MATHEMATICS

I. A. SHISHMAREV

AN A PRIORI ESTIMATE OF SOLUTIONS OF THE DIRICHLET PROBLEM FOR AN ELLIPTIC OPERATOR WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician S. L. Sobolev on 19 XI 1959)

Let there be given a certain open (N)-dimensional domain (g) with boundary manifold (\Gamma_2), and inside it an ((N-1))-dimensional geometrically closed surface (\Gamma_1), dividing the domain (g) into subdomains (g_1) and (g_2).

Consider in the closed domain ((g+\Gamma_2)) the following Dirichlet problem:

[
L_1u=0 \quad \text{in the domain } g_1;
]

[
L_2u=0 \quad \text{in the domain } g_2;
]

[
[u]\big|{\Gamma_1}=0,\qquad
\left[\frac{\partial u}{\partial \nu}\right]\bigg|=k,\qquad
u\big|_{\Gamma_2}=0.
\tag{1}
]

Here

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

is a linear differential operator of elliptic type defined in the domain (g_l), i.e. such that for (x=(x_1,\ldots,x_N)\in g_l) the conditions

[
a_{ij}^{(l)}=a_{ji}^{(l)},\qquad
\sum_{i,j=1}^{N} a_{ij}^{(l)}\xi_i\xi_j
\ge \alpha^{(l)}\sum_{i=1}^{N}\xi_i^2
\quad (\alpha^{(l)}=\operatorname{const}>0)
\tag{3}
]

are satisfied for any real (\xi_1,\xi_2,\ldots,\xi_N); it is also assumed that

[
c^{(l)}\le 0 \quad \text{in } g_l \quad (l=1,2);
\qquad
[u]\big|{\Gamma_1}
=
u\big|
-
u\big|_{x\to \Gamma_1+0};
]

[
\left[\frac{\partial u}{\partial \nu}\right]\bigg|{\Gamma_1}
\equiv
\frac{\partial u}{\partial \nu_1}\bigg|
+
\frac{\partial u}{\partial \nu_2}\bigg|_{x\to \Gamma_1+0},
]

where (\dfrac{\partial}{\partial \nu_l}) denotes differentiation in the direction of the conormal, equal to

[
\sum_{i,j=1}^{N} a_{ij}^{(l)}\cos(n^{(l)},x_j)\frac{\partial}{\partial x_i},
]

and the symbols (\Gamma_1-0) and (\Gamma_1+0) mean that limiting values are taken, respectively, from the inner and from the outer side of the surface (\Gamma_1) with respect to the domain (g_1); (k) is a given function defined on (\Gamma_1).

We assume that (L_1\not\equiv L_2), i.e. that the limiting values of the coefficients of the operators on the boundary (\Gamma_1) do not coincide identically; thus problem (1) is the Dirichlet problem for an elliptic operator with discontinuous coefficients.

Everywhere in what follows we assume that the coefficients of the operators (L_l),
(a_{ij}^{(l)}(x)), (b_i^{(l)}(x)), and (c^{(l)}(x)) ((l=1,2)), belong to the class* (C^{(0,\mu)}) ((\mu>0)), and that the surfaces (\Gamma_1) and (\Gamma_2) belong to the Lyapunov class.

We prove that for any solution of problem (1) belonging to the class (C^{(0)}) in the closed domain ((g+\Gamma_2)), to the class (C^{(1)}) in the closed domains ((g_1+\Gamma_1)) and ((g_2+\Gamma_1+\Gamma_2)), and to the class (C^{(2)}) in the open domains (g_1) and (g_2), the following a priori estimate is valid:
[
\max_{x\in(g+\Gamma_2)} |u(x)| \leq C \max_{x\in\Gamma_1} |k(x)|.
\tag{4}
]

Here the constant (C) depends only on the coefficients of the operators (L_l) and on the form of the domains (g_1) and (g_2); moreover, for all elliptic operators with uniformly bounded Hölder coefficients (a_{ij}^{(l)}), (b_i^{(l)}), and (c^{(l)}), and with uniformly bounded values of
[
\frac{1}{a^{(l)}} \qquad (l=1,2),
]
the constant (C) may be taken to be one and the same.

(1^\circ). Lemma 1 (Giraud). Let (g) be a bounded (N)-dimensional domain with boundary (\Gamma) belonging to the Lyapunov class; let (u(x)) be a solution of the equation (Lu=f) ((L) is any of the operators (L_l), see (2)), continuous and not identically constant in ((g+\Gamma)), regular in (g).

If (c\leq 0), (f\leq 0) ((f\geq 0)), (\min_{x\in\Gamma} u\leq 0) (\bigl(\max_{x\in\Gamma} u\geq 0\bigr)), then for each point (y) on (\Gamma) at which the function (u(x)) attains its minimal (maximal) value, and for each ray (l) issuing from the point (y) such that (\cos(l,n)<0) ((n) is the inner normal to (\Gamma)), there exists a positive constant (\gamma) such that, for (x\in l) and sufficiently small (r_{xy}) ((r_{xy}) is the distance between the points (x) and (y)),
[
u(x)-u(y)>\gamma r_{xy}\quad (<-\gamma r_{xy}).
]

For the proof of Lemma 1 see ((^1)), p. 14, or ((^2)); see also ((^3)).

(2^\circ). First of all we establish estimate (4) for the case when
(c^{(l)}\equiv 0) ((l=1,2)).

Lemma 2. Let the boundary (\Gamma_1) of the domain (g_1) belong to the Lyapunov class, and let the coefficients of the operator (L_1) belong in the domain ((g_1+\Gamma_1)) to the class (C^{(0,\mu)}).

Then there exists a function (\xi(x)) satisfying the following conditions:

1) (\xi(x)\in C^{(2)}) in the open domain (g_1) and satisfies in (g_1) the equation
[
L_1\xi(x)=0.
]

2) (\xi(x)\in C^{(1)}) in the closed domain ((g_1+\Gamma_1)) and on the boundary (\Gamma_1) is equal to (C=\mathrm{const}>0).

3) On the boundary (\Gamma_1),
[
\partial \xi(x)/\partial \nu<0,\qquad |\partial \xi(x)/\partial \nu|\geq 1.
]

Proof. Consider in the domain ((g_1+\Gamma_1)) the following Dirichlet problem:
[
L_1\bar{\xi}(x)=0 \quad \text{in the domain } g_1;
]
[
\bar{\xi}(x)\big|_{\Gamma_1}=\bar{C}>0 \quad (\bar{C}\text{ an arbitrary constant}).
]

Since the boundary surface (\Gamma_1) belongs to the Lyapunov class, by a result of Giraud ((^4)) this problem has a unique solution with first derivatives continuous in the closed domain ((g_1+\Gamma_1))

* A function (f(x)), defined in a bounded closed (N)-dimensional domain (T), belongs in this domain to the class (C^{(k,\mu)}) ((C^{(k)})), if its derivatives of order (k) satisfy a Hölder condition with exponent (\mu) in (T) (are continuous). A function (f(x)), defined in an open domain (C), belongs in this domain to the class (C^{(k,\mu)}) ((C^{(k)})), if it belongs to this class in every bounded closed domain contained in (C).

negative. Hence, for the solution on (\Gamma_1) the quantity (\partial \bar{\xi}/\partial \nu) is defined and continuous. Since (c^{(1)}(x)\ne 0) in the domain (g_1), (\bar{\xi}(x)) is not constant in ((g_1+\Gamma_1)). By Hopf’s theorem, (|\bar{\xi}(x)|<\bar C) for (x\in g_1).

Applying now Lemma 1, one may assert that (\partial \bar{\xi}/\partial \nu<0) everywhere on (\Gamma_1), and then, by virtue of the continuity of (\partial \bar{\xi}(x)/\partial \nu) and the closedness of the set of boundary points (the boundary (\Gamma_1)), there is a number (\alpha>0) such that (|\partial \bar{\xi}/\partial \nu|>\alpha) on (\Gamma_1).

Setting (\zeta(x)=\dfrac{1}{\alpha}\bar{\xi}(x)), we obtain the required function.

Theorem. Let (K_0) denote (\max\limits_{x\in\Gamma_2}|k(x)|); let (U_0) denote (\max\limits_{x\in(g+\Gamma_2)}|u(x)|). Let the boundary surfaces (\Gamma_1) and (\Gamma_2) belong to Lyapunov’s class, and let the coefficients of the operator (L_1) (respectively (L_2)) belong in the domain ((g_1+\Gamma_1)) (respectively ((g_2+\Gamma_1+\Gamma_2))) to the class (C^{(0,\mu)}).

Then for every solution of problem (1) belonging to the class (C^{(0)}) in the closed domain ((g+\Gamma_2)), to the class (C^{(1)}) in the closed domains ((g_1+\Gamma_1)) and ((g_2+\Gamma_1+\Gamma_2)), and to the class (C^{(2)}) in the open domains (g_1) and (g_2), the estimate

[
U_0\le CK_0 .
\tag{5}
]

is valid.

Proof. We note that it is sufficient to establish the estimate

[
\max_{x\in(g_1+\Gamma_1)} |u(x)|\le CK_0,
\tag{6}
]

since, by the maximum principle, formula (5) will follow from this. Denote by (\sigma_1) and (\sigma_2) two functions defined in the domain ((g_1+\Gamma_1)):

[
\sigma_1=K_0\zeta-u,\qquad \sigma_2=K_0\zeta+u,
]

where (\zeta) is the function from Lemma 2; (u(x)) is the solution of problem (1).

We shall show that (\sigma_1\ge 0) and (\sigma_2\ge 0) in the domain ((g_1+\Gamma_1)). We carry out the argument, for example, for the function (\sigma_1). Suppose the contrary, i.e. let (\sigma_1<0) at some point of the domain ((g_1+\Gamma_1)). Since (L_1\sigma_1=K_0L_1\zeta-L_1u=0) in the domain (g_1), the negative minimum value of the function (\sigma_1) is attained on the boundary (\Gamma_1) (by the maximum principle). Since the function (\zeta(x)) is, by construction, equal to a constant on (\Gamma_1), at the point where (\sigma_1) attains its absolute negative minimum the function (u(x)) has an absolute positive maximum. Taking into account the conditions ([u]{\Gamma_1}=0), (u|=0) and applying Lemma 1 to the function (u(x)) in the domains (g_1) and (g_2), we conclude that at that point of the boundary (\Gamma_1), where (u(x)) attains its maximum positive value,

[
\partial u/\partial \nu_1<0
]

and

[
\partial u/\partial \nu_2<0.
]

Since (\partial u/\partial \nu_1+\partial u/\partial \nu_2=k), at this point

[
|\partial u/\partial \nu_l|<|k|\quad (l=1,2).
]

By Lemma 1, at the point of the absolute negative minimum of the function (\sigma_1(x)) the relation

[
\frac{\partial\sigma_1}{\partial\nu_1}
=
K_0\frac{\partial\zeta}{\partial\nu_1}
-
\frac{\partial u}{\partial\nu_1}

0
]

must hold, but this is impossible, since, by Lemma 2, (\partial\zeta/\partial\nu_1<0), (|\partial\zeta/\partial\nu_1|\ge 1), and, as has just been proved, (|\partial u/\partial\nu_1|<|k|\le K_0).

The contradiction obtained proves that (\sigma_1\ge 0) everywhere in the domain ((g_1+\Gamma_1)). Similarly one establishes that (\sigma_2\ge 0). Therefore (|u|\le |\zeta|K_0) in the domain ((g_1+\Gamma_1)). By virtue of the maximum principle applied to the function (\zeta(x)), we finally obtain

[
\max_{x\in(g_1+\Gamma_1)} |u(x)|\le CK_0,
]

as was required to prove.

(3^\circ). We now consider the case when (C^{(l)}=0) in the domain (g_l) ((l=1,2)). In this case we define the function (\zeta(x)) in the domain (g_2) by means of the conditions

[
L_2\zeta(x)=0\quad \text{in } g_2,\qquad
\zeta(x)|{x\in\Gamma_1}=C,\qquad
\zeta(x)|=0.
]

For the function (\zeta(x)) thus defined in the domain ((g_2+\Gamma_1+\Gamma_2)), all the conditions of Lemma 2 are fulfilled. Carrying out

the proof of the theorem for the domain ((g_2+\Gamma_1+\Gamma_2)), we obtain the required estimate (5).

Remark 1. If the domain (g) is divided by means of (n-1) boundary surfaces of Lyapunov class into (n) nonintersecting domains, then for the solutions of the corresponding Dirichlet problem estimate (5) is valid, in which (K_0) is to be understood as
[
\max_i {K_0(g_1), \ldots, K_0(g_i), \ldots, K_0(g_n)}.
]

For the proof it suffices to carry out all the arguments for domains on whose boundary (u(x)) attains an absolute positive maximum and an absolute negative minimum.

Remark 2. Estimate (5) is uniform with respect to elliptic operators with uniformly bounded Hölder coefficients (a_{ij}^{(l)}, b_i^{(l)}, c^{(l)}) and uniformly bounded values of
[
\frac{1}{\alpha^{(l)}} \quad (l=1,2).
]

Indeed, the constant (C) entering estimate (5) depends only on the properties of the function (\zeta(x)), and for (\zeta(x)) the assertion formulated in the remark is obvious.

Remark 3. It follows from estimate (5) that problem (1) is stable with respect to the function (k).

Remark 4. We have restricted ourselves to considering problem (1), since the general Dirichlet problem for an elliptic operator with discontinuous coefficients:
[
\begin{aligned}
L_1u&=-f_1 &&\text{in the domain } g_1;\
L_2u&=-f_2 &&\text{in the domain } g_2;
\end{aligned}
\tag{7}
]
[
[u]\big|{\Gamma_1}=\varphi,\qquad
\left[\frac{du}{d\nu}\right]\bigg|=\psi,\qquad
u\big|_{\Gamma_2}=\chi,
]
where (\varphi,\psi,\chi,f_1) and (f_2) are some prescribed functions, can be reduced to problem (1).

Indeed, taking an arbitrary sufficiently smooth function (p), consider the problem
[
\begin{aligned}
L_1\tilde u&=-f_1 &&\text{in the domain } g_1;\
L_2\tilde u&=-f_2 &&\text{in the domain } g_2;\
\tilde u\big|{\Gamma_1,\,x\to\Gamma_1-0}&=p,\qquad
\tilde u\big|=p+\varphi,\qquad
\tilde u\big|_{\Gamma_2}=\chi.
\end{aligned}
]
The solution (\tilde u) of this problem certainly exists. Representing the solution of the general problem (7) in the form (u=\tilde u+\tilde{\tilde u}), we obtain for (\tilde{\tilde u}) problem (1).

I take this opportunity to express my gratitude to V. A. Il’in for posing the problem and for his attention to the work.

Received
18 XI 1959

References

1. K. Miranda, Equations with partial derivatives of elliptic type, IL, 1957.
2. G. Giraud, Bull. Sci. Math., 56, 343 (1932).
3. O. A. Oleinik, Matem. sborn., 30, 3 (1952).
4. G. Giraud, Bull. Soc. Math. de France, 61, 42 (1933).