UDC 517.946

MATHEMATICS

Academician A. D. ALEKSANDROV

THE MAXIMUM PRINCIPLE

1. In the present communication a general method is indicated for deriving theorems of maximum-principle type, generalizing, in particular, the results of \((^{2-5})\). It is based on considerations used in \((^1)\). As in \((^1)\), we consider functions \(u(x)\) in an \(n\)-dimensional domain \(G\), bounded and lower semicontinuous, and possessing an absolutely continuous support mapping. Suppose that such a function satisfies, at least at almost all of its points of convexity, the inequality

\[ F(u_{ij}, u_i, u, x) \leqslant 0 \]

where the derivatives \(u_i, u_{ij}\) may be understood as coefficients of approximating differentials (they include both ordinary and generalized derivatives and, as shown in \((^1)\), exist at almost all points of convexity).

Let \(E^m = E\) be an \(m\)-dimensional plane passing through the origin of coordinates \(O\), \(1 \leqslant m \leqslant n\), \(x_E\) the projection of the point \(x\), and \(M_E\) the projection of the set \(M\). Suppose that at the point \(x_0\), \(u_{ij}\,dx^i dx^j \geqslant 0\). The projection \(f_E\) of the function \(f(x) = u_{ij}(x_0)x^i x^j\), i.e.

\[ f_E(x') = \inf_{x_E=x'} f(x), \qquad x' \in E, \]

will be a quadratic form of rectangular coordinates in \(E\). We denote its determinant by \(w_E(x_0)\). If \(m=n\), then \(u_E=u\), \(w_E=\det(u_{ij})\).

By a pencil we shall mean a set of planes \(E^m\) passing through some given \(E^{m-1}\) (a point, if \(m=1\)). If \(m=n\), then the pencil consists of the single “plane” \(E^n\), and then all further qualifications concerning pencils are omitted.

We introduce the condition:

\((F)\). From the fact that the function \(u\) satisfies inequality (1), it follows that for the planes \(E\) of some pencil \(\{E\}\), at almost all points of convexity of \(u\), at which for some \(\varepsilon, \varepsilon' > 0\)

\[ u(x) < \inf u + \varepsilon, \qquad |\nabla u(x)| < \varepsilon' \]

and \(\nabla u\) is parallel to \(E\), the inequalities

\[ w_E(x) \leqslant X_E(x) U_E(\nabla u). \tag{2} \]

hold.

Here the functions \(X_E \geqslant 0\) are defined in \(G\) and are such that in a neighborhood of any point \(x_0 \in G\) each of them is majorized by a summable function of \(x_E\). (The indices \(E\) in \(X, U\) signify only that these functions depend on \(E\).) In deriving (2) from (1), substitutions \(u=u(x)\), \(u_i=u_i(x)\), \(u_{ij}=u_{ij}(x)\) are, of course, allowed. The generality of condition \((F)\) is clarified in \((^1)\); it is satisfied, one may say, for any elliptic \(F\).

Extend \(u\) to \(G+\partial G\), putting, for \(x \in \partial G\),

\[ u(x)=\liminf u(x'), \qquad x' \to x,\ x' \in G. \]

Introduce the set

\[ N \in \{\,u(x)=\inf_x u,\ x \in G+\partial G\,\}. \tag{3} \]

We shall call a cone a finite solid cone with vertex at \(O\), and shall say that it is a cone around the direction \(v\) if it contains a segment issuing from \(O\) in the direction \(v\).

Theorem 1. If, under all the conditions listed, the set \(N\) has a supporting plane \(P\) not intersecting \(N \cap \partial G\), then there exists su-

cone \(V\) about the outer normal \(\nu\) to \(P\), such that for almost all planes \(E\) of the pencil \(\{E\}\), under condition (F),

\[ \int_{V\cap E} \frac{dy}{U_E(y)} < \infty \]

(\(U_E\) is a function of a vector, and here, correspondingly, \(y\) is the point that is its endpoint).

The proof is based on two lemmas, in which the conditions of Theorem 1 are meant.

Lemma 1. The supporting image of any neighborhood of the set \(N\cap P\) contains a cone about \(\nu\).

The proof is obvious from the geometric consideration of supporting planes to the graph of the function \(u\) in \((n+1)\)-dimensional space.

Lemma 2. For every \(M\subset G\) and almost all \(E\in\{E\}\),

\[ \int_{\psi(M)\cap E} U_E^{-1}(y)\,dy \leq \int_{M_E} Y_E(y)\,dy, \]

where \(\psi(M)\) is the supporting image of \(M\), and \(Y_E\) is such a function that on \(M\), \(X_E(x)\leq Y_E(x_E)\).

The proof is contained in the derivations of § 1 of [1]. Combining Lemmas 1 and 2 obviously gives Theorem 1. This theorem also serves as the basis for studying the properties of the set \(N\), and thereby for deriving theorems of maximum-principle type.

Addition to Theorem 1. If \(u(x)\) is differentiable at all points of convexity and twice differentiable at them except, at most, for a countable set of them, and if inequalities (2) are also satisfied, then the assertion of Theorem 1 holds for all \(E\in\{E\}\); in particular, the pencil \(\{E\}\) may consist of a single plane.

Accordingly, under these conditions, in the corollaries of Theorem 1 one may mean one given plane \(E\).

1. From Theorem 1 one easily derives the following result (Theorem 3 of [1]), where, as below, a pencil \(\{E\}\) of positive measure is meant (in the sense of measure in the complete pencil with the same axis \(E^{m-1}\)).

Theorem 2. If, for almost all \(E\in\{E\}\), the functions \(U_E\) are such that the integrals of \(U_E^{-1}\) over any neighborhood of the origin \(O\) in \(E\) are infinite, then \(\inf u\) is attained on \(\partial G\).

The simplest example is the linear inequality

\[ a^{ij}u_{ij}+b^i u_i+cu\leq 0,\qquad a^{ij}\xi_i\xi_j\geq 0. \tag{4} \]

It is enough to consider it for \(u\leq \inf u+\varepsilon\), and therefore the term \(cu\) may be discarded if \(\inf u\ne 0\) and \(c\inf u\geq 0\).

For the given plane \(E\), define the quantity \(a_E\): if by a rotation of axes \(E\) is made the plane \((x^1,\ldots,x^m)\), then \(a_E=\det(a^{ij})\), \(ij\leq m\). Then, for \(d^2u\geq 0\),

\[ a^{ij}u_{ij}\geq m(a_E w_E)^{1/m}. \tag{5} \]

Therefore, if \(a_E\ne 0\), the term \(cu\) is absent and \(\nabla u\) is parallel to \(E\), then from (4) it follows that

\[ w_E\leq m^{-m}a_E^{-1}|b_E|^m|\nabla u|^m, \tag{6} \]

where \(b_E\) is the projection of the vector \((b^1,\ldots,b^n)\) onto \(E\). This is inequality (2) with

\[ X=m^{-m}a_E^{-1}|b_E|^m,\quad U=|\nabla u|^m. \]

The integral of \(U^{-1}(y)=|y|^{-m}\) over the whole cone in \(E\) with vertex \(y=0\) is infinite.

Let \(\inf u=0\) and let it be attained at a point \(x'\in G\). If \(x\) is a point of convexity of \(u\), then for every \(x'\)
\[ u(x)-u(x')\le (x-x')\nabla u(x). \]
Therefore, if \(u(x')=0\) and \(\nabla u(x)\) is parallel to \(E\), then
\[ u(x)\le |x_E-x'_E|\,|\nabla u(x)|. \tag{7} \]

Therefore from (4) and (5) we easily obtain:
\[ w_E\le m^{-m}a_E^{-1}\bigl(|b_E|+|c_-|\,|x_E-x'_E|\bigr)^m|\nabla u|^m. \tag{8} \]

This is again inequality (2) with \(U=|\nabla u|^m\). Applying to it, and also to (6), Theorem 2, we obtain:

Theorem 3. Let \(u(x)\) satisfy inequality (4), and suppose that for the planes \(E\) of some pencil the functions \(a_E^{-1}|b_E|^m\) are locally majorized by summable functions of \(x_E\). Suppose, moreover, that either \(\inf u\ne0\) and \(c\inf u\ge0\), or \(\inf u=0\) and, for every \(x'\in G\), in some neighborhood of it the functions
\[ a_E^{-1}|c_-|^m|x_E-x'_E|^m \]
are majorized by summable functions of \(x_E\). Then \(\inf u\) is attained on \(\partial G\).

1. Let us now assume the following:

(A). All the conditions of item 1 hold locally, i.e. for the restriction of \(u\) to any subdomain \(D\subset G\).

(B). This property is preserved under any regular mapping (analytic with Jacobian \(\ne0\)) of the space onto itself.

Let us note that, with regard to the absolute continuity of the supporting map, this is true, in particular, for functions having generalized derivatives \(u_{ij}\), summable with the \(n\)-th power, and also, for example, for differentiable functions having everywhere, except possibly at a countable set of points, finite upper derivatives \(u_{ii}\). As for condition (F), it is obviously true for inequality (4).

Under condition (A) all the preceding conclusions hold for each \(D\subset G\), which gives substantially greater information about the set \(N\). In particular, from Theorem 1 it follows immediately: if in condition (F) \(m=n\), i.e. (2) reduces to
\[ \det(u_{ij})\le XU, \]
and if the integral of \(U^{-1}\) over any cone is infinite, then the set \(M=N\cap G\) has no points of local strict convexity (i.e. no such \(x\) for which there exist half-spaces \(R\) and neighborhoods \(W\) such that \(M\cap R\cap W=(x)\)).

It follows from this that, if condition (B) is also satisfied, then either \(M\) is empty or \(M=G\). Indeed, suppose that \(M\) is nonempty and \(\ne G\). Then there is a point \(x\in M\) which is a vertex of some solid paraboloid \(P\) containing in a neighborhood of \(x\) no other points of \(M\). A mapping that carries \(P\) into a half-space makes \(x\) a point of local strict convexity of the transformed set \(M\), and, according to the preceding, this is impossible.

We apply this conclusion to inequality (4), since for \(m=n\) in (6) and (8) \(U=|\nabla u|^n\). Under transformation of inequality (4) the coefficients \(b^i\) acquire addends that are linear with respect to \(a^{ij}\). Therefore we obtain the theorem:

Theorem 4. Let \(u(x)\), under conditions (A), (B), with regard to the absolute continuity of the supporting map, satisfy inequality (4) with locally summable
\[ a^{-1}|b^i|^n,\qquad a^{-1}|a^{ij}|^n, \]
where \(a=\det(a^{ij})\). Then, if \(\inf u\ne0\) and \(c\inf u\ge0\), or \(\inf u=0\) and, in a neighborhood of every \(x'\in G\), the function
\[ a^{-1}|c_-|^n|x-x'|^n \]
is summable, then either the set \(M=N\cap g\), where \(u=\inf u\), is empty, or \(M=G\).

1. The application of similar, more delicate considerations in the general case \(m\le n\) leads to the following result.

Suppose that, for a given \(m\), to almost every \(x\in G\) there is assigned a plane \(E(x)\) passing through it in such a way that: 1) \(E(x)\) is determined by the eigenvectors of the matrix \(a^{ij}(x)\) corresponding to positive eigenvalues.

values \(\alpha_1,\ldots,\alpha_k,\ k \ge m\); 2) to each \(x_0 \in G\) there corresponds a neighborhood \(W(x_0)\) and a number \(a(x_0)>0\) such that for almost every \(x \in W(x_0)\) the product of any \(m\) of \(\alpha_1,\ldots,\alpha_k\) is not less than \(a(x_0)\). Then we say that there is a field of \(m\)-ellipticity \(\mathscr E^m\). (According to this definition, the dimensions \(k\) of the planes \(E(x)\) may be different at different \(x\). In the simplest case \(k=m\).)

We shall say that a plane \(P(x_0)\) passing through \(x_0\) is tangent at \(x_0\) to the field \(\mathscr E^m\) if almost all \(E(x)\in \mathscr E^m\) at points \(x\) close to \(x_0\) form with \(P(x_0)\) arbitrarily small angles. Finally, let us call a nonempty closed set \(S\) a generalized \(l\)-integral manifold of the field \(\mathscr E^m\) if at every point \(x\in S\) the \(l\)-dimensional plane \(P(x)\), tangent to \(\mathscr E^m\), is contained in the contingence of the set \(S\) at the point \(x\).

Theorem 5. Let a function \(u(x)\), with the same property as in Theorem 4, satisfy inequality (4), which has a field of \(m\)-ellipticity \(\mathscr E^m\). Suppose, moreover, that one of the following conditions is fulfilled: (I) \(\inf u=0\), and under every regular transformation \((x^i)\to(y^i)\), for each point \(x_0\in G\) there is a neighborhood in which the functions
\[ a^{ij}|m,\ |b^i|m,\ |c_-|m\left[\sum_{1}^{m}(y^i-y_0^i)^2\right]^{m/2} \]
are majorized by summable functions of \(y^1,\ldots,y^m\) only; (II) \(\inf u\ne0,\ c\inf u\ge0\), and \(a^{ij}, b^i\) satisfy the same condition.

Then the set \(M=N\cap G\) is either empty or is a generalized \(m\)-integral manifold of the field \(\mathscr E^m\).

For \(m=n\) this gives Theorem 4, since if (almost everywhere) \(a\ne0\), then, dividing (4) by \(a^{1/n}\), we obtain inequality (4) with \(\det(a^{ij})=1\).

The condition on \(|a^{ij}|m\), etc., in Theorem 5 looks ineffective. But it is certainly fulfilled for a function \(g(x)\) if, in a neighborhood of any point, \(|g|\) is estimated by a series \(\sum h_i(|x-x_i|)\), convergent for all \(x\) except singular points \(x_i\), where \(h_i(r)\ge0,\ r\in(0,\infty)\), are decreasing functions such that the series
\[ \sum \int_0^r h_i(r)^{m-1}\,dr \]
converges for every \(r\).

Theorems 4 and 5, in view of the invariance of their conditions with respect to changes of variables, are applicable to functions on analytic manifolds. For example, if a function \(u\in W_n^n\) on a compact manifold satisfies a homogeneous linear equation with \(a^{ij}\xi_i\xi_j\ge0\), for which \(a^{-1}|a^{ij}|n,\ a^{-1}|b^i|n\) are summable, then \(u=\mathrm{const}\) if \(c\le0\), and either \(u\) changes sign or \(u=0\) if \(c\ge0\) and somewhere \(c>0\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
9 XI 1966

CITED LITERATURE

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