UDC 517.946

MATHEMATICS

V. P. GLUSHKO

A PRIORI ESTIMATES IN \(\mathscr L_2\) OF SOLUTIONS OF GENERAL BOUNDARY-VALUE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

(Presented by Academician I. G. Petrovskii on January 27, 1969)

The results obtained in \((^1)\) are extended here to a broad class of degenerate elliptic operators of second order, defined in a domain \(D\) with smooth boundary.

Let \(D\) be a bounded domain of the Euclidean space \(E_n\), and
\[ S=\bigcup_{q=1}^{N} S_q \]
the boundary of \(D\), consisting of closed nonintersecting \((n-1)\)-dimensional manifolds \(S_q\) (components).

Condition 1. Each component \(S_q\) of the boundary \(S\) belongs to the class \(C^m\) \((m\ge 2)\). This means, in particular, that for each point \(x^0\in S_q\) there is a neighborhood \(A_{x^0}(\delta)\) and a transformation \(T_{x^0}(x)=(t,y)\) \((y\in E_{n-1})\) such that the intersection \(A_{x^0}(\delta)\cap D\) is mapped by the transformation \(T_{x^0}\) into the half-ball \(K_\delta: t^2+|y|^2\le \delta^2,\ t>0\).

We choose the transformation \(T_{x^0}(x)\) in a special way so that, in the new local coordinates \((t,y)\), for points \(x\in A_{x^0}(\delta)\cap D\), \(t\) is the coordinate of \(x\) along the inward normal to \(S\), and \((y_1,y_2,\ldots,y_{n-1})=y\) are the local coordinates on \(S\) of the point of \(S\) through which this normal passes.

With such a choice of the transformation \(T(x)\), the coordinates \(t\) of a point \(x\) in two local coordinate systems corresponding to the neighborhoods \(A_{x^0}(\delta_0)\) and \(A_{x^1}(\delta_1)\) \((x^0,x^1\in S_q)\) will coincide on the intersection of these neighborhoods. Thus, in some \(d\)-neighborhood of \(S_q\) a function \(t=t(x)\in C^m\) will be defined. Together with it, on the intersection of the \(d\)-neighborhood with \(D\), the function
\[ a(x)=\sum_{i,j=1}^{n} a_{ij}(x)\,\partial_{x_i}t\,\partial_{x_j}t, \tag{1} \]
will be defined, where \(a_{ij}(x)\) are the coefficients of the highest derivatives of the differential operator defined in \(D\),
\[ \mathscr L=\sum_{i,j=1}^{n} a_{ij}(x)\partial_{x_i}\partial_{x_j} +\sum_{i=1}^{n} a_i(x)\partial_{x_i}+a_0(x). \]

Condition 2. The coefficients \(a_{ij}(x), a_i(x)\) \((1\le i,j\le n)\) of the operator are real and continuous in \(\overline D\). The operator \(\mathscr L\) is elliptic at every interior point \(x\in D\). For definiteness we shall assume that the characteristic polynomial
\[ \sigma(x,\lambda)=\sum_{i,j=1}^{n} a_{ij}(x)\lambda_i\lambda_j>0 \]
for \(x\in D\) and \(\lambda\ne 0\). The boundary manifolds \(S_q\) are characteristic for \(\mathscr L\) \(\bigl(\sigma(x^0,\nu)=0\) for \(x^0\in S_q\), \(\nu\) being the inward normal to \(S\) at the point \(x^0\bigr)\).

Considering the operator \(\mathscr L\) in \(A_{x^0}(\delta)\cap D\), after the transformation \(T_{x^0}(x)=(t,y)\) we can write it in the form

\[ L_0=\partial_t\bigl(a_{nn}^0(t,y)\partial_t\bigr) +\sum_{k=1}^{n-1} a_{nk}^0(t,y)\partial_t\partial_{y_k} +\sum_{k,l=1}^{n-1} a_{kl}^0(t,y)\partial_{y_k}\partial_{y_l} +a_n^0(t,y)\partial_t+\sum_{k=1}^{n-1}a_k^0(t,y)\partial_{y_k}+a_0^0(t,y). \]

Obviously,

\[ a_{nn}^0=a\bigl(T_{x^0}^{-1}(t,y)\bigr),\qquad a_{nn}^0(t,y)>0\quad \text{for }t>0 \quad\text{and}\quad a_{nn}^0(0,y)\equiv0. \]

Condition 3. For every point \(x^0\in S_q\) the limits exist

\[ \lim_{(t,y)\to(+0,0)}\frac{a_{nn}^0(t,y)}{a_{nn}^0(t,0)}=1; \tag{2} \]

\[ \lim_{(t,y)\to(+0,0)} \frac{a_{nk}^0(t,y)}{\sqrt{a_{nn}^0(t,0)}}=\gamma_k \qquad (1\leq k\leq n-1); \tag{3} \]

\[ \lim_{(t,y)\to(+0,0)} \frac{\left|\operatorname{grad}_y a_{nn}^0(t,y)\right|} {\sqrt{a_{nn}^0(t,0)}}=0. \]

Introducing the function \(\alpha(t)=\sqrt{a_{nn}(t,0)}\) and denoting \(\alpha\partial_t=\alpha(t)\partial/\partial t\), we write \(L_0\) in the form \(L_0=L_0'+L_0''\), where

\[ L_0'=\frac{a_{nn}^0(t,y)}{\alpha^2(t)}(\alpha\partial_t)^2 +\sum_{k=1}^{n-1}\frac{a_{nk}^0(t,y)}{\alpha(t)}\alpha\partial_t\partial_{y_k} +\sum_{k,l=1}^{n-1} a_{kl}^0(t,y)\partial_{y_k}\partial_{y_l}, \]

and \(L_0''\) contains derivatives of order \(<2\).

Definition. We shall say that the operator \(\mathscr L\) is \(\alpha\)-elliptic in \(\overline D\) if it is elliptic in \(D\) and if at every point \(x^0\) of the boundary characteristic manifold \(S_q\) \((q=1,2,\ldots,N)\), under conditions (2), (3), the quadratic form in \(\eta,\xi=(\xi_1,\xi_2,\ldots,\xi_{n-1})\)

\[ \zeta(\eta,\xi)=\eta^2+\sum_{k=1}^{n-1}\gamma_k\eta\xi_k +\sum_{k,l=1}^{n-1}a_{kl}^0(0,0)\xi_k\xi_l\neq0 \tag{4} \]

does not vanish for any real \((\eta,\xi)\neq0\). Since we have agreed to assume \(\sigma(x,\lambda)>0\), it follows from (4) that \(\zeta(\eta,\xi)>0\) for \((\eta,\xi)\neq0\).

Let a boundary differential operator \(\mathscr B_q\) be given on \(S_q\), which in the neighborhood \(A_{x^0}(\delta)\) can be written in the form

\[ B_0(-\sqrt{-1}\,\partial_y,\partial_t) =\sum_{s=0}^{r}\sum_{|\tau|\leq \rho_s} b_{\tau,s}(y)(-\sqrt{-1}\,\partial_y)^\tau \partial_t^s, \]

where \(\tau=(\tau_1,\tau_2,\ldots,\tau_{n-1})\), \(|\tau|=\tau_1+\tau_2+\cdots+\tau_{n-1}\), and \(b_{\tau,s}(y)\) are sufficiently smooth complex-valued functions.

Definition. We shall say that the degenerate order of \(\mathscr B_q\) at the point \(x^0\in S_q\) is equal to \(m^*\), if the degree of the polynomial

\[ \sum_{s=0}^{r}\sum_{|\tau|\leq \rho_s} b_{\tau,s}(0)\xi^\tau\eta^{2s} \]

is equal to \(m^*\).

For every point \(x^0\in S_q\) define the functions \(b_\mu(t)=a_n^0(t,0)+\mu\partial_t a_{nn}^0(t,0)\), where \(\mu\) is a real parameter, and construct the homogeneous

polynomial in \(\xi\) of degree \(m^*\)

\[ \vartheta_{x^0}(\xi)= \sum_{s=1}^{r}\frac{b_0^s(0)}{b_1(0)b_2(0)\ldots b_s(0)} \Lambda_s(\xi)\left(-\frac{c(\xi)}{b_0(0)}\right)^s+\Lambda_0(\xi), \]

where

\[ \Lambda_s(\xi)=\sum_{|\tau|=m^*-2s} b_{\tau,s}(0)\xi^\tau,\qquad c(\xi)=-\sum_{k,l=1}^{n-1} a_{kl}^0(0,0)\xi_k\xi_l . \]

Definition. We shall say that the boundary operator \(\mathscr{B}_q\) of degenerate order \(m^*\) satisfies at the point \(x^0\in S_q\) the condition of complementarity with respect to \(\mathscr{L}\), if for any \(\xi\ne0\), \(\vartheta_{x^0}(\xi)\ne0\).

Condition 4. We shall assume that for the given integer \(m\ge2\) the components \(S_q\) of the boundary \(S\) can be divided into three sets (respectively \(Q_1,Q_2(m)\), and \(Q_3(m)\)) according to the following rule:

I. \(q\in Q_1\), if \(b_0(0)>0\) for every point \(x^0\in S_q\).

II. \(q\in Q_2(m)\), if \(b_{\frac12(m-1)}(0)<0\) for every point \(x^0\in S_q\).

III. \(q\in Q_3(m)\), if \(b_0(0)<0\) and there exists an integer \(\widetilde m_q\ge1\) such that \(b_{\frac12(\widetilde m_q-1)}(0)<0\), \(b_{\frac12\widetilde m_q}(0)\ge0\) for all \(x^0\in S_q\), and moreover \(m\ge \widetilde m_q+2\) if \(\widetilde m_q\) is even, and \(m\ge \widetilde m_q+3\) if \(\widetilde m_q\) is odd.

As shown in \((^2,^3)\), one can give a criterion for membership in \(Q_1,Q_2(m),Q_3(m)\) that does not depend on the choice of a particular coordinate system at the point \(x^0\). We note that any of the sets \(Q_1,Q_2(m),Q_3(m)\) may be empty.

Condition 5. Let, for an integer \(m\ge2\), the coefficients \(a_{ij}(x)\) belong to \(C^{m-1}(\overline D)\); \(a_i(x)\) \((0\le i\le n)\) belong to \(C^{m-2}(\overline D)\); the coefficients of the boundary operators \(\mathscr{B}_q\) (of degenerate order \(m_q^*\)) belong to \(C^{m-m_q^*}\) (where they are defined).

On the set \(C^m(K_d)\) define the norm

\[ \|v\|_{m,a}^{2}= \sum_{r+2s+|\tau|\le m}\int_{K_d}|a^i(t,y)\,\partial_t^{\,i+s}\partial_y^{\tau}v(t,y)|^2\,dt\,dy, \]

where \(a=a(t,y)\) is a weight function.

Construct a finite covering of \(\overline D\), consisting of a strictly interior subdomain \(D_{\delta_0}\) of the domain \(D\) and a system of neighborhoods \(A_{x^p}(\delta_p)\), where \(x^p\in S\), \(p=1,2,\ldots,P\). Let \(\varphi_p(x)\) \((p=0,1,2,\ldots,P)\) be functions belonging to \(C_0^\infty(E_n)\) and forming a partition of unity corresponding to this covering,

\[ \sum_{p=0}^{P}\varphi_p(x)=1\quad \text{in }\overline D,\qquad \operatorname{supp}\varphi_0\subset D_{\delta_0},\qquad \operatorname{supp}\varphi_p\subset A_{x^p}(\delta_p). \]

Definition. Let \(m\ge1\) be an integer. We shall say that a function \(v(x)\in\mathscr{L}_2(D)\), having generalized derivatives in \(D\) up to order \(m\), belongs to \(H_a^m(D)\), if there exists such a covering \(\overline D\{D_{\delta_0}, A_{x^p}(\delta_p),\,p=1,2,\ldots,P\}\) that the norm

\[ \|v\|_{m,a}= \left\{ \sum_{|\beta|\le m}\int_D|\partial_x^\beta(\varphi_0 v)|^2\,dx + \sum_{p=1}^{P}\|(\varphi_p v)_p\|_{m,a_p}^{2} \right\}, \]

is finite, where \((v)_p=v(T_{x^p}^{-1}(t,y))\); \(a_p(t,y)=\sqrt{(a)_p}\), and the function \(a(x)\) is defined by formula (1).

In the usual way we introduce the boundary norms \(\langle\cdot\rangle_{S_q}^{q,m}\) on the manifolds \(S_q\), putting for a finite function \(v(y)\) \((y\in E_{n-1})\)

\[ \langle v\rangle_{E_{n-1},m}= \int_{E_{n-1}}|\xi|^{2m}|\widetilde v(\xi)|^2\,d\xi, \]

where \(\widetilde v(\xi)\) is the Fourier transform in \(E_{n-1}\) of the function \(v(y)\).

Theorem. Let, for a given integer \(m \ge 2\), the operator \(\mathscr L\) and the domain \(D\) with boundary
\[ S=\bigcup_{q=1}^{N} S_q \]
satisfy conditions 1–5. On each boundary manifold \(S_q\), for \(q\in Q_2(m)\), let a boundary operator \(\mathscr B_q\) be given, satisfying condition 5 and having on \(S_q\) a pronounced order equal to
\[ m_q^* \le 2\left[\frac{m-2}{2}\right]. \]
Then, for the inequality
\[ \|v\|_{m,\alpha}\le c_m\left\{\|\mathscr Lv\|_{m-2,\alpha} +\sum_{q\in Q_2(m)}\langle \mathscr B_q v\rangle_{S_q,\,m-m_q^*-1} +\|v\|_0\right\}, \tag{5} \]
to hold, where \(v(x)\) is any function in \(H_\alpha^m(D)\), it is necessary and sufficient that the operator \(\mathscr L\) be \(\alpha\)-elliptic in \(\overline D\) and that the boundary operators \(\mathscr B_q\) satisfy the complementing condition with respect to \(\mathscr L\) on \(S_q\) for \(q\in Q_2(m)\).

One may consider the following boundary value problem: find a function
\[ v(x)\in H_\alpha^{m-2}(D) \]
satisfying the conditions
\[ \begin{aligned} \mathscr L v&=f &&\text{in }D;\\ \mathscr B_q v&=g_q &&\text{on }S_q\text{ for }q\in Q_2(m), \end{aligned} \tag{6} \]
where \(f\in H_\alpha^{m-2}(D)\) and \(g_q\bigl(\langle g_q\rangle_{S_q,\,m-m_q^*-1}<\infty\bigr)\) are prescribed functions.

If, for the given \(m\ge 2\), the hypotheses of the theorem under which inequality (5) holds are fulfilled, and if \(Q_3(m)\) is the empty set, then the operator corresponding to the boundary value problem (6) is Noetherian.

Voronezh State University

Received
18 I 1969

REFERENCES

1. V. P. Glushko, Functional Analysis and Its Applications, 2, 3, 1968, p. 87.
2. O. A. Oleinik, Mat. sbornik, 69 (111), 1 (1966).
3. J. J. Kohn, L. Nirenberg, Comm. Pure and Appl. Math., 18, No. 3, 443 (1965); Russian transl.: J. J. Kohn, L. Nirenberg, in the collection Pseudodifferential Operators, Moscow, 1967, p. 88.