Abstract
Full Text
V. P. DIDENKO
ON SOME ELLIPTIC SYSTEMS DEGENERATING ON THE BOUNDARY OF A DOMAIN
(Presented by Academician S. L. Sobolev on 18 XII 1961)
In this paper we consider systems of linear partial differential equations of the second order of the form:
\[ Lu=-y^m u_{xx}-u_{yy}+au_x+bu_y+cu=h, \tag{1} \]
\[ Lu=-u_{xx}-y^m u_{yy}+au_x+bu_y+cu=h, \tag{2} \]
where \(m\) is a positive real number; \(a=\|a_{i,k}\|\), \(b=\|b_{i,k}\|\), \(c=\|c_{i,k}\|\) are given square matrices of order \(n\); \(h=(h_1,\ldots,h_n)\) is a given vector; \(u=(u_1,\ldots,u_n)\) is the unknown vector.
Systems (1) and (2) are strongly elliptic for \(y>0\), with parabolic degeneration for \(y=0\).
Let \(\Omega\) be a simply connected domain lying in the half-plane \(y>0\), whose boundary \(\Gamma\) consists of a segment \(AB\) of the axis \(y=0\) and a smooth arc \(\sigma\) with endpoints at the points \(A\) and \(B\).
Equations of the form (1) and (2) in the domain \(\Omega\), in the case \(n=1\), have by now been well studied \((^{1-7})\). Attention was drawn to the importance of studying systems of the form (1) and (2), in particular, in the work \((^1)\).
In the present paper the first boundary-value problem for system (1) in the domain \(\Omega\) is investigated. For system (2), as also in the case of a single equation (\(n=1\)), the first boundary-value problem in this domain is not always correct. In the paper, for this system, conditions for the correctness of the first boundary-value problem and conditions for the correctness of the so-called problem \(E\) are established (see \((^3)\)).
Denote by \(\overset{\circ}{C}{}^{\,2}\) the set of twice continuously differentiable vectors that vanish in a boundary strip of the domain \(\Omega\). On this set we define a norm by the formula
\[ \|u\|_{+}=\int_{\Omega}\{y^m u_y^2+u_x^2\}\,d\Omega. \]
The closure of the set \(\overset{\circ}{C}{}^{\,2}\) in the plus-norm will be denoted by \(W^1_{(2,m)}\). By \(W^0_{(2,\mu)}\) we denote the set of all vectors \(f\) for which the integral
\[ \int_{\Omega}\mu^{-1} f^2\,d\Omega \]
is finite, where \(\mu(x,y)>0\) is a sufficiently smooth function in the domain \(\Omega\), with
\[ \mu= \begin{cases} O\bigl(y^{m-2}|\ln y|^{-1-\varepsilon}\bigr), & \text{for } m\ne 1,\\ O\bigl(y^{-1}|\ln y|^{-2-\varepsilon}\bigr), & \text{for } m=1. \end{cases} \]
We define a negative norm for each element \(f\in W^0_{(2,\mu)}\) by the formula
\[ \|f\|_{-}=\sup \frac{(f\cdot v)}{\|v\|_{+}}, \qquad v\in W^1_{(2,m)}, \]
where
\[ (f\cdot v)=\int_{\Omega}\{f_1v_1+\cdots+f_nv_n\}\,d\Omega \]
is the scalar product in the pro-
space \(W_2^0\). We denote by \(W_{(2,m)}^{-1}\) the closure, with respect to the negative norm, of the set \(W_{(2,\mu)}^0\).
Suppose that the matrices \(a, b, a_x, b_y, c\) satisfy the following conditions: a) in the case of system (1), for any \(m>0\), and for system (2), when \(m\leqslant 1\) or \(m\geqslant 2\), they are continuous in the closed domain \(\overline{\Omega}\); b) in the case of system (2), when \(1<m<2\), they are continuous in the closed domain \(\overline{\Omega}_\delta\), where \(\Omega_\delta=\Omega\cap(y=\delta>0)\), and belong to \(W_{(2,\mu)}^0\); c) \(a\) and \(b\) are symmetric; d) in any domain \(\Omega'\) interior with respect to \(\Omega\), the matrices \(a, b, c\) have generalized derivatives up to the third order, square-summable, and the vector \(h\) has generalized derivatives up to the second order, square-summable; e) for system (1), for any \(m>0\), and for system (2), when \(m\leqslant 1\), the inequality
\[
c-\frac{1}{2}a_x-\frac{1}{2}b_y \geqslant 0,
\]
holds, while for system (2), when \(m>1\), the inequality
\[
c-\frac{1}{2}a_x-\frac{1}{2}\,[mEy^{m-1}+b]_y \geqslant 0,
\]
holds, where \(E\) is the identity matrix; here the matrix inequalities are understood in the sense of inequalities for the corresponding quadratic forms constructed on arbitrary vectors of nonzero length.
Theorem 1. The system (1), for \(h\in W_2^0\), has a unique solution \(u\), twice continuously differentiable in the domain \(\Omega\), which vanishes on the boundary \(\Gamma\) in the mean, i.e.
\[
\lim_{\delta_1\to 0}\int_{\Gamma_{\delta_1}} u^2\,d\Gamma_{\delta_1}=0,
\]
where \(\delta_1\) is the width of the boundary strip of the domain \(\Omega\).
Theorem 2. If \(h\in W_2^0\) when \(m\leqslant 1\) or \(m\geqslant 2\), and \(h\in W_{(2,m)}^{-1}\) when \(1<m<2\), then, provided that in a neighborhood of \(y=0\) one of the following conditions is satisfied: 1) \(m<1\), \(b\) arbitrary; 2) \(m=1\), \(b+E>0\); 3) \(m>1\), \(b>0\), the system (2) has a unique solution \(u\), twice continuously differentiable in the domain \(\Omega\), which vanishes on the boundary \(\Gamma\) of the domain \(\Omega\), where vanishing on \(\Gamma\) in the case \(m<1\) is understood in the mean, and for \(m\geqslant 1\)—in the mean on \(\sigma\) and in the weak sense on \(\overline{AB}\), i.e.
\[
\lim_{AB_\delta}\int u\cdot v\,dx=0.
\]
Theorem 3. If \(h\in W_{(2,m)}^{-1}\), then, provided that in a neighborhood of \(y=0\) one of the following conditions is satisfied: 1) \(m=1\), \(b+E<0\); 2) \(1<m<2\), \(b+mEy^{m-1}<0\); 3) \(m\geqslant 2\), \(b\leqslant 0\), there exists a unique solution \(u\) of system (2), twice continuously differentiable in the domain \(\Omega\), belonging to \(W_{(2,m)}^1\) and vanishing only on \(\sigma\).
We briefly outline the proof of Theorem 2 in the case \(m>1\). We shall call a vector \(u\) a weak solution of the first boundary-value problem for system (2), satisfying the boundary conditions in the weak sense, if the relation
\[
(v\cdot h)=(L^*v\cdot u)
\tag{3}
\]
holds for all vectors \(v\), twice continuously differentiable in the closed domain \(\overline{\Omega}\), which vanish on the boundary strip \(\sigma\). Here \(L^*\) denotes
\[
L^*v=-(y^m v)_{yy}-v_{xx}-(av)_x-(bv)_y+vc.
\]
It is easy to prove the validity of the inequalities
\[ \|L^*v\|_- \geq \mathrm{const}\,\|v\|_+, \qquad \mathrm{const}>0; \tag{4} \]
\[ \|L^*v\|_{W_2^0} \geq \mathrm{const}\,\|v\|_{W_2^0}, \qquad \mathrm{const}>0. \tag{5} \]
Lemma 1. Every linear functional on the Hilbert space \(W_{(2,m)}^{-1}\) can be represented in the form
\[ l(f)=(f\cdot v), \]
where \(v\in W_{(2,m)}^1,\ f\in W_{(2,m)}^{-1}\).
To prove the existence of a weak solution of the first boundary-value problem, consider the linear functional
\[ l(v)=(v\cdot h). \tag{6} \]
Using (4), we easily obtain:
\[ l(v)=(v\cdot h)\leq \mathrm{const}\,\|L^*v\|_-, \qquad \mathrm{const}>0. \]
In view of this inequality, we may regard expression (6) as a linear functional in \(L^*v\) in the space \(W_{(2,m)}^{-1}\). Extending this functional, by the Banach–Hahn theorem, to the whole space and using the general form of linear functionals in \(W_{(2,m)}^{-1}\), we conclude that there exists an element \(u\in W_{(2,m)}^1\) such that our functional on \(W_{(2,m)}^{-1}\) can be represented in the form
\[ (v\cdot h)=(u\cdot L^*v). \]
This is precisely the definition of a weak solution. Thus, for the case \(1<m<2\) we have proved the existence of a weak solution of the first boundary-value problem. For sufficiently large \(m\), the element \(L^*v\) need no longer belong to the space \(W_{(2,m)}^{-1}\), i.e., the arguments given above are not always applicable. Therefore, in the case \(m\geq 2\), we proceed as follows: estimate expression (6), using (5):
\[ l(v)=(v\cdot h)\leq \mathrm{const}\,\|L^*v\|_{W_2^0}, \qquad \mathrm{const}>0. \]
Now we may regard (6) as a linear functional in \(L^*v\) in the space \(W_2^0\). Extending this linear functional to all of \(W_2^0\) and using the Riesz theorem on the general form of a linear functional, we obtain the existence of a weak solution of the first boundary-value problem belonging to the space \(W_2^0\); then we show that this solution also belongs to the space \(W_{(2,m)}^1\).
Using a priori estimates for the operator \(L^*\Delta^s\) (here \(\Delta\) is the Laplace operator, \(s\) is a positive integer), by arguments analogous to those given above, we obtain the required differentiability of the weak solution.
Denote by \(W_2^0(\Omega_{\delta_1})\) the set of vectors square-integrable in the domain \(\Omega\), equal to zero in the boundary strip of width greater than \(\delta_1\).
Lemma 2. For every vector \(f\in W_2^0(\Omega_{\delta_1})\) there exists a vector \(v\), equal to zero in the \(\delta_1\)-strip and belonging to the space \(W_{(2,m)}^1\), for which the equality holds:
\[ (a\cdot f)=(v\cdot La), \]
where \(a\) are twice continuously differentiable vectors equal to zero in a strip of width greater than \(\delta_1\).
From this lemma and the theorem on differentiability of weak solutions it follows that every element \(f \in W_2^0(\Omega_{\delta_1})\) can be approximated in the norm \(W_2^0\) by elements of the form \(L^*v\), where \(v \in C_0^2\).
Lemma 3. The set of all vectors \(f\) belonging to the sum
\[
\bigcup_{\delta>0} W_2^0(\Omega_\delta)
\]
is dense in the space \(W_{(2,m)}^{-1}\).
From this lemma there follows immediately
Lemma 4. Elements of the form \(L^*v\), where \(v \in C_0^2\), are dense in the space \(W_{(2,m)}^{-1}\).
It is now easy to prove the uniqueness of the weak solution of the first boundary-value problem. Indeed, let the vector \(u(x,y)\) be a weak solution, i.e., let equality (3) hold. We shall consider (3) only for elements \(v \in C_0^2\). Applying Schwarz’s inequality and inequality (4) to the right-hand side, we obtain
\[
(v\cdot h) \leq \mathrm{const}\,\|L^*v\|_{-},\qquad \mathrm{const}>0.
\]
Since the vectors \(L^*v\) are dense in the space \(W_{(2,m)}^{-1}\), applying Lemma 1 we obtain that there exists a unique element \(\psi \in W_{(2,m)}^1\) such that
\[
(v\cdot h)=(L^*v\cdot \psi).
\]
The vector \(\psi\) and the weak solution \(u\) coincide, since the linear functional \((L^*v\cdot \psi-u)\) vanishes on a set dense in the space \(W_{(2,m)}^{-1}\), i.e.,
\[
\|\psi-u\|_{+}=0.
\]
Thus, the uniqueness of the weak solution is proved.
The membership of the weak solution \(u\) in the domain \(\Omega\) in the space \(W_{(2,m)}^1\) and the differentiability theorem make it possible to conclude that \(u\) satisfies the stated boundary conditions.
In conclusion, I consider it a pleasant duty to express my deep gratitude to A. V. Bitsadze, S. V. Uspenskii, and S. A. Tersenov for the attention I received in writing the present work.
Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR
Received
28 XI 1961
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