Abstract Generated abstract
The paper studies the planar Poincare boundary value problem for a second order elliptic equation whose first order coefficients have first order polar singularities and whose zeroth order coefficient has a second order polar singularity. After showing conformal invariance, the problem is reduced in the unit disk to an auxiliary Riemann Hilbert problem for a system of two complex functions, and then to systems of singular integral equations in weighted Banach spaces. Under explicit smallness conditions on integral norms of the coefficients, the paper proves solvability results and determines the dimension of the homogeneous solution space when the boundary index is at least minus one. For negative index below minus one, it derives an equivalent transformed system and states finite necessary and sufficient solvability conditions under a corresponding contraction inequality.
Full Text
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).