Full Text
UDC 517.944
MATHEMATICS
Yu. P. Krasovskii
PROPERTIES OF GREEN’S FUNCTIONS AND GENERALIZED SOLUTIONS OF ELLIPTIC BOUNDARY-VALUE PROBLEMS
(Presented by Academician S. L. Sobolev on 13 V 1968)
Generalized solutions of such problems for which the system of boundary operators is normal have been well studied. Such a system makes it possible to construct the adjoint problem and to obtain estimates of generalized solutions by means of Green’s formula and estimates of classical solutions of the adjoint problem (see \((^{1-3})\)). Here Green’s functions will be used for the construction of generalized solutions. This makes it possible to consider generalized solutions of elliptic boundary-value problems without the assumption that the system of boundary operators is normal, and also to enlarge the class of admissible solutions and right-hand sides of the problem.
We consider the boundary-value problem:
\[ \mathcal{L}u=\varphi_0(x)\quad \text{for } x\in\Omega;\qquad B_j u=\varphi_j(x)\quad \text{for } x\in S . \tag{1} \]
Here \(j=1,\ldots,m\); \(\Omega\) is a closed bounded domain of \(n\)-dimensional space \((n\ge 2)\); \(x=(x_1,\ldots,x_n)\); \(S\) is the boundary of \(\Omega\); the function \(\varphi_0(x)\) is defined in \(\Omega\), \(\varphi_j(x)\) are defined on \(S\); \(\mathcal{L}\) is an elliptic operator of order \(2m\), defined in \(\Omega\); \(B_j\) are differential operators of order \(m_j\le 2m-1\), defined on \(S\):
\[ \mathcal{L}\equiv \sum_{0\le |\beta|\le 2m} a_\beta(x)D_x^\beta,\qquad B_j\equiv \sum_{0\le |\beta|\le m_j} b_{j\beta}(x)D_x^\beta \left( D_x^\beta=\frac{\partial^{|\beta|}}{\partial x_1^{\beta_1}\cdots \partial x_n^{\beta_n}} \right). \tag{2} \]
Our assumptions concerning the operators \(\mathcal{L}, B_j\) and the boundary \(S\) are as follows: \(\mathcal{L}\) is properly elliptic; the \(B_j\) cover \(\mathcal{L}\) \((^{1})\); all coefficients \(a_\beta(x)\in C^{l_1+1}(\Omega)\); \(b_{j\beta}(x)\in C^{l_1+1}(S)\); the boundary \(S\in C^{l_1+2m+1}\) in the sense of \((^{4})\); \(l_1\) is an integer subject to the condition \(l_1\ge \max(2l_0-1,\,2m)\), where
\[ l_0=\max_j(2m-m_j). \]
We introduce some notation. If \(u(x)\) is a function defined in \(\Omega\), then by \(Lu\) we denote the vector-function \(Lu=(\mathcal{L}u,B_1u,\ldots,B_m u)\); by \(\varphi\) we denote the vector-function \(\varphi=(\varphi_0,\ldots,\varphi_m)\), where \(\varphi_0(x),\mathcal{L}u(x)\) are defined in \(\Omega\); \(B_j u(x),\varphi_j(x)\) \((j\ge 1)\) are defined on \(S\). We shall write problem (1) in the form \(Lu=\varphi\). In addition, put:
\[ (\varphi\cdot\psi)= \int_{\Omega}\varphi_0(x)\psi_0(x)\,dx + \sum_{j=1}^{m}\int_{S}\varphi_j(x)\psi_j(x)\,dx_S; \]
\[ (u\cdot v)_\Omega=\int_{\Omega}u(x)v(x)\,dx;\qquad (u\cdot v)_S=\int_{S}u(x)v(x)\,dx_S; \]
\(C^{k+\alpha}(\Omega), C^{k+\alpha}(S)\) are Hölder spaces, \(\left|\ \right|_{k+\alpha}^{\Omega}\), \(\left|\ \right|_{k+\alpha}^{S}\) are the norms in these spaces; \(0<\alpha<1\); \(k\ge 0\) is an integer (see, for example, \((^{4})\)).
It is known that elliptic problems have a finite number of solvability conditions. The following lemma shows that these conditions can be written with the aid of a certain system of smooth functions.
Lemma 1. There exists a certain number \(N\) of vector-functions
\[ \bar\alpha_k=(\alpha_{0k},\alpha_{1k},\ldots,\alpha_{mk}),\qquad k=1,\ldots,N, \]
\[ \alpha_{0k}(x)\in C^{l_1+\alpha}(\Omega),\qquad \alpha_{jk}(x)\in C^{l_1-2m+m_j+1+\alpha}(S),\qquad j=1,\ldots,m, \]
such that the problem \(Lu=\bar\varphi,\ u\in C^{2m+\alpha}(\Omega)\), is solvable if and only if the conditions
\[ (\bar\varphi\cdot \bar\alpha_k)=0,\qquad k=1,\ldots,N \tag{3} \]
are satisfied.
The proof of this lemma is based on the representation of any solution of the problem \(Lu=\varphi\) in the form of the sum of the principal part and the remainder, obtained in \((^7)\).
Denote by \(\bar P\) the projection operator onto the subspace generated by the vectors \(\bar\alpha_k\), and by \(P\) the projection operator onto the subspace of solutions of the homogeneous problem \(Lu=0\),
\[ \bar P\bar\varphi=\sum_{k=1}^{N}(\bar\varphi\cdot\bar\alpha_k)\,\tilde{\bar\alpha}_k,\qquad Pu=\sum_{i=1}^{N'}(u\cdot\tilde u_i)_\Omega u_i. \tag{4} \]
Here \(u_i\in C^{2m+l_1-l_0+\alpha}(\Omega)\) are solutions of the homogeneous problem \(Lu_i=0\), all \(u_i\) are orthonormal, and every solution \(u(x)\in C^{2m+\alpha}(\Omega)\) of the problem \(Lu=0\) is a linear combination of the \(u_i\); all \(\bar\alpha_k\) are also assumed orthonormal, i.e.
\[ (\bar\alpha_i\cdot\tilde{\bar\alpha}_k)=\delta_{ik},\qquad (u_i\cdot\tilde u_k)_\Omega=\delta_{ik}, \]
and \(\bar a,\tilde u\) denote the quantities complex conjugate to \(a,u\).
Definition. We shall call the vector-function
\[ \mathscr G(x,y)=\{\mathscr G_0(x,y),\mathscr G_1(x,y),\ldots,\mathscr G_m(x,y)\}, \]
where \(\mathscr G_0(x,y)\) is defined for \(x,y\in\Omega,\ x\ne y\), and \(\mathscr G_1(x,y),\ldots,\mathscr G_m(x,y)\) are defined for \(x\in\Omega,\ y\in S,\ x\ne y\), the Green vector-function of the problem \(Lu=\bar\varphi\), if the function
\[ u(x)=\int_{\Omega}\mathscr G_0(x,y)\varphi_0(y)\,dy +\sum_{j=1}^{m}\int_{S}\mathscr G_j(x,y)\varphi_j(y)\,dy_S \tag{5} \]
for arbitrary \(\varphi_0\in C^\alpha(\Omega)\), \(\varphi_j\in C^{2m-m_j+\alpha}(S)\), belongs to \(C^{2m+\alpha}(\Omega)\) and is a solution of the problem
\[ Lu=\bar\varphi-\bar P\bar\varphi,\qquad Pu=0. \]
The functions \(\mathscr G_j(x,y)\), \(j=0,1,\ldots,m\), are assumed integrable in \(y\) for every \(x\) lying inside \(\Omega\).
Theorem 1. The Green vector-function of the problem \(Lu=\bar\varphi\) exists and is unique.
The definition given here of the Green vector-function is a generalization of the definition given in \((^6,{}^7)\), where the problem \(Lu=\varphi\) was assumed to be uniquely solvable. The construction of \(\mathscr G(x,y)\) differs only slightly from the analogous construction in \((^7)\) with the aid of the principal part \(\bar{\mathscr G}^{(r)}(x,y)\), which remains the same, and a smooth remainder. Thus it is easy to show that the functions \(\mathscr G_j(x,y)\) have derivatives \(D_x^sD_y^t\mathscr G_j(x,y)\) wherever they are defined, for \(s\le 2m+l_1-2l_0+1,\ t\le l_1\) \((j=0)\), \(t\le l_1-2m+m_j+1\) \((j\ge 1)\). These derivatives satisfy the estimates given in \((^7)\) (see the corollary to Theorem 3.3) for the derivatives of Green’s functions in the case of unique solvability of the problem \(Lu=\varphi\), if in those estimates the constant \(M_0\) is taken equal to
\[ M_0=\sup |u|_{2m+1}^{\Omega}\quad \text{for}\quad |\mathcal L u|_{\alpha}^{\Omega}+\sum_{j=1}^{m}|B_j u|_{2m-m_j+\alpha}^{S}=1, \tag{6} \]
where the supremum is taken over those \(u\in C^{2m+\alpha}(\Omega)\) for which \(Pu=0\).
The following property of Green’s functions is derived on the basis of the study of their principal part, carried out in \((^5,{}^7)\).
Theorem 2. Let \(y\in S\) be some point; let \(m'\) be the maximal order of the derivative in the direction of the normal to \(S\) that is contained in the operators \(B_j\) \((j=1,\ldots,m)\) in some neighborhood of \(y\).
If \(m'\le 2m-2\), then
\[ D_y^k \mathscr G_0(x,y)=0 \quad \text{for } 0\le k\le 2m-m'-2, \]
where \(x\) is an arbitrary point of \(\Omega\), \(x\ne y\); \(D_y^k\) is any derivative of order \(k\) at the point \(y\).
Corollary. Suppose the conditions of Theorem 2 are satisfied. Then the functions \(\alpha_{0i}(y)\), \(i=1,\ldots,N\), entering into the solvability conditions (3), satisfy the equality
\[ D_y^k\alpha_{0i}(y)=0,\quad \text{where } y\in S,\; 0\le k\le 2m-m'-2. \]
Let us refine the definition of the number \(m'\). Suppose that in some neighborhood of the point \(y\in S\) a local coordinate system \(x_1',\ldots,x_n'\) is chosen, with \(x_n'=0\) the equation of \(S\); let \(B_j'\) be the operator \(B_j\) written in the coordinate system \(x_1',\ldots,x_n'\). By \(m'(y)\) we denote the maximal order of the derivative with respect to \(x_n'\) which occurs in the operators \(B_j'\) \((j=1,\ldots,m)\) at the point \(y\). Obviously, \(m'(y)\) does not depend on the choice of the local coordinate system \(x_1',\ldots,x_n'\). By \(m'\) in Theorem 2 is denoted \(\max_z m'(z)\), where \(z\) belongs to the indicated neighborhood of \(y\). In what follows \(m'\) will denote \(\max_y m'(y)\), when \(y\in S\) is any point.
We turn to the consideration of generalized solutions. For this purpose we introduce the necessary functional spaces. Define the norms:
\[ |u|_{-(k+\alpha)}^{\Omega} = \sup_{\Phi}\left|(u\cdot\Phi)_{\Omega}\right| \quad \text{provided } |\Phi|_{k+\alpha}^{\Omega}=1; \]
\[ |v|_{-(k+\alpha)}^{S} = \sup_{F}\left|(v\cdot F)_{S}\right| \quad \text{provided } |F|_{k+\alpha}^{S}=1; \]
\[ |u|_{-(k+\alpha)}^{\Omega,r} = \sup_{\Phi'}\left|(u\cdot\Phi')_{\Omega}\right| \quad \text{provided } |\Phi'|_{k+\alpha}^{\Omega}=1, \]
where the supremum is taken over all (normalized) \(\Phi\in C^{k+\alpha}(\Omega)\), \(F\in C^{k+\alpha}(S)\), \(\Phi'\in C^{k+\alpha}(\Omega)\); \(\Phi'\) vanishes on \(S\) together with all derivatives up to order \(r\) inclusive, \(r\le k\). Denote by \(C^{-(k+\alpha)}(\Omega)\), \(C_r^{-(k+\alpha)}(\Omega)\) the closures of the set of infinitely differentiable functions defined in \(\Omega\), respectively in the norms \(|\cdot|_{-(k+\alpha)}^{\Omega}\), \(|\cdot|_{-(k+\alpha)}^{\Omega,r}\). Denote by \(C^{-(k+\alpha)}(S)\) the closure of the set of functions belonging to \(C^{2m+\alpha}(S)\) in the norm \(|\cdot|_{-(k+\alpha)}^{S}\).
It is easy to see that \(C^{-(k+\alpha)}(\Omega)\), \(C_r^{-(k+\alpha)}(\Omega)\), \(C^{-(k+\alpha)}(S)\) are complete linear normed spaces with the norms defined above. Such spaces and norms are usually called negative. In the present case the corresponding positive spaces are \(C^{(k+\alpha)}(\Omega)\), \(C_r^{k+\alpha}(\Omega)\), \(C^{k+\alpha}(S)\), where \(C_r^{k+\alpha}(\Omega)\) is the subspace of \(C^{k+\alpha}(\Omega)\) consisting of functions that vanish on \(S\) together with all derivatives up to order \(r\) inclusive.
Let \(u\) be a function defined in \(\Omega\); by \(\partial^j u/\partial \nu^j\) we denote the function defined on \(S\) and equal to the derivative of order \(j\) in the direction of the inner normal to \(S\) of \(u\).
Introduce the norm \(|u|_{-(k+\alpha),t}^{\Omega}\), \(t\ge 0\) an integer, by setting
\[ |u|_{-(k+\alpha),t}^{\Omega} = |u|_{-(k+\alpha)}^{\Omega} + \sum_{j=0}^{t} \left| \frac{\partial^j u}{\partial \nu^j} \right|_{-(k+j+1+\alpha)}^{S}, \]
and define the space \(\widetilde C^{-(k+\alpha),t}\) as the completion of the set of infinitely differentiable functions defined in \(\Omega\) with respect to the norm \(|\cdot|_{-(k+\alpha),t}^{\Omega}\). Obviously, \(\widetilde C^{-(k+\alpha),t}\) is a complete linear normed space.
We introduce also the space of vector-functions \(\bar\varphi=\{\varphi_0,\varphi_1,\ldots,\varphi_m\}\), associated with the operator \(L=\{\mathcal L,B_1,\ldots,B_m\}\). We shall say that \(\bar\varphi\in \bar C_L^{-(k+\alpha)}\) if \(\varphi_0\in C_r^{-(k+2m+\alpha)}(\Omega)\) for \(2m-2-m'\ge 0\); \(\varphi_0\in C^{-(k+2m+\alpha)}(\Omega)\) for \(m'=2m-1\); \(\varphi_j\in C^{-(k+m_j+1+\alpha)}(S)\) \((j=1,\ldots,m)\).
\[ |\bar\varphi|_{-(k+\alpha)}^{L}=|\varphi_0|+\sum_{j=1}^{m}|\varphi_j|^{S}{}_{-(k+m_j+1+\alpha)}, \]
\[ |\varphi_0|= \begin{cases} |\varphi_0|_{-(k+2m+\alpha)}^{\Omega,\,2m-2-m'} & \text{for } 2m-2-m'\ge 0,\\[4pt] |\varphi_0|_{-(k+2m+\alpha)}^{\Omega} & \text{for } m'=2m-1; \end{cases} \]
\[ |\ |_{-(k+\alpha)}^{L}\text{ is the norm in the space } \bar C_L^{-(k+\alpha)}. \]
Definition. We shall call \(u\) a generalized solution of the problem \(Lu=\varphi\), where \(\bar\varphi\in \bar C_L^{-(k+\alpha)}\) \((k\le l_1-2m)\), if \(u\in C^{-(k+\alpha)}(\Omega)\) and there exists a sequence of functions \(u_i\in C^{2m+\alpha}(\Omega)\) such that the relations
\[ \lim_{i\to\infty}|u-u_i|_{-(k+\alpha)}^{\Omega}=0,\qquad \lim_{i\to\infty}|\bar\varphi-Lu_i|_{-(k+\alpha)}^{L}=0 \]
hold.
In other words, we call \(u\) a generalized solution if it can be obtained as the limit of classical solutions in the corresponding negative norms.
We note that the operators \(\bar P, P\) (4) are easily defined by continuity on the negative spaces, if one takes into account the smoothness of the functions \(a_{jk}(x)\) according to Lemma 1 and of the homogeneous solutions \(u_i(x)\) of the problem \(Lu=0\), and also the consequence of Theorem 2. In this way we obtain that the operator \(\bar P\) is a bounded operator in the space \(\bar C_L^{-(k+\alpha)}\) for \(k\le l_1-2m\), and the operator \(P\) is a bounded operator in the space \(C^{-(k+\alpha)}(\Omega)\) for \(k\le l_1+2m-l_0\).
Theorem 3. If \(\bar\varphi\in \bar C_L^{-(k+\alpha)}\), \(\bar P\bar\varphi=0\), then there exists a unique generalized solution of the problem \(Lu=\varphi\) such that \(Pu=0\). This generalized solution belongs to \(C^{-(k+\alpha),m'}\), and the estimates
\[ M^{-1}|\bar\varphi|_{-(k+\alpha)}^{L}\le |u|_{-(k+\alpha),m'}^{\Omega}\le M|\bar\varphi|_{-(k+\alpha)}^{L} \]
are valid.
Here \(M\) depends only on the coefficients of the operators \(\mathcal L,B_j\), the boundary \(S\), \(M_0\) (6), and \(\alpha\); more precisely, \(M\) depends only on the constant \(c\) defined in (5), on \(M_0\) (6), and on \(\alpha\).
The proof of this theorem is based on the representation of the Green functions as the sum of the principal part and a remainder and on the properties of the operators giving the principal part of the Green function, which were considered in \((5,7)\). Here one also uses the property of the Green functions formulated in Theorem 2.
It follows from Theorem 3 that the operator \(L\), defined by continuity on the space \(C^{-(k+\alpha),m'}\), realizes a one-to-one and continuous mapping of the subspace \(\widetilde C^{-(k+\alpha),m'}\), containing those \(u\in C^{-(k+\alpha),m'}\) for which \(Pu=0\), onto the subspace \(\bar C_L^{-(k+\alpha)}\), containing those \(\bar\varphi\in \bar C_L^{-(k+\alpha)}\) for which \(\bar P\bar\varphi=0\).
Rostov-on-Don Institute
of Agricultural Machine Building
Received
11 V 1968
CITED LITERATURE
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