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UDC 517.9
MATHEMATICS
Ya. A. SOVIN
ELLIPTIC BOUNDARY-VALUE PROBLEMS
FOR PLANE DOMAINS WITH CORNERS AND DISCONTINUITIES
REACHING THE BOUNDARY
(Presented by Academician I. N. Vekua, 20 I 1969)
In the present note we state existence theorems for boundary-value problems satisfying the Lopatinskii condition in weighted spaces, analogous to those considered in \((^1)\).
1°. Notation, formulation of the problem. Let \(\Omega\) be a bounded plane domain with piecewise smooth boundary \(\Gamma\), containing a finite number of nonzero angles, divided by a finite number of smooth curves \(\gamma_\nu\) into parts \(\Omega_\mu\). In \(\Omega_\mu\) let operators \(L_\mu(x,y,\partial/\partial x,\partial/\partial y)\) be given, and on \(\Gamma,\gamma_\nu\), respectively, operators \(B(x,y,\partial/\partial x,\partial/\partial y)\) and \(C_\nu^\pm(x,y,\partial/\partial x,\partial/\partial y)\) in the form of matrices of dimensions respectively \(N\times N\), \(m\times N\), and \(2m\times N\) (the number \(m\) will be defined below), whose elements \(l_{ij}^{(\mu)}\), \(b_{ij}\), and \(c_{ij\nu}^{\pm}\) are homogeneous linear differential operators of orders respectively \(s_i+t_j\), \(\sigma_i+t_j\), \(\sigma_i+t_j\), \(\max_i s_i=0\) (see \((^2)\)). The operators \(L_\mu\) are uniformly elliptic and satisfy the additional condition on \(L\) (see \((^2)\)). Then the number \(m\) introduced above is equal to half the order of \(\det L_\mu\). The coefficients entering into \(l_{ij}^{(\mu)}\), \(b_{ij}\), \(c_{ij\nu}^{\pm}\) are assumed sufficiently smooth (at angular points and points of intersection of the curves this assumption can be weakened).
Consider the problem (see \((^3)\))
\[ L_\mu(x,y,\partial/\partial x,\partial/\partial y)U=f(x,y), \qquad (x,y)\in\Omega_\mu; \tag{1} \]
\[ B(x,y,\partial/\partial x,\partial/\partial y)U=\varphi(x,y), \qquad (x,y)\in\Gamma; \tag{2} \]
\[ C_\nu^+U^+ - C_\nu^-U^-=\psi_\nu(x,y), \qquad (x,y)\in\gamma_\nu, \tag{3} \]
in the functional spaces
\[ \|U_j\|_{\dot W_a^k(\Omega)}^2 = \sum_{|\alpha|\le k} \iint_\Omega r^{a-2(k-|\alpha|)}|D^\alpha U_j|\,dx\,dy, \qquad j=1,\ldots,N, \]
\[ \|\varphi\|_{\dot W_a^{k-1/2}(\partial\omega)} = \inf_{\Pi}\|\Pi\varphi\|_{\dot W_a^k(\omega)}, \]
where \(\Pi\varphi\in \dot W_a^k(\omega)\) is an extension of \(\varphi\) from the boundary \(\partial\omega\) into the domain \(\omega\), and \(r\) is the product of the distances from the point \((x,y)\) to all angular points and points of intersection of the pairs \((\Gamma,\gamma_\nu)\) and \((\gamma_\mu,\gamma_\nu)\). The direct product of the spaces
\[ \dot W_a^{\,k-s_1}\times\cdots\times \dot W_a^{\,k-s_N} \times \dot W_a^{\,k-\sigma_1-1/2}\times\cdots\times \dot W_a^{\,k-\sigma_m-1/2} \]
will be denoted by \(\dot H_a^k\). Suppose that the operators \(B,C_\nu^\pm\) jointly cover the operators \(L_\mu\) in the sense of \((^3)\).
2°. Problems with one angular point. First consider problem (1)—(2) in a domain not divided by the curves \(\gamma_\nu\), with boundary \(\Gamma\) containing only one angular point (for example, the origin), and pose the auxiliary problem
\[ L(0,0;\partial/\partial x,\partial/\partial y)U=f, \qquad (x,y)\in\Omega_0; \tag{4} \]
\[ B(0,0,\partial/\partial x,\partial/\partial y)U=\varphi, \qquad (x,y)\in\Gamma_0, \tag{5} \]
where \(\Gamma_0\) consists of two rays issuing from the angular point tangent to \(\Gamma\), and \(\Omega_0\) —
the interior of the angle between them. After the change of variables
\[ x=e^{-\tau}\cos\omega,\qquad y=e^{-\tau}\sin\omega,\qquad U_j=e^{-t_j\tau}V_j(\omega,\tau),\qquad j=1,\ldots,N, \]
and the Fourier transform with respect to \(\tau\)
\[ \widetilde F(\lambda)=\int_{-\infty}^{+\infty} F(\tau)e^{-i\lambda\tau}\,d\tau \]
the auxiliary problem (4)—(5) is transformed into a boundary-value problem in the strip \(0<\omega<\omega_0\) for a system of ordinary differential equations with respect to the functions \(V_j(\omega,\tau)\), \(j=1,\ldots,N\),
\[ \widetilde L(\omega,d/d\omega,i\lambda)\widetilde V=\widetilde F(\omega,\lambda); \tag{6} \]
\[ \widetilde B(\omega,d/d\omega,i\lambda)\widetilde V\big|_{\omega=0;\,\omega_0}=\widetilde\Phi(\lambda), \tag{7} \]
where \(\widetilde L\) and \(\widetilde B\) are the Fourier transforms with respect to \(\tau\) of matrices consisting, respectively, of the elements
\[ e^{-(s_i+t_j)\tau}l_{ij}(0,0,\partial/\partial x,\partial/\partial y),\qquad e^{-(\sigma_i+t_j)\tau}b_{ij}(0,0,\partial/\partial x,\partial/\partial y) \]
and, by construction of the system, independent of \(\tau\). If problem (4)—(5) is properly elliptic and satisfies the Lopatinskii condition, then problem (6)—(7) satisfies conditions analogous to Conditions I and II for systems formulated in (4) for the case \(s_i=0\), \(t_j=2m\). Applying the method set forth in (4), one can show that for sufficiently large \(\lambda\) there exists a resolving operator \(R(\lambda)\), acting from the intersection of the Sobolev spaces
\[ W_2^{k-s_1}\times\cdots\times W_2^{k-s_N}\times W_2^{k-\sigma_1-1/2}\times\cdots\times W_2^{k-\sigma_m-1/2} \]
to the space
\[ H^k=W_2^{k-t_1}\times\cdots\times W_2^{k-t_N}, \]
which is a meromorphic function of \(\lambda\) and is such that \(\widetilde V=R(\lambda)(\widetilde F,\widetilde\Phi)\) is the solution of problem (6)—(7) satisfying the estimate
\[ \int_0^{\omega_0}\sum_{q=0}^{k+t_j} \left[1+|\lambda|^{2(k+t_j-q)}\right] \left|\frac{d^q\widetilde V_j}{d\omega^q}\right|^2\,d\omega \le C\left\{ \int_0^{\omega_0}\sum_{i=1}^{N} \left( \left|\frac{d^{\,k-s_i}\widetilde F_i}{d\omega^{\,k-s_i}}\right|^2 + |\lambda|^{k-s_i}\widetilde F_i|^2 \right)d\omega +\right. \]
\[ \left. +\sum_{i=1}^{m}|\lambda|^{2(k-\sigma_i)-1} \left( |\widetilde\Phi_i|_{\omega=0}|^2+ |\widetilde\Phi_i|_{i=\omega_0}|^2 \right) \right\}. \tag{8} \]
Theorem 1. Suppose that the resolving operator \(R(\lambda)\) for problem (4)—(5) has no poles on the line \(\operatorname{Im}\lambda=h\). Then for \((f,\varphi)\in \dot H_2^k{}_{(k-h-1)}\) there exists a unique solution of problem (4)—(5), satisfying the estimate
\[ \|U_j\|_{\dot W_2^{k+t_j}{}_{(k-h-)}} \le C\|f,\varphi\|_{\dot H_2^k{}_{(k-h-1)}}. \tag{9} \]
Theorem 2. If \(R(\lambda)\) for the auxiliary problem (4)—(5) has no poles on the line \(\operatorname{Im}\lambda=h\), then for \((f,\varphi)\in \dot H_2^k{}_{(k-h-1)}\), under the fulfillment of a finite number of conditions of the type of functionals in \(\dot H_2^k{}_{(k-h-1)}\), \(F_p(f,\varphi)=0\), there exists a solution of problem (1)—(2)
\[ U_j\in \dot W_2^{k+t_j}{}_{(k-h-1)},\qquad j=1,\ldots,N. \]
If this solution is unique, then the estimate (9) is valid for it.
3°. Adjoint problems. We now consider problem (1)—(3) with one line of discontinuity \(\gamma\), intersecting \(\Gamma\) at nonzero angles at the origin and at the point \(Q\). Rotate the coordinate axes so that the \(x\)-axis becomes tangent to \(\Gamma\), and the \(y\)-axis coincides with the inward normal. Then the equations of the curves \(\Gamma,\gamma\) in a neighborhood of the origin can be represented in the form \(y=y_\Gamma(x)\), \(y=y_\gamma(x)\). Since the angle between \(\Gamma\) and \(\gamma\) is nonzero, the transformation \(x_1=y-y_\Gamma,\ y_1=y-y_\gamma\) is a smooth nondegenerate one, carrying problem (1)—(3) into the corresponding problem for the half-plane. Putting further, for \(y<0\), \(z=-y,\ U_j(x,-z)=V_j(x,z)\), we see that,
that problem (1)—(3) is equivalent to the “doubled” boundary-value problem (1)—(2) with one straight angle, i.e., to a problem with twice as many equations and boundary conditions; moreover, proper ellipticity and the covering condition are preserved.
Consider the auxiliary problem
\[ L(0,0,\partial/\partial x,\partial/\partial y)U=f,\qquad (x,y)\in\Omega_0; \tag{10} \]
\[ B(0,0,\partial/\partial x,\partial/\partial y)U=\varphi,\qquad (x,y)\in\Gamma_0; \tag{11} \]
\[ C^+(0,0,\partial/\partial x,\partial/\partial y)U^+ - C^-(0,0,\partial/\partial x,\partial/\partial y)U^- =\psi,\qquad (x,y)\in\gamma_0, \tag{12} \]
where \(\Omega_0,\Gamma_0\) are the same as in problem (4)—(5), and \(\gamma_0\) is the tangent to \(\gamma\). Passing to the “doubled” problem, we construct \(R(\lambda)\).
Theorem 3. If \(R(\lambda)\) of the “doubled” problem (10)—(12) has no poles on the line \(\operatorname{Im}\lambda=h\), then for \((f,\varphi,\psi)\in \dot H^{k}_{2(k-h-1)}\) there exists a unique solution of problem (10)—(12) satisfying the estimate
\[ \|U_j\|_{\dot W^{k+t_j}_{2(k-h-1)}} \le C\|f,\varphi,\psi\|_{\dot H^{k}_{2(k-h-1)}} . \tag{13} \]
If the domain \(\Omega\) is divided by two intersecting curves \(\gamma_1\) and \(\gamma_2\), which do not have a common tangent at the point of intersection, then, after performing the transformations \(x_1=y-y_{\gamma_1}\), \(y_1=y-y_{\gamma_2}\), by analogous reasoning we obtain a “quadrupled” boundary-value problem with one corner. Consider the auxiliary problem
\[ L(0,0,\partial/\partial x,\partial/\partial y)U=f,\qquad (x,y)\in\Omega_0; \tag{14} \]
\[ C_\nu^+(0,0,\partial/\partial x,\partial/\partial y)U^+ - C_\nu^-(0,0,\partial/\partial x,\partial/\partial y)U^- =\varphi_\nu,\qquad (x,y)\in\gamma_{\nu0},\quad \nu=1,2. \tag{15} \]
Theorem 4. If \(R(\lambda)\) of the “quadrupled” problem (14)—(15) has no poles on the line \(\operatorname{Im}\lambda=h\), then for \((f,\varphi)\in \dot H^{k}_{2(k-h-1)}\) there exists a unique solution of problem (14)—(15) satisfying the estimate
\[ \|U_j\|_{\dot W^{k+t_j}_{2(k-h-1)}} \le C\|f,\varphi\|_{\dot H^{k}_{2(k-h-1)}} . \]
Theorems 1, 3, and 4 make it possible to construct a regularizer for problem (1)—(3) (see \((^1,^4,^5)\)); therefore the following is true.
Theorem 5. If \(R_\nu\) of the auxiliary problems (4)—(5) for each corner point, \(R_\mu\) of the “doubled” auxiliary problems for each point of intersection of \(\Gamma,\gamma\), and \(R_{\mu\nu}\) of the “quadrupled” auxiliary problems for each point of intersection of the curves \((\gamma_\mu,\gamma_\nu)\) have no poles on one and the same line \(\operatorname{Im}\lambda=h\), then for \((f,\varphi,\psi)\in \dot H^{k}_{2(k-h-1)}\), upon satisfaction of a finite number of conditions of the type of functionals in \(\dot H^{k}_{2(k-h-1)}\), there exists a solution of problem (1)—(3), \(U_j\in \dot W^{k+t_j}_{2(k-h-1)}\), \(j=1,\ldots,N\). If it is unique, then an estimate of type (13) is valid.
Remark. The curves \(\gamma_\nu\) may close up without reaching the boundary. If they do not intersect, then normal solvability holds (see \((^3)\)).
In conclusion I express my gratitude to Prof. S. D. Eidelman for posing the problem and for his constant attention to the present work, and also to Prof. Kondrat’ev for useful discussions and valuable advice.
Chernivtsi State University
Received
19 XII 1968
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