UDC 517.544

MATHEMATICS

E. M. SAAL

ON THE STABILITY OF THE DIRICHLET AND NEUMANN PROBLEMS

(Presented by Academician M. V. Keldysh, June 4, 1970)

Let \(C(F)\) be the capacity \((^{1})\) of a bounded closed set \(F\) in three-dimensional Euclidean space. Following the concept of analytic capacity \(((^{2}), p. 103)\), we shall call the capacity \(C(E)\) of an arbitrary set \(E\) in this space the number
\[ C(E)=\sup_{F\subset E} C(F). \]
We shall say that an open set \(E\) has stable capacity if \(C(\overline E)=C(E)\).

Theorem 1. A domain \(E\) may have stable capacity despite the presence of irregular \((^{1})\) points of the complement to its closure.

The proof consists in constructing an example. In doing so, the construction of Lebesgue’s well-known example of the non-solvability of the Dirichlet problem \(((^{3}), p. 203)\) is used.

Let \(E\) be a bounded simply connected domain and \(x\in E\). Denote by \(\widehat E_x\) the complement of the inversion of the domain \(\overline E\) with respect to the sphere with center at \(x\) and fixed sufficiently large radius \(R\).

Theorem 2 (main). Let the boundary of the domain \(E\) have no irregular points and let the complement of \(\overline E\) be connected. In order that the Dirichlet problem be stable in the closed domain \(\overline E\) \((^{1})\), it is necessary and sufficient that, for every \(x\in E\), the capacity of the domain \(\widehat E_x\) be stable.

Remark. As was shown in \((^{1})\), regularity of the boundary alone is insufficient for the stability of the Dirichlet problem. From this example there now follows the existence in three-dimensional space of a Jordan rectifiable domain with regular boundary, but with unstable capacity.

Other criteria for the stability of the Dirichlet problem can be found in \((^{1,4})\). The proof of Theorem 2 is based on the following lemmas and theorems.

Lemma 1. For every \(x\in E\), the potential \((^{1})\) \(W_x(y)\) of the closure of the domain \(\widehat E_x\) and the regular part \(g_E(x,y)\) of the Green function of the domain \(E\) are related by
\[ g_E(x,y)=\frac{1}{|x-y|}\, W_x\!\left(x+R^2\frac{y-x}{|y-x|^2}\right),\qquad y\in E, \]
\[ W_x(y)=\frac{R^2}{|x-y|}\, g_E\!\left(x,x+R^2\frac{y-x}{|y-x|^2}\right),\qquad y\in \widehat{\overline E}_x . \tag{1} \]

Lemma 2. Let \(\{E_r\}\) be an increasing sequence of domains converging to \(E\) in the sense that \(\overline E_r\subset E\), but \(\bigcup E_r=E\). In order that the potentials of the sets \(\overline E_r\) converge to the potential of the set \(\overline E\) uniformly on every closed set outside \(\overline E\), it is necessary and sufficient that
\[ C(\overline E)=C(E). \]

Denote by \(HL_2'(E)\) the Hilbert space of classes of functions \(u(x), v(x)\), \(x\in E\), summable over \(\overline E\), harmonic in \(E\), with scalar product
\[ \langle u,v\rangle_E=\frac{1}{4\pi}\int_{\overline E}(\nabla u\cdot \nabla v)\,dE, \]
where \(\nabla u=\operatorname{grad}u\), and the integration is carried out with respect to Lebesgue measure. All functions differing only by a constant term belong to the same class.

Lemma 3. Let the complement of \(\bar E\) be connected and let \(\{E_\rho\}\) be a decreasing sequence of domains converging to \(E\) in the sense that \(E_\rho \supset \bar E\), but \(\bigcap E_\rho=\bar E\).

In order that the functions \(g_{E_\rho}(x,y)\), as functions of \(y\in E\), converge weakly in the space \(HL_2'(E)\) to the function \(g_E(x,y)\), it is necessary and sufficient that the capacity of the domain \(\widetilde E_x\) be stable.

Proof. The family of functions \(g_{E_\rho}(x,y)\) is weakly compact in \(HL_2'(E)\) by the well-known Dirichlet principle. Its weak convergence to the function \(g_E(x,y)\) now follows from Lemmas 1 and 2. This proves sufficiency. The necessity of the condition is easily established with the aid of the equality
\[ C(\widetilde E_x)=R^2 g_E(x,x), \tag{2} \]
which follows from (1).

Let \(f(x)\) be an everywhere infinitely smooth function. We define the solution \(u_f\) of the Dirichlet problem in the space \(HL_2'(E)\) with boundary condition \(f(x)\) by the equality
\[ \langle u,f\rangle_E=\langle u,u_f\rangle_E, \tag{3} \]
where \(u\in HL_2'(E)\) is an arbitrary element.

The basis for such a definition is given by the following

Lemma 4. If \(E\) is a sufficiently smooth domain, then \(u_f\), up to an additive constant, coincides with the classical solution of the Dirichlet problem.

Proof. The operation
\[ A_E f(x)\equiv f(x)+\langle f(y),G_E(x,y)\rangle_E =f(x)-\frac{1}{4\pi}\int_E \Delta f(y)\cdot G_E(x,y)\,dE, \tag{4} \]
where \(G_E(x,y)\) is the Green function of the domain \(E\), and \(\Delta\) is the Laplace operator, which gives the classical solution in the case of a sufficiently smooth domain, projects \(f(x)\) into the space \(HL_2'(E)\). But \(u_f\), defined by equality (3), is also the projection of \(f(x)\) onto \(HL_2'(E)\). Consequently, \(u_f\) and \(A_E f(x)\) are equal as elements of the space \(HL_2'(E)\).

Remark. It can be proved that the equality \(u_f=A_E f\) is valid if all the domains \(\widetilde E_x\) have stable capacity.

Let \(E\) and \(\{E_\rho\}\) satisfy the conditions of Lemma 3, and let \(u_{f,\rho}\in HL_2'(E)\) be the solution of the Dirichlet problem in the domain \(E_\rho\).

If \(u_f\) is the weak limit of \(u_{f,\rho}\) in the space \(HL_2'(E)\) as \(E_\rho\to E\) for every infinitely smooth function \(f(x)\), then we shall call the Dirichlet problem weakly stable in the domain \(E\) in the sense of \(HL_2'(E)\).

Theorem 3. Let the complement of \(E\) be connected. For the weak stability of the Dirichlet problem in the domain \(E\) in the sense of \(HL_2'(E)\), it is necessary and sufficient that, for every \(x\in E\), the domain \(\widetilde E_x\) have stable capacity.

Proof. Necessity is a consequence of Lemma 3. To prove sufficiency, define the operation \(A_{E_\rho}\) by equality (4), with \(E\) replaced by \(E_\rho\), and consider the difference
\[ A_{E_\rho}f(x)-A_E f(x) =\langle f(y),G_{E_\rho}(x,y)\rangle_{E_\rho\setminus E} +\langle f(y),[g_E(x,y)-g_{E_\rho}(x,y)]\rangle_E, \]
\[ x\in E. \]
Hence, by Lemma 3, we have
\[ \lim_{E_\rho\to E} A_{E_\rho}f(x)=A_E f(x),\qquad x\in E. \]
But the sequence \(A_{E_\rho}f(x)\) is weakly compact in \(HL_2'(E)\) and, therefore, converges weakly to \(A_E f(x)\) when \(E_\rho\to E\).

Theorem 4. Let the complement of \(\bar E\) be connected. For the stability of the Dirichlet problem inside the domain \(E\) (see the definition in \((*)\)) it is necessary and sufficient that, for every \(x\in E\), the domain \(\widetilde E_x\) have stable capacity.

Proof. Necessity is established by means of equality (2), sufficiency by means of Theorem 3.

Theorem 2 now follows from Theorem 4 and one of the results of [1].

The space \(HL_2'(E)\) has a reproducing function \(P_E(x,y,x_0)\), defined by the relation

\[ u(x)=u(x_0)+\langle u(y), P_E(x,y,x_0)\rangle_E, \tag{5} \]

where \(u\) is an arbitrary element of \(HL_2'(E)\), \(x,x_0\in E\), \(P_E(x_0,y,x_0)=0\). The function \(P_E(x,y,x_0)\) has the symmetry property \(P_E(x,y,x_0)=P_E(y,x,x_0)\).

Theorem 5. Let \(E\) and \(\{E_\rho\}\) satisfy the conditions of Lemma 3. In order that, for any \(x\in E\), the functions \(P_{E_\rho}(x,y,x_0)\), as functions of \(y\in E\), converge in the space \(HL_2'(E)\) to the function \(P_E(x,y,x_0)\), it is necessary and sufficient that the capacities of the domains \(\widehat E_x\) be stable.

Proof. It is established that the convergence of the reproducing functions \(P_{E_\rho}(x,y,x_0)\) to \(P_E(x,y,x_0)\) is equivalent to the completeness of harmonic polynomials in the space \(HL_2'(E)\). Next it is proved that the latter, in turn, is equivalent to the weak convergence of the functions \(g_{E_\rho}(x,y)\) to \(g_E(x,y)\), when \(E_\rho\to E\). In doing so one uses the relation

\[ \left\langle u(y), \frac{1}{|x-y|}\right\rangle_E = \langle u(y), g_E(x,y)\rangle_E,\qquad u\in HL_2'(E). \]

The proof is completed by reference to Lemma 3.

Let \(\varphi(u)\) be a linear functional on the space \(HL_2'(E)\). We define the solution \(u^\varphi\) of the Neumann problem in the domain \(E\) with boundary condition \(\varphi\) by means of the equality

\[ \varphi(u)=\langle u,u^\varphi\rangle_E, \]

where \(u\) is an arbitrary element of \(HL_2'(E)\).

This formulation of the Neumann problem is a natural generalization of the classical one (cf. the definition of the Neumann problem in [5]).

Let \(E\) and \(\{E_\rho\}\) satisfy the conditions of Lemma 3, and let \(u_\rho^\varphi\in HL_2'(E_\rho)\) be the solution of the Neumann problem in the domain \(E_\rho\) with boundary condition \(\varphi\). If, for any linear functional \(\varphi\), the elements \(u_\rho^\varphi\) converge weakly in the space \(HL_2'(E)\) to the element \(u^\varphi\), then the Neumann problem will be called stable in the domain \(E\).

Theorem 6. Let the complement of \(\overline E\) be connected. For stability of the Neumann problem in the domain \(E\) it is necessary and sufficient that the domains \(\widehat E_x\), \(x\in E\), have stable capacity.

Proof. In view of (5) we have

\[ \varphi(u)=\varphi\langle u(y),P_E(x,y,x_0)\rangle_E = \langle u(y),\varphi(P_E(x,y,x_0))\rangle_E. \]

Consequently,

\[ u^\varphi(y)=\varphi(P_E(x,y,x_0)),\qquad u_\rho^\varphi(y)=\varphi(P_{E_\rho}(x,y,x_0)). \tag{6} \]

Therefore, by Theorem 5 and the symmetry of the function \(P_E(x,y,x_0)\), the elements \(u_\rho^\varphi(y)\) converge pointwise to \(u^\varphi(y)\) in the domain \(E\), when \(E_\rho\to E\). To prove sufficiency it remains to observe that, in view of the boundedness of the functional \(\varphi\), the family \(u_\rho^\varphi(y)\) is weakly compact in the space \(HL_2'(E)\). Necessity of the condition is established with the aid of formulas (6) and Theorem 5.

In conclusion I express my gratitude to A. A. Gonchar for useful discussions.

Taganrog Radio Engineering Institute

Received
4 VI 1970

REFERENCES

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