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UDC 517.946
MATHEMATICS
V. P. GLUSHKO
ON DEGENERATING ELLIPTIC EQUATIONS OF SECOND ORDER IN ARBITRARY SMOOTH DOMAINS
(Presented by Academician I. N. Vekua on 30 IV 1966)
Here the results of papers \((^{1,2})\) are extended to the case of a domain with an arbitrary smooth boundary.
Let the differential expression \(Lu \equiv a^{ij}u_{x_i x_j}+a^i u_{x_i}+au\) be elliptic everywhere in the bounded domain \(D \subset R_n\) with boundary \(\dot D\), except for a closed set \(D_0 \subset \dot D\). The coefficients \(a^{ij}(x)\), \(a^i(x)\), \(a(x)\) are real functions belonging to \(C^1\) in any closed subdomain \(\bar D \setminus D_0\). In addition, we shall assume that \(L\varphi-a\varphi\), for any function \(\varphi \in C^\infty(R_n)\), is bounded above by a constant \(c=c(\varphi)\).
Let the domain \(D\) belong to the class \(A^{(3)}\) (see \((^3)\)) and let \(D_0=\bar D' \cup \bar D''\), where \(\bar D'\) and \(\bar D''\) have no common points, and their boundaries \(\Gamma'\) and \(\Gamma''\) on \(\dot D\) belong to \(A^{(2)}\). As is known, for sufficiently small \(\delta>0\) there exists a finite covering of the \(2\delta\)-neighborhood of \(\dot D\) such that the intersection with \(D\) of each of the domains of the covering, by means of a nonsingular (for each domain of the covering, generally speaking, its own) transformation
\[ y_m=y_m(x_1,x_2,\ldots,x_n), \qquad m=1,2,\ldots,n, \tag{1} \]
is transformed into a domain of special form (for example, a half-ball of radius \(\rho \leq 1\)); moreover, the quantities \(y_n(x)\) corresponding to different domains of the covering coincide on the intersection of these domains, and the quantities \(y_{n-1}(x)\), in addition, coincide on the intersection of domains covering the \(2\delta\)-neighborhoods of \(\Gamma'\) and \(\Gamma''\). Depending on this, we shall denote by: \(T_\alpha'\) \((\alpha \in \mathfrak A')\) the domains of the covering having common points only with \(D'\); \(T_\alpha''\) \((\alpha \in \mathfrak A'')\) those having common points only with \(D''\); \(E_k'\) \((k \in K')\) those having common points with \(\Gamma'\); \(E_k''\) \((k \in K'')\) those having common points with \(\Gamma''\). By \(G_\beta\) \((\beta \in \mathfrak B)\) we shall denote the domains forming a finite covering of the part of \(\bar D\) remaining after removing the \(2\delta\)-neighborhood of \(D_0\). It may be assumed that the distances of \(\bigcup_{\alpha} T_\alpha'\) to \(\Gamma'\), of \(\bigcup_{\alpha} T_\alpha''\) to \(\Gamma''\), and of \(\bigcup_{\beta} G_\beta\) to \(D_0\) are not less than \(\delta\). We shall denote by
\[ \widetilde L v = b^{ml}v_{y_m y_l}+b^m v_{y_m}+av \]
the differential expression into which \(L\) is transformed after the transformation (1).
- Consider the boundary-value problem:
\[ Lu=f \text{ in } D; \qquad u=0 \text{ on } D_1=\dot D\setminus D_0; \qquad u=0 \text{ on } \bar D''. \tag{2} \]
It follows from \((^2)\) that a solution of problem (2) exists and is unique if in \(\bar D \setminus D_0\) there exist a homogeneous majorant \(H_0(x)\) and an inhomogeneous majorant \(H(x)\) of the equation \(Lu=f\), possessing certain properties. The following conditions ensure the existence of such majorants.
1.1. For every \(\alpha \in \mathfrak A'\) there exist functions \(p_\alpha(s)\), \(q_\alpha(s)\), continuous on \((0,1]\), such that
\[ b^n(y)/b^{nn}(y) \geq p_\alpha(y_n); \]
\[ \int_0^1 e^{P_\alpha(s)}\,ds=\infty,\quad \text{where } P_\alpha(s)=\int_s^1 p_\alpha(t)\,dt;\quad \int_0^1 e^{-P_\alpha(r)}q_\alpha(r)\,dr<\infty; \]
\[ \eta_n(x)\equiv \max_{1\le i\le n} a^{ij}(x)(y_n)_{x_j}\le c_\alpha e^{-P_\alpha(y_n(x))}\int_{y_n(x)}^1 e^{P_\alpha(s)}\,ds, \]
\[ |f(x)|\le b^{nn}(y(x))q_\alpha(y_n(x)), \]
where \(x\in T_\alpha'\cap D\) and \(c_\alpha>0\) is a constant.
1.2. For each \(\alpha\in \mathfrak A''\) there exist functions \(p_\alpha(s)\), \(q_\alpha(s)\), continuous on \((0,1]\), such that
\[ b^n(y)/b^{nn}(y)\le p_\alpha(y_n); \]
\[ \int_0^1 e^{P_\alpha(s)}\,ds<\infty,\quad \text{where } P_\alpha(s)=\int_s^1 p_\alpha(t)\,dt; \]
\[ \int_0^1\int_s^1 e^{P_\alpha(s)-P_\alpha(r)}q_\alpha(r)\,dr\,ds<\infty,\quad \eta_n(x)\le c_\alpha e^{-P_\alpha(y_n(x))}\int_0^{y_n(x)} e^{P_\alpha(s)}\,ds, \]
\[ |f(x)|\le b^{nn}(y(x))q_\alpha(y_n(x)), \]
where \(x\in T_\alpha''\cap D\) and \(c_\alpha>0\) is a constant.
1.3. For each \(k\in K'\) there exist functions \(p_k(\sigma,t)\) and \(q_k(t)\), continuous for \(t\in[0,1]\) and \(\sigma\in[-2,2]\), except for the set \(t=0\), \(0\le\sigma\le2\), and moreover
\[ \frac{b^n(y)}{b^{nn}(y)}\ge \max_{-2\le\tau\le y_{n-1}}\int_{-2}^{\tau} p_k(\sigma,y_n)\,d\sigma; \]
\[ -\frac{b^{n-1}(y)}{b^{n-1\,n-1}(y)}\ge \max_{y_n\le s\le1}\int_s^1 p_k(y_{n-1},t)\,dt; \]
\[ -\frac{b^{n-1}(y)}{b^{n-1\,n-1}(y)} \ge \sup_{\substack{0<r<s\\ y_n<s<1}} \left\{-\int_r^s p_k(y_{n-1},t)\,dt\right\}. \]
Denote
\[ Q_k(\tau,s)=\int_{-2}^{\tau}\int_s^1 p_k(\sigma,t)\,dt\,d\sigma \]
and suppose that
\[ \overline{\lim}_{s\to0}\ \sup_{-2<\tau<y_{n-1}<2} \left\{-\int_\tau^{y_{n-1}}\int_s^1 p_k(\sigma,t)\,dt\,d\sigma\right\}<\infty; \]
\[ \int_0^1\int_{-2}^{0} e^{Q_k(\tau,s)}\,d\tau\,ds=\infty;\quad \int_0^1\int_{-2}^{\sigma} e^{Q_k(\tau,s)}\,d\tau\,ds<\infty \quad \text{for } \sigma<0; \]
for \(\sigma^*\in[0,2]\), \(\sigma<\sigma^*\), \(s>0\) \((s\ge0,\ \text{if } \sigma^*=0)\)
\[ \overline{\lim}_{\rho\to0}\int_{-2}^{\sigma^*} e^{Q_k(\tau,s)}\,d\tau \bigg/ \int_{-2}^{\sigma} e^{Q_k(\tau,s)}\,d\tau \le c_k,\quad \rho=\sqrt{(\sigma-\sigma^*)^2+s^2}; \]
the integral
\[ \int_0^1 e^{-Q_k(\tau,r)}q_k(r)\,dr \]
converges uniformly for \(\tau\in[-2,2]\);
\[ \Theta_n(x)\le c_k\min_{-2\le\tau\le y_{n-1}(x)} e^{-Q_k(\tau,y_n(x))}\int_{y_n(x)}^1 e^{Q_k(\tau,s)}\,ds; \]
\[ \Theta_{n-1}(x)\leq c_k\min_{y_n(x)\leq s\leq 1} e^{-Q_k(y_{n-1}(x),s)} \int_{-2}^{y_{n-1}(x)} e^{Q_k(\tau,s)}\,d\tau; \]
\[
|f(x)|\leq b^{nn}(y(x))q_k(y_n(x)),
\]
where
\[
\Theta_i=\max\{\eta_i(x),\sqrt{\eta_i(x)}\}\quad (i=n-1,n);
\]
\[
\eta_{n-1}(x)\equiv \max_{1\leq i\leq n}a^{ij}(x)(y_{n-1})_{x_j},\quad x\in E_k'\cap D,
\]
\(c_k>0\) is a constant.
1.4. For each \(k\in K''\) there exist functions \(p_k(\sigma,t)\) and \(q_k(t)\), continuous for all \(t\in[0,1]\) and \(\sigma\in[-2,2]\), except, possibly, for the set \(t=0,\ 0\leq\sigma\leq 2\), and moreover
\[ \frac{b^n(y)}{b^{nn}(y)}\leq \min_{y_{n-1}\leq\tau\leq 2}\int_{-2}^{\tau}p_k(\sigma,y_n)\,d\sigma;\quad \frac{b^{n-1}(y)}{b^{\,n-1\,n-1}(y)}\leq \inf_{\substack{s<r<1\\ 0<s<y_n}}\int_s^r p_k(y_{n-1},t)\,dt, \]
and such a function \(\omega_k(\sigma)\), continuous for \(\sigma\in[-2,2]\), equal to zero for \(\sigma\in[0,2]\) and positive for \(\sigma\in[-2,0)\), that the condition
\[
\omega_k(y_{n-1})\leq \vartheta_k(y_{n-1})\int_{y_{n-1}}^0\omega_k(\sigma)\,d\sigma,
\]
where
\[
\vartheta(y_{n-1})\leq b^{n-1}(y)/b^{\,n-1\,n-1}(y)
\]
and \(\vartheta(y_{n-1})\) is a bounded function for \(y_{n-1}\in[-2,0]\), is satisfied.
We shall assume that
\[ \overline{\lim_{s\to 0}}\ \sup_{-2<y_{n-1}<\tau<2} \int_{y_{n-1}}^\tau\int_s^1 p_k(\sigma,t)\,dt\,d\sigma<\infty; \tag{3} \]
\[ \int_0^1\int_{-2}^2\int_s^1 e^{Q_k(\tau,s)-Q_k(\tau,r)}q_k(r)\,dr\,d\tau\,ds<\infty;\quad \sup_{-2<\tau<0}\int_0^1 e^{-Q_k(\tau,r)}q_k(r)\,dr<\infty; \]
\[ \Theta_n(x)\leq c_k \min_{y_{n-1}(x)\leq\tau\leq 2} e^{-Q_k(\tau,y_n(x))} \int_0^{y_n(x)} e^{Q_k(\tau,s)}\,ds+ \]
\[ {}+c_k e^{-Q_k(y_{n-1}(x),y_n(x))} \int_{y_{n-1}(x)}^0\int_s^0 \omega_k(\sigma)\,d\sigma; \]
\[ \Theta_{n-1}(x)\leq c_k \inf_{0<s<y_n(x)} e^{-Q_k(y_{n-1}(x),s)} \int_{y_{n-1}(x)}^2 e^{Q_k(\tau,s)}\,d\tau; \]
\[ \Theta_{n-1}(x)\leq c_k'\quad \text{for }y_{n-1}(x)<0;\qquad |f(x)|\leq b^{nn}(y(x))q_k(y_n(x)), \]
where \(x\in E_k''\cap D\) and \(c_k,c_k'>0\) are constants. Note that from condition (3) there follows the existence of the integral
\[ \int_0^1\int_{-2}^2 e^{Q_k(\tau,s)}\,d\tau\,ds. \]
1.5. \(f(x)\) belongs to \(C^\alpha\ (0<\alpha<1)\) in any closed subdomain \(\overline D\setminus D_0\), and there exist constants \(q_\beta\) \((\beta\in\mathfrak B)\) such that
\[
|f(x)|\leq |a(x)|q_\beta
\]
for \(x\in G_\beta\cap D\).
1.6. The functions \(p_\alpha(s), q_\alpha(s), Q_k(\tau,s), q_k(s)\) satisfy the following “compatibility” conditions:
\[ \overline{\lim_{s\to 0}}\,|p_{\alpha_1}(s)-p_{\alpha_2}(s)|<\infty;\qquad \frac{1}{c_{\alpha_1\alpha_2}}q_{\alpha_2}(s)\leq q_{\alpha_1}(s)\leq c_{\alpha_1\alpha_2}q_{\alpha_2}(s), \]
where \(0<s<1,\ T_{\alpha_1}'\cap T_{\alpha_2}'\neq\varnothing\ (T_{\alpha_1}''\cap T_{\alpha_2}''\neq\varnothing)\) and \(c_{\alpha_1\alpha_2}>0\) is a constant;
\[ \frac{1}{c_{k\alpha}'}e^{P_\alpha(s)-P_\alpha(r)} \leq \int_{-2}^{y_{n-1}} e^{Q_k(\tau,s)-Q_k(\tau,r)}\,d\tau \leq c_{k\alpha}' e^{P_\alpha(s)-P_\alpha(r)}; \]
\[ \frac{1}{c_{k\alpha}'}q_\alpha(s)\leq q_k(s)\leq c_{k\alpha}'q_\alpha(s), \]
where \(0<s<r\leqslant 1;\ \frac12<y_{n-1}<1;\ k\in K',\ \alpha\in \mathfrak A';\ T_\alpha'\cap E_k'\ne\varnothing\) and \(c'_{k\alpha}>0\) is a constant;
\[ \frac{1}{c''_{k\alpha}}e^{P_\alpha(s)-P_\alpha(r)} \leqslant \int_{y_{n-1}}^{2} e^{Q_k(\tau,s)-Q_k(\tau,r)}\,d\tau \leqslant c''_{k\alpha}e^{P_\alpha(s)-P_\alpha(r)}; \]
\[ \frac{1}{c''_{k\alpha}}q_\alpha(s)\leqslant q_k(s)\leqslant c''_{k\alpha}q_\alpha(s), \]
where \(0<s<r\leqslant 1;\ \frac12<y_{n-1}<1;\ k\in K'',\ \alpha\in \mathfrak A'',\ T_\alpha''\cap E_k''\ne\varnothing\) and \(c''_{k\alpha}>0\) is a constant;
\[ \lim_{s\to 0}\sup_{-1<\tau<2}\left|{}^*Q_{k_1}(\tau,s)-Q_{k_2}(\tau,s)\right|<\infty;\qquad \frac{1}{c_{k_1k_2}}q_{k_2}(s)\leqslant q_{k_1}(s)\leqslant c_{k_1k_2}q_{k_2}(s); \]
\[ \frac{1}{c_{k_1k_2}}\omega_{k_2}(y_{n-1}) \leqslant \omega_{k_1}(y_{n-1}) \leqslant c_{k_1k_2}\omega_{k_2}(y_{n-1}), \]
where
\[
0<s<1;\quad -2<y_{n-1}<0;\quad
E'_{k_1}\cap E'_{k_2}\ne\varnothing\ (E''_{k_1}\cap E''_{k_2}\ne\varnothing)
\quad\text{and}\quad c_{k_1k_2}>0
\]
is a constant.
Theorem 1. Suppose conditions 1.1–1.6 are satisfied. Then, for \(a(x)\leqslant -a_0\), where \(a_0\) is a sufficiently large positive constant, there exists a majorant \(H(x)\) of the equation \(Lu=f\) in \(\bar D\setminus D_0\) such that \(H(x)\geqslant 0\) in \(\bar D\) and \(H(x)\to 0\) as \(x\to \bar D''\).
Theorem 2. Suppose the conditions of Theorem 1 are satisfied. Then there exists a majorant \(H_0(x)\) of the equation \(Lu=0\), moreover \(H_0(x)\geqslant 0\) in \(\bar D\), and for every \(\varepsilon>0\) there exists \(\rho=\rho(\varepsilon)\) such that \(H(x)/H_0(x)\leqslant \varepsilon\) for \(x\) belonging to the \(\rho\)-neighborhood of \(D_0\).
Theorem 3. Suppose the conditions of Theorem 1 are satisfied. Then there exists a unique solution of problem (2) satisfying the estimate \(|u(x)|\leqslant H(x)\) \((x\in D)\).
- Consider the boundary-value problem:
\[ Lu=f\ \text{in }D;\qquad Ru=0\ \text{on }D_1=\bar D\setminus D_0;\qquad u=0\ \text{on }\bar D'', \tag{4} \]
where the boundary operator \(R=-A\,\partial/\partial\gamma+B\) satisfies the conditions:
2.1. \(A(x)\) and \(B(x)\) are positive and belong to \(C^{1+\alpha}\) in any closed subdomain of \(D_1\).
2.2. The direction \(\gamma\) forms an acute angle with the inner normal to \(D_1\), and in a \(\delta\)-neighborhood \(\Gamma'[\Gamma'']\) the inequalities hold
\[ -A(x)\partial y_n/\partial\gamma\leqslant c\eta_n(x)\,[A(x)\partial y_n/\partial\gamma\leqslant c\eta_n(x)]; \]
\[ A(x)\partial y_{n-1}/\partial\gamma\leqslant c\eta_{n-1}(x)\,[-A(x)\partial y_{n-1}/\partial\gamma\leqslant c\eta_{n-1}(x)]. \]
Theorems analogous to Theorems 1 and 2 are valid, from which the following follows.
Theorem 4. Suppose the conditions of Theorem 1, 2.1, 2.2 are satisfied and \(B(x)\geqslant b_0\), where \(b_0\) is a sufficiently large positive constant. Then there exists a unique solution of problem (4), satisfying the estimate
\[ |u(x)|\leqslant H(x)\quad (x\in D). \]
- The consideration of problems (2) and (4) with inhomogeneous boundary conditions is connected only with questions of extending functions given on the boundary into the interior of the domain in a certain class of functions. Therefore the corresponding results of work (2) carry over almost without change to the case of an arbitrary smooth domain.
The author expresses his gratitude to Prof. S. G. Krein for valuable discussions.
Voronezh State University
Received
14 IV 1966
CITED LITERATURE
- M. Schechter, Comm. Pure and Appl. Math., 13, No. 2, 321 (1960).
- V. P. Glushko, DAN, 163, No. 1, 22 (1965).
- C. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.