Mathematics

WANG TONG

THE INCREASING PROPERTY OF DIRECT VALUES OF GENERALIZED POTENTIALS FOR THE COMPARISON FUNCTION AND THE PRINCIPAL FUNDAMENTAL SOLUTION OF A GENERAL SECOND-ORDER ELLIPTIC EQUATION

(Presented by Academician I. G. Petrovskii on 22 V 1963)

Theorems on the increasing property of direct values of a potential play a large role in mathematical physics \((^1)\). However, up to now these theorems have been proved only for potentials of the Laplace operator \((^2)\). In the present work the increasing property is established for the direct values of the double-layer potential and the direct values of the conormal derivative of the single-layer potential for the comparison function and the principal fundamental solution of a general second-order elliptic equation. By the direct value of a double-layer potential is meant this same potential, considered as a function of the points of the surface on which its density is distributed. The direct values of the conormal derivative of a single-layer potential are defined analogously.

\(1^\circ\). Let, in the \(N\)-dimensional Euclidean space \(R_N\), there be given an open domain \(g\), whose boundary is a closed surface \(\Gamma\). In the domain \(g\) we consider a general uniformly elliptic equation of second order

\[ Mu=\sum_{i,j=1}^{N} a_{ij}(x)\frac{\partial^2 u(x)}{\partial x_i\partial x_j} +\sum_{i=1}^{N} b_i(x)\frac{\partial u(x)}{\partial x_i} +c(x)u(x)=f(x), \tag{1} \]

i.e., such that at any points \(x=(x_1\ldots x_N)\in g\) one has

\[ a_{ij}=a_{ji},\qquad \sum_{i,j=1}^{N} a_{ij}\xi_i\xi_j \geq \alpha \sum_{i=1}^{N}\xi_i^2 \tag{2} \]

for arbitrary real \(\xi_1,\ldots,\xi_N\), where \(\alpha>0\) is a certain constant.

Suppose now that the surface \(\Gamma\) belongs to the class \(A^{(n,\alpha)}\), \(n>2,\ 0<\lambda\leq 1\), the coefficients of equation (1) belong to the class \(C^{(n-1,\lambda)}(g+\Gamma)\), and \(c\leq 0\) in \(g+\Gamma\). Then, by virtue of a result of Gevrey (see \((^3)\), p. 77), the coefficients of equation (1) can be continued through \(\Gamma\) to the whole space \(R_N\) so that equation (1) is uniformly elliptic in the whole space, the coefficient \(c(x)\) is nonpositive everywhere and strictly negative outside some bounded domain.

Definition. The function \(G(x,y)\) is called the principal fundamental solution of equation (1), if it is a fundamental solution of equation (1)** in the whole space \(R_N\) and if there exist two constants \(a>0\) and \(R>0\) such that for \(\gamma\geq R\)

\[ G(x,y)=O(e^{-a\gamma}),\qquad \frac{\partial G(x,y)}{\partial x_i}=O(e^{-a\gamma}),\qquad i=1,\ldots,N, \]

where \(\gamma\) is the distance between the points \(x\) and \(y\).

* For the definition of these classes, see \((^3)\), p. 10.
** For the definition of a fundamental solution, see \((^3)\), p. 26.

It is known that under the conditions formulated above the principal fundamental solution \(G(x,y)\) of equation (1) exists \((^4)\).

Consider the double-layer potential for \(G(x,y)\):

\[ W(x)=\int_{\Gamma}\mu(y)P_yG(x,y)\,ds_y, \tag{3} \]

where \(\mu(x)\) is the density of this potential,

\[ P_yG(x,y)=\frac{\partial}{\partial \nu_y}G(x,y) = \frac{1}{a(y)} \sum_{i,j=1}^{N}a_{ij}(y)\frac{\partial G(x,y)}{\partial y_i}\cos(n,y_i) \]

is the conormal derivative of \(G(x,y)\), while \(n\) is the exterior normal to \(\Gamma\) and

\[ a(y)= \left[ \sum_{i=1}^{N} \left( \sum_{j=1}^{N}a_{ij}(y)\cos(n,y_j) \right)^2 \right]^{1/2}. \]

The direct value \(W_{\mathrm{pr}}\) on \(\Gamma\) of the double-layer potential \(W(x)\) is the function of the points of the surface \(\Gamma\)

\[ W_{\mathrm{pr}}=W(x)\big|_{x\in\Gamma} = \int_{\Gamma}\mu(y)\,[P_yG(x,y)]_{x\in\Gamma}\,ds_y. \tag{4} \]

We have succeeded in proving that the direct value \(W_{\mathrm{pr}}\) raises the smoothness of \(\mu(x)\) by almost one unit, i.e. the following holds.

Theorem 1. If the density \(\mu(x)\) belongs to the class \(C^{(n-2,\lambda)}\) on \(\Gamma\), then the direct value \(W_{\mathrm{pr}}\) of the double-layer potential belongs to the class \(C^{(n-1,\lambda')}\) on \(\Gamma\), where \(0<\lambda'<\lambda\) is arbitrary; moreover, the norm \(W_{\mathrm{pr}}\) in \(C^{(n-1,\lambda')}(\Gamma)\) is estimated in terms of the norm of the density \(\mu(x)\) in \(C^{(n-2,\lambda)}(\Gamma)\):

\[ \|W_{\mathrm{pr}}\|_{C^{n-1,\lambda'}} = O\bigl(\|\mu\|_{C^{n-2,\lambda}}\bigr), \tag{5} \]

where the constant entering the \(O\)-term depends only on the norms of the coefficients of the equation
\(\|a_{ij}\|_{C^{n-1,\lambda}(g+\Gamma)}\),
\(\|b_i\|_{C^{n-1,\lambda}(g+\Gamma)}\),
\(\|c\|_{C^{n-1,\lambda}(g+\Gamma)}\), on the ellipticity constant \(\alpha\), on the choice of \(\lambda'\), and on the property of the surface \(\Gamma\) *, but does not depend on the density \(\mu(x)\).

Remark 1. Theorem 1 remains valid if in (4) \(P\) takes the more general form:

\[ P_xu(x)=\alpha(x)\frac{\partial u(x)}{\partial \nu_x}+\beta(x)u(x), \]

where \(\alpha(x)\) and \(\beta(x)\) are any two functions of the class \(C^{(n-2,\lambda)}(\Gamma)\). Then the constant entering the \(O\)-term on the right-hand side of (5) also depends on the norms
\(\|\alpha\|_{C^{n-2,\lambda}(\Gamma)}\) and
\(\|\beta\|_{C^{n-2,\lambda}(\Gamma)}\).

\(2^\circ\). Suppose now that \(\Gamma\) belongs to the class \(A^{(n,\lambda)}\), \(0<\lambda\leqslant 1\), \(n\geqslant2\), that \(a_{ij}\) belong to the class \(C^{(n-1,\lambda)}\) in \(g+\Gamma\), and that \(b_i\) and \(c\) belong to the class \(C^{(n-2,\lambda)}\) in \(g+\Gamma\), with \(c\leqslant0\). In this case one can also construct the principal fundamental solution \(G(x,y)\) of equation (1), by virtue of the result of Gevrey noted above. For \(G(x,y)\) consider the simple-layer potential:

\[ V(x)=\int_{\Gamma}\nu(y)G(x,y)\,ds_y, \tag{6} \]

where \(\nu(x)\) is the density of this potential.

* That is, it depends on the norm of the function defining the surface \(\Gamma\) in local coordinates.

The direct value \(V_{\mathrm{pr}}\) on \(\Gamma\) of the conormal derivative of the simple-layer potential \(V(x)\) is the following function of the points of the surface \(\Gamma\):

\[ V_{\mathrm{pr}}=[P_xV(x)]_{x\in\Gamma} =\int_{\Gamma}v(y)\,[P_xG(x,y)]_{x\in\Gamma}\,ds_y . \tag{7} \]

We have proved that the direct value of the conormal derivative of the simple-layer potential increases the smoothness of the density \(v(x)\) by almost one unit; that is, the following holds.

Theorem 2. If the density \(v(x)\) belongs to the class \(C^{(n-2,\lambda)}\) on \(\Gamma\), then the direct value \(V_{\mathrm{pr}}\) of the conormal derivative of the simple-layer potential belongs to the class \(C^{(n-1,\lambda')}\) on \(\Gamma\), where \(0<\lambda'<\lambda\) is arbitrary, and the norm of \(V_{\mathrm{pr}}\) in \(C^{(n-1,\lambda')}(\Gamma)\) is estimated in terms of the norm of the density \(v(x)\) in \(C^{(n-2,\lambda)}(\Gamma)\):

\[ \|V_{\mathrm{pr}}\|_{C_{n-1,\lambda'}}= O\bigl(\|v\|_{C_{n-2,\lambda}}\bigr), \tag{8} \]

where the constant occurring in the term \(O\) depends only on the norms of the coefficients
\(\|a_{ij}\|_{C_{n-1,\lambda}(g+\Gamma)}\),
\(\|b_i\|_{C_{n-2,\lambda}(g+\Gamma)}\),
\(\|c\|_{C_{n-2,\lambda}(g+\Gamma)}\), the ellipticity constant \(a\), the choice of \(\lambda'\), and the properties of the surface \(\Gamma\), but does not depend on the density \(v(x)\).

Remark 2. Theorem 2 remains valid if in (7) \(P\) has the more general form

\[ P_xu(x)=\gamma(x)\frac{\partial u(x)}{\partial \nu_x}+s(x)u(x), \]

where \(\gamma(x)\) and \(s(x)\) are any two functions of the class \(C^{(n-1,\lambda)}(\Gamma)\). Then the constant occurring in the term \(O\) on the right-hand side of (8) also depends on the norms
\(\|\gamma\|_{C_{n-1,\lambda}(\Gamma)}\) and
\(\|s\|_{C_{n-1,\lambda}(\Gamma)}\).

3°. Suppose now that \(a_{ij}, b_i\), and \(c\) belong to the class \(C^{(n,\lambda)}\) in \(R_N\), \(0<\lambda\leq 1\), \(0\leq n\), and outside some bounded domain \(c<-b^2\), where \(b>0\). Then there also exists the principal fundamental solution \(G(x,y)\) of equation (1). For \(G(x,y)\) consider the volume potential

\[ U(x)=\int_g Z(y)G(x,y)\,dy, \tag{9} \]

where \(g\) is any bounded domain, and \(Z(x)\) is the density of this potential. It has been proved that the volume potential for the principal fundamental solution increases the smoothness of the density by almost two orders; that is, the following holds.

Theorem 3. Let the density \(Z(x)\) belong to the class \(C^{(n,\lambda)}(g+\Gamma)\) in \(g+\Gamma\); then the volume potential \(U(x)\) for the principal fundamental solution \(G(x,y)\) belongs to the class \(C^{(n+2,\lambda')}(g)\), where \(0<\lambda'<\lambda\) is arbitrary, and the norm of \(U(x)\) in \(C^{(n+2,\lambda')}(g')\) is estimated in terms of the norm of \(Z(x)\) in \(C^{(n,\lambda)}(g+\Gamma)\):

\[ \|U\|_{C_{n+2,\lambda'}(g')}= O\bigl(\|Z\|_{C_{n,\lambda}(g+\Gamma)}\bigr), \tag{10} \]

where \(g'\) is an arbitrary interior subdomain of the domain \(g\), whose distance from the boundary \(\Gamma\) of the domain \(g\) is equal to \(d\), and the constant occurring in the term \(O\) depends only on the norms of the coefficients
\(\|a_{ij}\|_{C_{n,\lambda}(g+\Gamma)}\),
\(\|b_i\|_{C_{n,\lambda}(g+\Gamma)}\),
\(\|c\|_{C_{n,\lambda}(g+\Gamma)}\), the ellipticity constant \(a\), the choice of \(\lambda'\), and the number \(d\), but does not depend on the density \(Z(x)\).

Remark 3. Theorem 3 was proved by V. A. Il’in and I. A. Shishmarev for a smooth closed domain \(g+\Gamma\in A^{(n+2,\lambda)}\) (5).

4°. It is known that to each equation (1) one can put in correspondence its own comparison function, defined by the formula

\[ H(x,y)= \begin{cases} \dfrac{1}{(N-2)\omega_N\sqrt{A(y)}}\,\rho^{\,2-N}, & N \geqslant 3,\\[1.2em] \dfrac{1}{2\pi\sqrt{A(y)}}\,\ln\dfrac{1}{\rho}, & N=2, \end{cases} \]

where

\[ \rho^2=\sum_{i,j=1}^{N} A_{ij}(y)(x_i-y_i)(x_j-y_j), \]

and \(A_{ij}(x)\) is the ratio of the algebraic complement of the element \(a_{ij}(x)\) in the determinant \(A(x)=|a_{ij}(x)|\) to the determinant \(A(x)\) itself; \(\omega_N\) is the area of the unit spherical surface of \(N-1\) dimensions.

Let us now consider the potentials (3), (6), and (9), in which, instead of the principal fundamental solution \(G(x,y)\), the comparison function \(H(x,y)\) appears. Then Theorems 1, 2, and 3 are also valid; that is, more precisely, the following holds:

Theorem 4. Suppose that all the conditions of Theorem 1 (or, respectively, of Theorem 2 or 3) are satisfied, except for the smoothness requirements on the coefficients \(b_i(x)\) and \(c(x)\). Under these conditions Theorem 1 (or, respectively, Theorems 2 and 3) is valid for the corresponding potential in which, instead of \(G(x,y)\), the comparison function \(H(x,y)\) stands; moreover, in this case the constants entering the estimates (5), (8), and (10) do not depend on the coefficients \(b_i(x)\) and \(c(x)\).

Remark 4. The improving property of the direct values of potentials for the comparison function of the Laplace operator was proved by Kh. L. Smolitskii \((^2)\).

The author takes this opportunity to express sincere gratitude to V. A. Il’in for posing the problem and for guidance, to A. N. Tikhonov for discussion of the results, and to I. A. Shishmarev for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
10.V.1963

REFERENCES

\(^1\) N. M. Günter, Potential Theory and Its Application to the Basic Problems of Mathematical Physics, 1953.
\(^2\) Kh. L. Smolitskii, The limiting problem for the wave equation, Doctoral dissertation, L., 1950.
\(^3\) C. Miranda, Equazioni alle derivate parziali di tipo ellittico, IL, 1957.
\(^4\) G. Giraud, Ann. Éc. Norm. Sup., 49, 1 (1932).
\(^5\) V. A. Il’in, I. A. Shishmarev, DAN, 141, No. 3 (1961).