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A. DZHURAEV
ON THE POINCARÉ PROBLEM FOR A SECOND-ORDER ELLIPTIC EQUATION WITH SINGULAR COEFFICIENTS
(Presented by Academician I. N. Vekua on 23 IV 1962)
In the present note we study the plane boundary-value Poincaré problem (problem A) for an elliptic equation of the form
\[ \Delta u+a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}+cu=0 \tag{1} \]
in the case when the coefficients \(a(x,y)\), \(b(x,y)\) have a polar singularity of the first order, and the coefficient \(c(x,y)\) has a polar singularity of the second order at some fixed point of the domain\(^*\). In the case when \(a,b,c\in L_p\) for \(p>2\), this problem, both for simply connected and for multiply connected domains, has been well studied \((^{1-3})\). We also note that if \(c\equiv 0\), then equation (1), with respect to the function \(\partial u/\partial x-i\,\partial u/\partial y\), is transformed into a generalized Cauchy—Riemann complex equation \((^{1,4})\).
- Denote by \(D\) a bounded simply connected domain in the plane of the variables \(x,y\), with boundary \(\Gamma\) of class \((^{i})\) \(C_{\alpha}\), \(0<\alpha<1\). Let \(S_{\varepsilon}(D)\) be the Banach space of functions \(f(z)\) representable in the form
\[ f(z)=\frac{f_0(z)}{(x^2+y^2)^{\varepsilon/2}},\qquad f_0(z)\in S(D), \]
with norm
\[ \|f\|_{S_{\varepsilon}(D)}=\sup_{x,y\in D}(x^2+y^2)^{\varepsilon/2}|f(z)|, \]
where \(S(D)\) is the space of bounded measurable functions \(f_0\) with norm
\[ \sup_{x,y\in D}|f_0(z)|. \]
We shall say that \(u\in S_{\varepsilon}^{1}(D)\) if \(u\) has a generalized derivative, \(u\in S_{\varepsilon}(D)\), and
\[ \frac{\partial u}{\partial z}\in S_{1+\varepsilon}(D), \]
where
\[ \frac{\partial}{\partial z}=\frac12\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right). \]
Problem A. It is required to find a real-valued function \(u\in S_{\varepsilon}^{1}(D)\) satisfying in \(D\) the equation
\[ \Delta u+\frac{a}{r}\frac{\partial u}{\partial x}+\frac{b}{r}\frac{\partial u}{\partial y}+\frac{c}{r^2}u=0 \tag{1'} \]
and on \(\Gamma\) the condition
\[ \alpha u_x+\beta u_y=h, \tag{2} \]
where \(a,b,c\) are given real bounded measurable functions in \(D\); \(\alpha,\beta,h\) are real functions given on \(\Gamma\), Hölder continuous, and
\[
r=\sqrt{x^2+y^2}.
\]
Below, in order to study problem A, we shall relate it to a certain auxiliary Riemann—Hilbert boundary-value problem (problem P) for an elliptic system in two complex-valued functions, with coefficients having a polar singularity of the first order.
Introducing the operator
\[
\frac{\partial}{\partial \bar z}=\overline{\frac{\partial}{\partial z}},
\]
where the bar above indicates passage to the complex conjugate expression, problem A can be written in the form:
\[ \frac{\partial^2 u}{\partial \bar z\,\partial z} +\frac{A(z)}{|z|}\frac{\partial u}{\partial z} +\frac{\overline{A(z)}}{|z|}\frac{\partial u}{\partial \bar z} +\frac{c(x,y)}{4|z|^2}u=0; \tag{1''} \]
\[ \operatorname{Re}\lambda(t)\frac{\partial u}{\partial t}=h(t), \tag{2'} \]
where
\[
A(z)=(a+ib)/4,\qquad \alpha(t)+i\beta(t)=\lambda(t).
\]
\[ \text{\(^*\) This point can always be shifted to the origin of coordinates.} \]
Let \(z=\varphi(\zeta)\) be a holomorphic function conformally mapping the domain \(D\) onto the unit disk and satisfying the conditions
\[ \varphi(0)=0,\qquad \varphi'(0)>0. \tag{3} \]
It follows from (3) that
\[ \varphi(\zeta)=\zeta\varphi_0(\zeta),\qquad \varphi_0(\zeta)\ne 0 \text{ anywhere}. \tag{4} \]
By virtue of (4) it is not difficult to see that problem A is conformally invariant. Therefore, everywhere below, without restricting the generality of the problem, we shall assume that \(D\) is the unit disk: \(|z|<1\).
In this case equation (1) and condition (2) may be written in the form:
\[ \frac{\partial}{\partial \bar z}\left(z\frac{\partial u}{\partial z}\right) +\frac{A(z)}{|z|}\left(z\frac{\partial u}{\partial z}\right) +\frac{\overline{A(z)}}{|z|}\frac{z}{\bar z} \overline{\left(z\frac{\partial u}{\partial z}\right)} +\frac{c}{4z}u=0; \tag{1'''} \]
\[ \operatorname{Re}\,\bar t\cdot \lambda(t)\left(t\frac{\partial u}{\partial t}\right)=h(t). \tag{2''} \]
In what follows, an essential role is played by the study of the following problem:
Problem P. It is required to find a pair of functions \(u_1(z), u_2(z)\in S_\varepsilon(D)\), satisfying in \(D\) the system of equations
\[ \frac{\partial u_1}{\partial \bar z}=\frac{1}{\bar z}\,\bar u_2,\qquad \frac{\partial u_2}{\partial \bar z} +\frac{A(z)}{|z|}u_2 +\frac{\overline{A(z)}}{|z|}\frac{z}{\bar z}\bar u_2 +\frac{c}{4z}u_1=0 \tag{5} \]
and on \(\Gamma\) the conditions
\[ \operatorname{Re}\, i u_1=0,\qquad \operatorname{Re}\,\bar t\cdot \lambda(t)u_2=h(t). \tag{6} \]
The connection between problems A and P is established by the following:
Theorem 1. If \(u(x,y)\in S_\varepsilon^1(D)\) is a solution of problem A, then the pair of functions
\[ u_1=u,\qquad u_2=z\frac{\partial u}{\partial z} \]
of class \(S_\varepsilon(D)\) is a solution of problem P.
Conversely, if a pair of functions \(u_1,u_2\) of class \(S_\varepsilon(D)\) is a solution of problem P and if, in addition, the homogeneous Dirichlet problem
\[ \Delta u+\frac{c(x,y)}{r^2}u=0,\qquad u\big|_{\Gamma}=0 \tag{7} \]
has only the trivial solution from \(S_\varepsilon^1(D)\), then the function \(u_1\) belongs to the class \(S_\varepsilon^1(D)\) and is a solution of problem A.
Denote by \(l_P\) the number of linearly independent (over the field of real numbers) solutions of the homogeneous problem \(\mathrm P^0\) (\(h\equiv0\)), and by \(l_A\) the number of linearly independent solutions of the corresponding homogeneous problem \(\mathrm A^0\) (\(h\equiv0\)). Let, in addition, \(l_D\) be the number of linearly independent solutions of the Dirichlet problem (7). Then the relation \((^2,^3)\) holds:
\[ l_P-l_A=q, \tag{8} \]
where \(q\) is an integer nonnegative number not exceeding \(l_D\).
- Suppose that
\[ \chi=\frac{1}{2\pi}\{\arg \lambda(t)\}_{\Gamma}\ge -1. \]
Then, proceeding in the same way as in \((^1,^3)\), we verify that problem P is equivalent to the following system of integral equations:
\[ u_1-T_1u_2=\tilde c_1; \tag{9} \]
\[ u_2-T_2u_2-T_3u_1=\Phi'(z), \tag{10} \]
where \(T_1,T_2,T_3\) are linear bounded operators in the Banach space \(S_\varepsilon(\bar D)\) (cf. (4)), which have the form
\[ T_1u_2=-\frac{1}{\pi}\iint_D \left\{ \frac{e^{i\varphi(\zeta)}\overline{u_2(\zeta)}}{\bar\zeta(\zeta-z)} -\frac{z e^{-i\varphi(\zeta)}u_2(\zeta)}{\zeta(1-\bar\zeta z)} \right\}\,d\xi\,d\eta; \tag{11} \]
\[ T_2 u_2=\frac{1}{\pi}\iint_D\left\{\frac{A(\zeta)u_2+\overline{A(\zeta)}\dfrac{\zeta e^{i\varphi(\zeta)}}{\zeta e^{i\varphi(\zeta)}}\bar u_2}{|\zeta|(\zeta-z)} +z^{2\chi+3}\frac{\overline{A(\zeta)}u_2+A(\zeta)\dfrac{\overline{\zeta e^{i\varphi}}}{\zeta e^{i\varphi}}\bar u_2}{|\zeta|(1-\bar\zeta z)}\right\}\,d\xi\,d\eta; \tag{12} \]
\[ T_3 u_1=\frac{1}{4\pi}\iint_D\left\{\frac{e^{i\varphi(\zeta)}\overline{u_1(\zeta)}}{\zeta(\zeta-z)} +z^{2\chi+3}\frac{e^{-i\varphi(\zeta)}u_1}{\zeta(1-\bar\zeta z)}\right\}c(\xi,\eta)\,d\xi\,d\eta, \tag{13} \]
where \(\varphi(z)\) is a completely determined single-valued holomorphic function in \(D\), and
\[ \Phi(z)=\frac{1}{2\pi i}\int_\Gamma h(t)e^{\Omega(t)}\frac{t+z}{t-z}\frac{dt}{t} +i\tilde{\alpha}_0 z^{\chi+1} + \]
\[ +\sum_{k=0}^{\chi}\{\alpha_k(z^k-z^{2\chi+2-k})+i\beta_k(z^k+z^{2\chi+2-k})\}. \tag{14} \]
Here \(\Omega=\operatorname{Im}\varphi;\ c_1,\ \tilde{\alpha}_0,\ \alpha_k,\beta_k\ (k=0,1,2,\ldots,\chi)\) are arbitrary real constants.
Let
\[ K_{a,b}=\sup_{x,y\in D}\frac{r^\varepsilon}{2\pi}\iint_D\left(\frac{1}{|\zeta-z|}+\frac{r^{2\chi+3}}{|1-\bar\zeta z|}\right) \sqrt{\frac{a^2+b^2}{(\xi^2+\eta^2)^{1+\varepsilon}}}\,d\xi\,d\eta; \tag{15} \]
\[ K_{\nu,\chi}=\sup_{x,y\in D}\frac{r^\varepsilon}{\pi}\iint_D\left(\frac{1}{|\zeta-z|}+\frac{r^{2\chi+3}}{|1-\bar\zeta z|}\right)|\nu(\xi,\eta)|\,d\xi\,d\eta. \tag{16} \]
Then, if the inequality
\[ K_{a,b}+K_{1/4,-1}K_{c/4,\chi}<1, \tag{17} \]
is satisfied, the system (9)—(10) is always uniquely solvable. The system (9)—(10) corresponding to the homogeneous problem \(\mathrm{P}^0\) \((h\equiv0)\) then has exactly \(2\chi+4\) linearly independent (over the field of real numbers) solutions. Since every solution of problem (7) is also a solution of the problem
\[ \Delta u+\frac{c}{r^2}u=0,\qquad x'(s)u_x+y'(s)u=0, \tag{18} \]
it is not difficult to see that, when inequality (17) is fulfilled, the number \(l_D\) does not exceed two. Then, taking into account relation (8), we arrive at the following result:
Theorem 2. If \(\chi\ge -1\), then, when inequality (17) is fulfilled, problem \(A\) is always solvable, and the corresponding homogeneous problem \(A^0\) \((h\equiv0)\) has exactly \(2\chi+4-q\) linearly independent solutions, where \(q\) is a nonnegative integer not exceeding two.
3. Let \(\chi<-1\). In this case, following (1) (p. 298), put
\[ iz^m u_1=v_1,\qquad z^m u_2=v_2,\qquad m=-(\chi+1)>0. \tag{19} \]
Then the functions \(v_1,\ v_2\) will satisfy the problem
\[ \frac{\partial v_1}{\partial\bar z} -\frac{\overline{e^{i\varphi(\zeta)}}}{z}\,i\left(\frac{z}{\bar z}\right)^m\bar v_2=0; \tag{20} \]
\[ \frac{\partial v_2}{\partial\bar z} +\frac{A(z)}{|z|}v_2 +\frac{\overline{A(z)}}{|z|}\frac{z^{m+1}e^{i\varphi(\zeta)}}{\bar z^{m+1}e^{i\varphi(\zeta)}}\bar v_2 +i\,\frac{c(x,y)e^{i\varphi(\zeta)}}{4\bar z}\left(\frac{z}{\bar z}\right)^m\bar v_1=0, \tag{21} \]
\[ \operatorname{Re}t^{-m}v_1=0,\qquad \operatorname{Re}v_2=e^\Omega h. \]
Since \(m>0\), one can, as above, construct the corresponding equivalent system of integral equations with respect to the functions \(v_1\) and \(v_2\).
Returning then by formulas (19) to the functions \(u_1, u_2\) (after transformations analogous to (1), (p. 298)) and taking into account that \(u_1, u_2 \in S_\varepsilon(D)\), we obtain the system of integral equations
\[ u_1-\hat T_1u_2=P_{2m-1}(z); \tag{22} \]
\[ u_2-\hat T_2u_2-\hat T_3u_1= \frac{1}{\pi i}\int_\Gamma \frac{h(t)e^{\Omega(t)}\,dt}{t^m(t-z)}, \tag{23} \]
where
\[ \hat T_1u_2=\frac{1}{\pi}\iint_D \left\{ \frac{i e^{i\varphi(\zeta)}\overline{u_2(\zeta)}}{\zeta(\zeta-z)} - z^{-2\varkappa-2} \frac{\bar\zeta^{-2\varkappa-3} i e^{-i\varphi(\zeta)}u_2(\zeta)} {\zeta(1-\bar\zeta z)} \right\}\,d\xi\,d\eta, \tag{24} \]
\[ \hat T_2u_2=\frac{1}{\pi}\iint_D \left\{ \frac{ A(\zeta)u_2+\overline{A(\zeta)} \dfrac{\zeta e^{i\varphi(\zeta)}}{\bar\zeta e^{i\varphi(\zeta)}}\overline{u_2(\zeta)} } {|\zeta|(\zeta-z)} + \bar\zeta^{-2\varkappa-3} \frac{ \bar A u_2+ A\dfrac{\bar\zeta e^{i\varphi(\zeta)}}{\zeta e^{i\varphi(\zeta)}} } {|\zeta|(1-\bar\zeta z)} \right\}\,d\xi\,d\eta; \tag{25} \]
\[ \hat T_3u_1=\frac{1}{4\pi}\iint_D \left\{ \frac{i e^{i\varphi(\zeta)}\overline{u_1(\zeta)}}{\bar\zeta(\zeta-z)} - \frac{\bar\zeta^{-2\varkappa-3}e^{-i\varphi(\zeta)}u_1(\zeta)} {\zeta(1-\bar\zeta z)} \right\}c(\xi,\eta)\,d\xi\,d\eta; \tag{26} \]
\[ P_{2m-1}(z)=\hat\alpha_0+ \sum_{k=1}^{m}(\hat\beta_{m-k}+i\hat\alpha_{m-k})z^k+ \sum_{k=m}^{2m-2}\mu_k z^{k+1}; \tag{27} \]
\[ \mu_k=\frac{1}{\pi}\iint_D \bar\zeta^{\,k}e^{-i\varphi(\zeta)}u_2(\zeta)\zeta^{-1}\,d\xi\,d\eta. \]
In order that the solution of the system (22)—(23) be a solution of problem (20)—(21), it is necessary and sufficient that \(l\) relations be satisfied. The number of these relations is determined by the inequality
\[ -2\varkappa-4\le l\le -4(\varkappa+1). \tag{28} \]
Thus we obtain the following result:
Theorem 3. Let \(\varkappa<-1\). Then, if the inequality
\[ \Lambda=\sup_{x,y\in D}\frac{r^\varepsilon}{2\pi}\iint_D \left\{ \frac{1}{|\zeta-z|} + \frac{|\zeta|^{-2\varkappa-3}}{|1-\bar\zeta z|} \right\} \frac{\sqrt{a^2+b^2}+\tfrac12|c|\tilde\Lambda} {\sqrt{(\xi^2+\eta^2)^{1+\varepsilon}}} \,d\xi\,d\eta<1, \tag{29} \]
where
\[ \tilde\Lambda=\sup_{x,y\in D}\frac{r^\varepsilon}{\pi}\iint_D \left\{ \frac{1}{|\zeta-z|} + \frac{r^{-2\varkappa-2}|\zeta|^{-2\varkappa-3}}{|1-\bar\zeta z|} \right\}\,d\xi\,d\eta, \]
then for the solvability of problem A it is necessary and sufficient that a finite number of conditions be fulfilled.
Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR
Received
6 IV 1962
References
- I. N. Vekua, Generalized analytic functions, Moscow, 1959.
- Ten En-cher, Candidate dissertation, Siberian Branch of the Academy of Sciences of the USSR, 1960.
- A. Dzhuraev, DAN, 142, No. 5 (1962).
- L. G. Mikhailov, DAN, 129, No. 3 (1959).