,y_0)\) such that \(\varphi_0-\psi_0<2\varepsilon\) for all \((x,y)\in\mathfrak A_0\). Choosing, by the Heine–Borel lemma, a finite number of neighborhoods \(\mathfrak A_i\subset\sigma\) \((i=0,1,\ldots,n)\) covering the whole curve \(\sigma\), we obtain upper functions \(\varphi_0,\varphi_1,\ldots,\varphi_n\) and lower functions \(\psi_0,\psi_1,\ldots,\psi_n\) such that \(\varphi_i-\psi_i<2\varepsilon\) for \((x,y)\in\mathfrak A_i\). Then the functions \(\varphi=\min(\varphi_0,\varphi_1,\ldots,\varphi_n)\) and \(\psi=\max(\psi_0,\psi_1,\ldots,\psi_n)\) are, respectively, upper and lower functions, and \(\varphi-\psi<2\varepsilon\) for \((x,y)\in\sigma\). Hence, in view of the arbitrariness of \(\varepsilon>0\), it follows that the lower envelope of the upper functions and the upper envelope of the lower functions coincide everywhere on \(\sigma\).
We shall show that the lower envelope of the upper functions and the upper envelope of the lower functions coincide also inside the domain. For this it is enough to construct some upper function \(\widetilde\varphi\) and lower function \(\widetilde\psi\) for which at each point \((x,y)\in D\), \(\widetilde\varphi-\widetilde\psi<\varepsilon\) for any \(\varepsilon>0\). Denote by \(\Lambda\) the maximum of the difference \(\varphi-\psi\) in the domain \(D\), where \(\varphi\) and \(\psi\) are the functions constructed above.
We construct an upper function \(\varphi^{(1)}\) and a lower function \(\psi^{(1)}\) such that, for an arbitrary domain \(D_1,\ \overline{D}_1\subset D\),
\[ \max_{\overline{D}_1}(\varphi^{(1)}-\psi^{(1)})<q\Lambda, \tag{4} \]
where \(0<q<1\) is a certain number independent of \(\Lambda\).
For some disk \(K_i\subset D\) we form the functions \(\varphi_i=\mathfrak M_{K_i}(\varphi)\) and \(\psi_i=\mathfrak M_{K_i}(\psi)\). We shall show that
\[ \max_{L_i}(\varphi_i-\psi_i)<q_i\Lambda, \tag{5} \]
where \(L_i\) is a disk concentric with \(K_i\), with \(L_i\subset K_i\), and \(0<q_i<1\) is a number depending only on \(L_i\).
In the disk \(K_i\) the function \(\chi=\varphi_i-\psi_i\) satisfies the equation
\[ F(\chi)\equiv y^m\frac{\partial^2\chi}{\partial y^2} +\frac{\partial^2\chi}{\partial x^2} -\frac{\partial \bar f}{\partial \chi_x}\frac{\partial \chi}{\partial x} -\frac{\partial \bar f}{\partial \chi_y}\frac{\partial \chi}{\partial y} -\frac{\partial \bar f}{\partial \chi}\chi=0, \]
where the notation adopted is
\[ \bar g=g\left\{x,y,\psi_i+\theta(\varphi_i-\psi_i), \frac{\partial}{\partial x}[\psi_i+\theta_1(\varphi_i-\psi_i)], \frac{\partial}{\partial y}[\psi_i+\theta_2(\varphi_i-\psi_i)]\right\}, \quad 0<\theta,\theta_1,\theta_2<1. \]
Consider the function
\[ \tilde\chi=\{\exp a(r^2-R^2)+\lambda[1-(1+y)^{-\nu}]\}\Lambda, \]
where \(r^2=(x-x_1)^2+(y-y_1)^2\), \((x_1,y_1)\) is the center, \(R\) is the radius of the disk \(K_i\), and \(0<\lambda=\operatorname{const}<1\). Choosing sufficiently small \(a>0\) and sufficiently large \(\nu>0\), one can ensure that \(F\tilde\chi\leqslant0\). Then choose \(\lambda\) from the inequality
\[ \exp a(\rho^2-R^2)+\lambda<1, \tag{6} \]
where \(\rho\) is the radius of the disk \(L_i\). Since inside the disk \(K_i\), \(F\tilde\chi\leqslant0\), and on \(\partial K_i\) \(\varphi_i-\psi_i<\tilde\chi\), it follows that everywhere in the disk \(K_i\) \(\varphi_i-\psi_i\leqslant\tilde\chi\). Consequently,
\[ \max_{L_i}(\varphi_i-\psi_i)<\max_{L_i}\tilde\chi\leqslant\max_{\partial L_i}\tilde\chi \]
\[ =\{\exp a(\rho^2-R^2)+\lambda[1-(1+Y)^{-\nu}]\}\Lambda=q_i\Lambda, \]
where \(Y\) is the maximum of the ordinates of points of \(\partial L_i\). Hence, and from (6), (5) follows.
Take a finite covering of the domain \(\bar D_1\) by disks \(L_i\subset D\) \((i=1,2,\ldots,l)\) and construct the functions \(\varphi_i\) and \(\psi_i\). For each function \(\varphi_i\) in \(\psi_i\) inside the disk \(L_i\) we obtain inequality (5).
The functions \(\varphi^{(1)}=\min(\varphi_1,\ldots,\varphi_l)\) and \(\psi^{(1)}=\max(\psi_1,\ldots,\psi_l)\) will be, respectively, an upper and a lower function. Since every point of the domain \(\bar D_1\) belongs to some disk \(L_j\), we have
\[ \max_{\bar D_1}(\varphi^{(1)}-\psi^{(1)})\leqslant \max_{L_j}(\varphi_j-\psi_j)<q_j\Lambda\leqslant q\Lambda, \]
where \(q=\max(q_1,\ldots,q_l)\).
Thus we have constructed functions \(\varphi^{(1)}\) and \(\psi^{(1)}\) satisfying inequality (4). Continuing this process, we construct a sequence of upper functions \(\varphi^{(1)},\varphi^{(2)},\ldots\) and lower functions \(\psi^{(1)},\psi^{(2)},\ldots\), whose difference \(\varphi^{(s)}-\psi^{(s)}\) is everywhere in the domain \(D_1\) less than \(q^s\Lambda\).
Since \(q<1\) and \(D_1\) is an arbitrary subdomain of the domain \(D\), we obtain that the lower envelope of the upper functions and the upper envelope of the lower functions coincide everywhere in the domain \(D\). Denote this common envelope by \(v(x,y)\).
We shall prove that \(v(x,y)\) is a solution of equation (1). To this end it suffices to show that for any disk \(K\subset D\),
\[ v=\mathfrak M_K(v). \]
Using the constructed upper functions \(\varphi^{(1)},\varphi^{(2)},\ldots\), form the functions
\[ v_n=\mathfrak M_K(\varphi^{(n)}). \]
Outside the disk \(K\) and on \(\partial K\), the sequence \(\{v_n\}\) converges uniformly to the function \(v\), while inside \(K\) it is a sequence of solutions of equation (1) converging uniformly on the boundary \(\partial K\). By Harnack’s theorem (see, for example, (1)), the sequence \(\{v_n\}\) converges uniformly inside \(K\) to the solution of equation (1) taking the value \(v\) on \(\partial K\). From the continuity of this limiting function in \(K+\partial K\) it follows that
\[ v=\mathfrak M_K(v). \]
The boundedness of the solution in the domain \(D\) follows from the inequality
\[ \psi(x,y)\leqslant v(x,y)\leqslant \varphi(x,y), \]
where \(\varphi\) and \(\psi\) are some upper and lower functions.
Remark. The uniqueness of the solution of problem E holds without the additional requirement of condition C on the curve \(\sigma\).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
17 XI 1966
REFERENCES
- M. I. Aliev, Differential Equations, 3, No. 1 (1967).
- M. V. Keldysh, DAN, 77, No. 2 (1951).
- O. Perron, Math. Zs., 18, 42 (1923).