MATHEMATICS

V. G. MAZ'YA

ON THE SOLVABILITY OF THE NEUMANN PROBLEM

(Presented by Academician V. I. Smirnov on 8 VI 1962)

In the present note conditions are formulated for the solvability of the Neumann problem for the Laplace operator in a generalized setting.

1°. The Neumann problem with a nonhomogeneous boundary condition. Let \(\Omega\) be a bounded domain in \(n\)-dimensional Euclidean space \(R_n\) with boundary \(\Gamma\). Find an element \(u(x)\in \overset{\circ}{L}{}_{2}^{(1)}(\Omega)\) *, for which the relation
\[ \int_{\Omega} \nabla v \nabla u\,dm_n(x)=\int_{\Gamma} v\mu\,(dx), \tag{1} \]
is satisfied, where \(v(x)\) is an arbitrary function from \(C^{(1)}(\overline{\Omega})\); \(m_n\) is \(n\)-dimensional Lebesgue measure and \(\mu(E)\) is a given completely additive function of Borel sets \(E\subset \Gamma\), satisfying the condition \(\mu(\Gamma)=0\).

It is natural to call the function \(\mu\) the flux of \(u(x)\) through \(\Gamma\).

Theorem 1. For the solvability of the generalized Neumann problem (1) it is necessary and sufficient that the flux \(\mu\) satisfy the condition
\[ \sup_{v\in C^{(1)}(\overline{\Omega})} \int_{0}^{\infty} \frac{\mu^2(E_t)} {\displaystyle \int_{S_t}|\nabla v|\,d\mu_{n-1}}\,dt<\infty, \tag{2} \]
where \(\mu_{n-1}\) is \((n-1)\)-dimensional Hausdorff measure; \(S_t=\{x;\ v(x)=t\}\cap\Omega,\ E_t=\{x,\ v(x)\ge t\}\cap\Gamma\).

From Theorem 1 one can derive the following sufficient condition for the solvability of problem (1). Following note \((^1)\), by \(F\) we denote an open subset of \(\Omega\), by \(E\) a subset of \(F\) closed in \(\Omega\), and by \(C_{2}^{(n)}(K)\) the \(n\)-dimensional 2-conductivity of the conductor \(K=F\setminus E\).

Theorem 2. Let \(\mu_{n-1}\Gamma<\infty\). If for any conductor \(K=F\setminus E\) in \(\Omega\), satisfying the condition \(\mu_{n-1}(\overline{F}\cap\Gamma)\le \frac12\mu_{n-1}\Gamma\), the inequality
\[ \mu(\overline{E}\cap\Gamma)\le \eta\,[c_{2}^{(n)}(K)] \]
holds, where the function \(\eta(t)\) is nondecreasing and
\[ \int_{0}^{\infty}\eta(t)\,\frac{dt}{t^2}<\infty, \]
then the solution of problem (1) exists and is unique.

Remark. If \(\Omega\) is the unit circle, then the necessary and sufficient condition for the solvability of problem (1) takes the simple form
\[ \iint_{\Gamma\Gamma}\ln\frac{1}{|x_1-x_2|}\,\mu(dx_1)\mu(dx_2)<\infty. \tag{3} \]
If
\[ \mu(E)=\int_E \varphi(s)\,ds, \]
where \(s\) is the arclength on \(\Gamma\) and \(\varphi\in L(\Gamma)\), then condition (3) can be rewritten in the following form:
\[ \iint_{\Gamma\Gamma} \left(\int_{x_1}^{x_2}\varphi(s)\,ds\right)^2 \frac{ds(x_1)\,ds(x_2)}{|x_1-x_2|^2}<\infty. \]

* Here and below we use the notation of note \((^1)\).

It is easy to show that the last inequality is necessary and sufficient for the solvability of the Neumann problem (1) in a plane domain with smooth boundary, if the modulus of continuity \(\omega(t)\) of the angle between the tangent to \(\Gamma\) and a fixed direction satisfies the condition

\[ \int_0 \frac{\omega(t)}{t}\,dt<\infty . \]

Definition 1. Let \(\mu_{n-1}\Gamma<\infty\). The boundary of the domain \(\Omega\) belongs to the class \(I_{2,\nu(t)}^{(n-1)}\) if there exists a function \(\nu(t)\) such that

\[ \sup \frac{\nu\bigl(\mu_{n-1}(\overline E\cap \Gamma)\bigr)}{c_2^{(n)}(K)}<\infty, \]

where the supremum is taken over conductors \(K=F\setminus E\) in \(\Omega\) such that \(\mu_{n-1}(\overline F\cap \Gamma)\le \tfrac12\mu_{n-1}\Gamma\). In the case \(\nu(t)=t^{2/q^*}\) \((q^*>0)\) we denote the class \(I_{2,\nu(t)}^{(n-1)}\) by \(I_{2,q^*}^{(n-1)}\).

Fig. 1

In the following theorem we shall consider problem (1) under the assumption

\[ \mu(E)=\int_E \varphi\,d\mu_{n-1}. \]

Theorem 3. 1) For the solvability of problem (1) for every \(\varphi\in L_q(\Gamma)\)
\[ \left(2\frac{n-1}{n}\le q\le 2,\ \text{if } n>2;\quad 1<q\le 2,\ \text{if } n=2\right) \]
it is necessary and sufficient that the boundary of the domain \(\Omega\) belong to the class
\[ I_{2,\frac{q}{q-1}}^{(n-1)}. \]
2) If
\[ \Gamma\in I_{2,\nu(t)}^{(n-1)}, \]
where \(\nu(t)\) is a nondecreasing absolutely continuous function satisfying the condition

\[ \int_0 \frac{dt}{[\nu'(t)]^{\frac{q}{q-2}}}<\infty \qquad (q>2), \]

and if \(\varphi\in L_q(\Gamma)\), then problem (1) has a unique solution.

Example 1. It can be shown that the boundary of the domain
\[ \Omega_1:\left\{\sum_{i=1}^{n-1}x_i^2<x_n^{2\beta},\ 0<x_n<1\right\} \]
\((\beta>1)\) belongs to the class \(I_{2,q^*}^{(n-1)}\), where
\[ q^*=\frac{\beta(n-2)+1}{\beta(n-1)-1}. \]
Consequently, the largest value of \(\beta\) for which problem (1) is solvable for all functions \(\varphi\in L_2(\Gamma)\) orthogonal to one on \(\Gamma\) is equal to two.

Example 2. For the domain \(\Omega_2\) shown in Fig. 1 (see (2)), problem (1) is solvable for all \(\varphi\in L_2(\Gamma)\) if \(\alpha\le 2\), and is not always solvable if \(\alpha>2\).

2°. The Neumann problem with homogeneous boundary condition

Let \(\Omega\) be an arbitrary open set in \(R_n\), \(\lambda=\mathrm{const}>0\), and \(f(x)\in L_q(\Omega)\). Find a function \(u(x)\in W_2^{(1)}(\Omega)\) satisfying the condition

\[ \int_\Omega \{(\nabla u\,\nabla v+\lambda uv)\,dm_n(x)\} = \int_\Omega fv\,dm_n(x), \tag{4} \]

where \(v\) is any function from \(W_2^{(1)}(\Omega)\).

Definition 2. The set \(\Omega\) belongs to the class \(I_{2,q^*}^{(n)}\) \((q^*>0)\) if there exists a positive constant \(M<m_n\Omega\) such that

\[ \sup \frac{m_n^{2/q^*}E}{c_2^{(n)}(K)}=\mathfrak{B}(M)<\infty, \]

where the supremum is taken over all conductors \(K=F\setminus E\) in \(\Omega\) satisfying the condition \(m_n F\leq M\).

Theorem 4. For the unique solvability of problem (4) for all
\(f(x)\in L_q(\Omega)\) \(\left(\dfrac{2n}{n+2}\leq q<2,\ \text{if } n>2;\ 1<q<2,\ \text{if } n=2\right)\), it is necessary and sufficient that
\(\Omega\in I_{2,\frac{q}{q-1}}^{(n)}\).

Theorem 4 follows easily from the results of note \((^3)\).

Example 3. It was noted in \((^2)\) that the domain \(\Omega_1\) belongs to the class
\(I_{2,\,2\frac{\beta(n-1)+1}{\beta(n-1)-1}}^{(n)}\). Hence the smallest exponent \(q\) for which problem (4) is solvable for any function \(f(x)\in L_q(\Omega_1)\) is equal to
\(2\dfrac{\beta(n-1)+1}{\beta(n-1)+3}\).

In the paper \((^4)\) Deny and Lions showed that problem (4) is always solvable in \(W_2^{(1)}(\Omega)\) if \(f(x)\in L_2(\Omega)\). In the same work it is proved that solvability in \(L_2^{(1)}(\Omega)\) of the problem

\[ \int_{\Omega}\nabla u\,\nabla v\,dm_n(x)=\int_{\Omega}fv\,dm_n(x), \tag{5} \]

where \(\Omega\) is a domain of finite measure, \(f(x)\) is any function in \(L_2(\Omega)\) orthogonal to unity, and \(v(x)\in W_2^{(1)}(\Omega)\), is equivalent to the validity of the Poincaré inequality

\[ \|u\|_{L_2(\Omega)}^2-\frac{1}{m_n\Omega}\left(\int_{\Omega}u\,dm_n(x)\right)^2 \leq c\|\nabla u\|_{L_2(\Omega)}^2. \]

By virtue of one of the theorems of note \((^3)\), the last inequality is true if and only if \(\Omega\in I_{2,2}^{(n)}\).

Example 4. Problem (5) for the domain
\[ \Omega_3:\left\{0<x_n<\infty,\ \sum_{i=1}^{n-1}x_i^2\leq e^{-\frac{2x_n}{n-1}}\right\} \]
is solvable for all \(f(x)\in L_2(\Omega_3)\) orthogonal to unity in \(\Omega_3\).

In conclusion we give a criterion for the discreteness of the spectrum of the Neumann problem for the Laplace operator.

Theorem 5. For the discreteness of the spectrum of the Neumann problem in \(L_2(\Omega)\), it is necessary and sufficient that \(\Omega\in I_{2,2}^{(n)}\) and \(\mathfrak{B}(M)\to 0\) as \(M\to 0\).

Example 5. For the domain \(\Omega_2\) (Fig. 1), the largest value of \(\alpha\) for which problem (5) is solvable for any function \(f(x)\in L_2(\Omega_2)\) orthogonal to unity is equal to three. For \(\alpha<3\) the spectrum of the Neumann problem is discrete.

All the results of the work carry over, without substantial changes, to equations with variable coefficients.

Received
5 VI 1962

References

\(^1\) V. G. Maz’ya, DAN, 140, No. 2 (1961).
\(^2\) R. Courant, D. Hilbert, Methods of Mathematical Physics, 2, 1951.
\(^3\) V. G. Maz’ya, DAN, 133, No. 3 (1960).
\(^4\) J. Deny, J. L. Lions, Ann. de l’Inst. Fourier, 5 (1953–1954).