MATHEMATICS

V. A. IL’IN and I. A. SHISHMAREV

SMOOTHNESS PROPERTIES OF GENERALIZED POTENTIALS OF AN ELLIPTIC OPERATOR

(Presented by Academician I. G. Petrovskii on 30 VI 1961)

In the present note the differential properties of generalized potentials of a second-order elliptic operator are investigated. Analogous results for potentials of the Laplace operator were obtained by Kh. L. Smolitskii \((^{1})\).

\(1^\circ\). A special problem. Here we shall consider a certain auxiliary problem, which, however, is also of independent interest, for example, in a number of questions of mechanics. Denote by \(g_1\) an arbitrary open \(N\)-dimensional domain lying, together with its boundary \(C\), inside the domain \(g\), bounded by the surface \(\Gamma\), and by \(g_2\) the domain \(g \setminus (g_1 + C)\). Consider the problem:

\[ \begin{gathered} Lu=0 \quad \text{in the domain } g_1,\\ Lu=0 \quad \text{in the domain } g_2,\\ [u]\big|_C=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]\Big|_C=\theta(x),\qquad u\big|_\Gamma=0. \end{gathered} \tag{1} \]

Here \(L\) is a linear elliptic operator of second order

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

defined in the domain \((g+\Gamma)\); \(c(x)\leq 0\); \([u]\big|_C=u\big|_{C-0}-u\big|_{C+0}\);

\[ \left[\frac{\partial u}{\partial \nu}\right]\Big|_C = \frac{\partial u}{\partial \nu}\Big|_{C-0} - \frac{\partial u}{\partial \nu}\Big|_{C+0}; \]

\(\partial/\partial\nu\) is differentiation along the conormal exterior with respect to the domain \(g_1\); the symbols \(C-0\) and \(C+0\) denote that limiting values are taken, respectively, from the inner and outer sides of the surface \(C\) (with respect to \(g_1\)).

By a solution of problem (1) from the class* \(C^{(n,\mu)}\) \((n\geq 2)\) we shall mean a function which: 1) belongs to the class \(C^{(0)}\) in the domain \((g+\Gamma)\), to the class \(C^{(n,\mu)}\) in each of the closed domains \((g_1+C)\) and \((g_2+C+\Gamma)\); 2) satisfies all the conditions of problem (1) in the usual classical sense.

Theorem 1. If the coefficients of the operator \(L\) belong to the class \(C^{(n-2,\mu)}\) in the closed domain \((g+\Gamma)\), while \(\theta(x)\) belongs to the class \(C^{(n-1,\mu)}\) on the surface \(C\), and the boundary surfaces \(C\) and \(\Gamma\) belong to the class** \(A^{(n,\mu)}\), then for every solution of problem (1) from the class \(C^{(n,\mu)}\) there holds

* For the definition of the classes \(C^{(n)}\) and \(C^{(n,\mu)}\), see \((^{2})\), p. 10.
** For the definition of the class \(A^{(n,\mu)}\), see \((^{2})\), pp. 10–11.

the following a priori estimate is valid*

\[ u_{n,\mu}=O\bigl(\theta_{n-1,\mu}+\theta_0\bigr)\quad (n\geqslant 2). \tag{3} \]

The constant bounding the growth of the \(O\)-terms depends on \(A_{n-2,\mu}\), \(B_{n-2,\mu}\), \(C_{n-2,\mu}\), \(A_0\), \(B_0\), \(C_0\), and also on the form of the domains \(g_1\) and \(g_2\).

For problem (1), the uniqueness theorem is proved without difficulty (see, for example, (³)). The following theorem (of existence) is also valid:

Theorem 2. Under the conditions of Theorem 1 there exists (and moreover a unique) solution of problem (1) of class \(C^{(n,\mu)}\) \((n\geqslant 2)\).

The proof of Theorem 2 is carried out by the method of continuation with respect to a parameter on the basis of estimate (3) and using the fact that, for the particular case of the operator \(L\), the existence of a solution of problem (1) of class \(C^{(n,\mu)}\) has already been proved (see (⁴)).

2°. Estimates of generalized potentials. The unique solution of problem (1) is representable in the form of a generalized simple-layer potential

\[ u(x)=\int_C F(x,y)\theta(y)\,dy, \tag{4} \]

where \(F(x,y)\) is the Green’s function of the Dirichlet problem for the operator \(L\) in the closed domain \((g+\Gamma)\). Thus, under the conditions of Theorem 1, the generalized simple-layer potential belongs to the class \(C^{(n,\mu)}\) and is estimated by formula (3) in each of the closed domains \((g_2+C)\) and \((g_2+C+\Gamma)\).

Let us now consider the following boundary-value problem:

\[ \begin{gathered} Lu=0 \quad \text{in the domain } g_1,\\ Lu=0 \quad \text{in the domain } g_2,\\ [u]\big|_C=\varphi,\qquad \left[\frac{\partial u}{\partial \nu}\right]\bigg|_C=0,\qquad u\big|_\Gamma=0. \end{gathered} \tag{5} \]

We represent the solution \(u(x)\) of problem (5) in the form \(u(x)=v(x)+w(x)\), where \(v(x)\) is the solution of the problem

\[ \begin{gathered} Lv=0 \quad \text{in the domain } g_1,\\ v\big|_{C-0}=\varphi,\quad v\equiv 0 \quad \text{in the domain } g_2; \end{gathered} \tag{6} \]

\(w(x)\) is the solution of the problem

\[ \begin{gathered} Lw=0 \quad \text{in the domain } g_1,\\ Lw=0 \quad \text{in the domain } g_2,\\ [w]\big|_C=0,\qquad \left[\frac{\partial w}{\partial \nu}\right]\bigg|_C =-\frac{\partial v}{\partial \nu}\bigg|_{C-0},\qquad w\big|_\Gamma=0. \end{gathered} \tag{7} \]

Using, for the estimate of \(v(x)\), the known results of Schauder and Caccioppoli (see (²), p. 144) and using formula (3) to estimate \(w(x)\), we arrive at the theorem:

Theorem 3. If the coefficients of the operator \(L\) belong to the class \(C^{(n-2,\mu)}\) in the closed domain \((g+\Gamma)\); \(\varphi(x)\) belongs to \(C^{(n,\mu)}\) on the surface \(C\); the boundary surfaces \(C\) and \(\Gamma\) belong to \(A^{(n,\mu)}\), then for every solution of problem (5) of class \(C^{(n,\mu)}\) the a priori estimate

\[ u_{n,\mu}=O(\varphi_{n,\mu}+\varphi_0)\quad (n\geqslant 2) \tag{8} \]

is valid.

With regard to the constant entering into this estimate, one may repeat what was said above about formula (3). From Theorem 3, as above, it follows:

* We shall denote the sum of the maxima of the moduli of the derivatives of order \(n\) of a function \(p(x)\) by \(p_n\), and the sum of the Hölder coefficients of these derivatives, taken for the exponent \(\mu\), by \(p_{n,\mu}\).

Theorem 4. Under the hypotheses of Theorem 3 there exists a (moreover unique) solution of problem (5) of the class \(C^{(n,\mu)}\) \((n \geqslant 2)\).

Representing this solution further in the form of a generalized double-layer potential*

\[ u(x)=\int_C Q_y F(x,y)\,\varphi(y)\,dy, \tag{9} \]

where \(F(x,y)\) is the Green’s function of the Neumann problem for the operator \(L\) in the closed domain \((g+\Gamma)\), we arrive at the assertion that, under the hypotheses of Theorem 3, the generalized double-layer potential belongs to the class \(C^{(n,\mu)}\) and is estimated by formula (8) in each of the closed domains \((g_1+C)\) and \((g_2+C+\Gamma)\).

Finally, let us consider the following boundary-value problem:

\[ \begin{aligned} Lu&=f &&\text{in the domain } g_1,\\ Lu&=0 &&\text{in the domain } g_2,\\ [u]\big|_C&=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]\bigg|_C=0,\qquad u\big|_\Gamma=0. \end{aligned} \tag{10} \]

We shall represent its solution \(u(x)\) in the form \(u(x)=v(x)+w(x)\), where \(v(x)\) is the solution of the problem

\[ \begin{aligned} Lv&=f &&\text{in the domain } g_1,\\ v\big|_{C-0}&=0,\quad v\equiv 0 &&\text{in the domain } g_2; \end{aligned} \tag{11} \]

and \(w(x)\) is the solution of the problem

\[ \begin{aligned} Lw&=0 &&\text{in the domain } g_1,\\ Lw&=0 &&\text{in the domain } g_2,\\ [w]\big|_C&=0,\qquad \left[\frac{\partial w}{\partial \nu}\right]\bigg|_C =-\frac{\partial v}{\partial \nu}\bigg|_{C-0},\qquad w\big|_\Gamma=0. \end{aligned} \tag{12} \]

In complete analogy with the preceding, from (11) and (12) we conclude that the following holds:

Theorem 5. If the coefficients of the operator \(L\) belong to the class \(C^{(n-2,\mu)}\) in the closed domain \((g+\Gamma)\); \(f(x)\) belongs to \(C^{(n-2,\mu)}\) in \((g_1+C)\); the boundary surfaces \(C\) and \(\Gamma\) belong to \(A^{(n,\mu)}\), then there exists a (moreover unique) solution of problem (10) of the class \(C^{(n,\mu)}\), and for it the estimate

\[ u_{n,\mu}=O\left(f_{n-2,\mu}+f_0\right)\qquad (n\geqslant 2). \tag{13} \]

is valid.

On the other hand, this solution of problem (10) can be represented in the form of a generalized volume potential:

\[ u(x)=\int_{g_1} F(x;y)\,f(y)\,dy, \tag{14} \]

where \(F(x,y)\) is the Green’s function of the Dirichlet problem for the operator \(L\) in the domain \((g+\Gamma)\). Thus, if the hypotheses of Theorem 5 are fulfilled, the generalized volume potential belongs to the class \(C^{(n,\mu)}\) in each of the closed domains \((g_1+C)\) and \((g_2+C+\Gamma)\), and estimate (13) is valid for it in these domains.

Remark. It is obvious that if in formulas (4), (9), (14), by \(F(x,y)\) we understand any principal fundamental solution of the equation \(Lu=0\), then one may assert that the potentials defined by these formulas will belong to the class \(C^{(n,\mu)}\), and for them there will be

* For the definition of the operator \(Q_y\), see (²), p. 21.

the estimates (3), (8), and (13) are valid in each of the closed domains \((g_1+C)\) and \((T+C+\Gamma_T)\), where \(T\) is an arbitrary domain such that \(g_1\subset T\subset g\); \(\Gamma_T\) is the boundary of \(T\), if the conditions of Theorems 1, 3, and 5 are satisfied with the requirement \(\Gamma\subset A^{(n,\mu)}\) replaced by the requirement \(\Gamma_T\subset A^{(n,\mu)}\).

\(3^\circ\). Proof of Theorem 1. We shall outline the proof of Theorem 1, the main theorem in this paper. We use Schauder’s method, developed for obtaining a priori estimates. Fix an arbitrary point \(x_0\) of the domain \(g_1\) and rewrite problem \(1^\circ\) in the form:

\[ \sum_{i,j=1}^{N} a_{ij}(x_0)\frac{\partial^2 u}{\partial x_i\partial x_j} = \sum_{i,j=1}^{N}\bigl[a_{ij}(x_0)-a_{ij}(x)\bigr]\frac{\partial^2 u}{\partial x_i\partial x_j} -\sum_{i=1}^{N} b_i\frac{\partial u}{\partial x_i}-cu \equiv f_1 \quad \text{in } g_1, \]

\[ \sum_{i,j=1}^{N} a_{ij}(x_0)\frac{\partial^2 u}{\partial x_i\partial x_j} = \sum_{i,j=1}^{N}\bigl[a_{ij}(x_0)-a_{ij}(x)\bigr]\frac{\partial^2 u}{\partial x_i\partial x_j} -\sum_{i=1}^{N} b_i\frac{\partial u}{\partial x_i}-cu \equiv f_2 \quad \text{in } g_2, \tag{15} \]

\[ [u]\big|_{C}=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]\bigg|_{C}=\theta(x),\qquad u\big|_{\Gamma}=0. \]

By means of a change of coordinates, in the left-hand side of equations (16) we pass from operators with constant coefficients to the Laplace operator; we obtain

\[ \Delta u=\bar f_1 \quad \text{in the domain } \bar g_1, \]

\[ \Delta u=\bar f_2 \quad \text{in the domain } \bar g_2, \tag{16} \]

\[ [u]\big|_{\bar C}=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]\bigg|_{\bar C}=\bar\theta,\qquad u\big|_{\bar\Gamma}=0. \]

We represent the solution of problem (16) in the form \(u(x)=v(x)+w(x)\), where \(w(x)\) is the solution of a problem of the type (16) in which zero stands in place of \(\bar\theta(x)\), and

\[ v(x)=\int_{\bar C} K(x,y)\,\mu(y)\,dy; \]

\(K(x,y)\) is the Green’s function of the Dirichlet problem for the Laplace operator in the domain \((\bar g+\bar\Gamma)\), and \(\mu(y)\) is chosen from the condition

\[ \left[\frac{\partial v}{\partial \nu}\right]\bigg|_{\bar C}=\bar\theta. \]

The function \(v(x)\) is easily estimated (see (1)); and for the estimate of \(w(x)\) one first considers the problem:

\[ \Delta\tau=F_1 \quad \text{in the domain } \bar g_1\cap \Omega(\bar x_0,2\rho), \]

\[ \Delta\tau=F_2 \quad \text{in the domain } \bar g_2, \tag{17} \]

\[ [\tau]\big|_{\bar C}=0,\qquad \left[\frac{\partial\tau}{\partial\nu}\right]\bigg|_{\bar C}=0,\qquad \tau\big|_{\bar\Gamma}=0; \]

\(\Omega(\bar x_0,2\rho)\) is the ball of radius \(2\rho\) with center at the point \(\bar x_0\). Having obtained an estimate of \(\tau_{n,\mu}\) in \(\Omega(\bar x_0,\rho)\), we estimate \(w(x)\) by a method close to that set forth in \((^2)\).

Moscow State University
named after M. V. Lomonosov

Received
15 VI 1961

REFERENCES

1. N. M. Günter, The Theory of Potential and Its Application to the Fundamental Problems of Mathematical Physics, 1953.
2. C. Miranda, Partial Differential Equations of Elliptic Type, IL, 1957.
3. V. A. Il’in, I. A. Shishmarev, Siberian Mathematical Journal, 2, No. 1 (1961).
4. I. A. Shishmarev, DAN, 137, No. 1 (1961).