MATHEMATICS

V. V. FUFAEV

ON THE DIRICHLET PROBLEM FOR DOMAINS WITH ANGLES

(Presented by Academician I. M. Vinogradov, 10 XI 1959)

This note is devoted to the study of the differential properties of a function \(U(x,y)\), harmonic in a plane domain \(\Omega\), in dependence on the differential properties of the boundary function \(f(s)\) in the case when the boundary \(\Gamma\) of the domain \(\Omega\) is piecewise smooth. The differential properties of \(U\) and \(f\) are determined in terms of membership of these functions, respectively, in the classes \(W_p^r H^\alpha(\Omega)\) and \(H_p(\Gamma)\) \((1 \leq p \leq \infty)\), whose definition is given in the papers of S. M. Nikol’skii \((^1)\).

If the contour \(\Gamma\) is sufficiently smooth, then the following assertion holds, proved in the papers of N. P. Mozzhezerova \((^2)\), S. M. Nikol’skii \((^3)\), and Ya. S. Bugrov \((^4)\); it represents a strengthening (for \(p=\infty\)) and an extension to the case of finite \(p\) of the corresponding results of Lichtenstein \((^5)\), N. M. Günter \((^6)\), Kh. L. Smolinskii, and others.

If the function \(f \in H_p^r(\Gamma)\), then for the harmonic function \(U\), for which \(U|_\Gamma=f\), one has \(U \in W_p^\rho H^\alpha(\Omega)\), \(\rho+\alpha=r+\frac1p\). In the case of piecewise smooth contours \(\Gamma\), this property is far from always preserved.

In the paper of S. M. Nikol’skii \((^3)\), necessary and sufficient conditions are given which the boundary function \(f\), prescribed on the boundary \(\Gamma\) of the square \(\Omega\), must satisfy in order that the corresponding function harmonic in \(\Omega\) belong to \(W_p^r H^\alpha(\Omega)\). Uolsh and Yang considered a similar problem for the square for \(p=\infty\) and \(r=0\) \((^7)\).

These investigations are extended in the present note to the case of arbitrary domains with nondegenerate angles, and arbitrary \(p\) and \(r\). It turns out that the angles \(\pi/j\), \(j=1,2,\ldots\) (and thus also the straight angle), are in a certain sense exceptional.

Let the domain \(\Omega\) have a boundary \(\Gamma\) which is a piece of a smooth curve of class \(H_\infty^{r+2+1/p}\), whose ends meet at an angle \(\omega\), and suppose that some finite parts of \(\Gamma\) adjoining the vertex of the angle \(P_0\) are straight-line segments. We shall assume that the value of the arc \(s=0\) corresponds to the vertex of the angle and that its length is equal to \(l\). The boundary function \(f\) will be given as a function \(f(s)\) of the arc \(s\) on the interval \(0 \leq s \leq l\).

Theorem 1. Let \(r, r-1/p\) be nonintegral positive numbers and let the harmonic function \(U \in W_p^{\bar r}H^\alpha(\Omega)\), \(\bar r+\alpha=r\).

Then the function \(f(s)=U|_\Gamma\) satisfies the following conditions:

1) \(f(s)\in H_p^{r-1/p}([0,l])\);

2) in the case \(\omega=\pi/j\), \(j=1,2,\ldots\),

\[ f^{(kj)}(+0)=(-1)^k f^{(kj)}(-0) \]

for all \(k=0,1,2,\ldots\), for which \(kj<\rho\), where \(\rho=r-1/p=\bar\rho+\beta\), \(\bar\rho\) is an integer, \(0<\beta<1\).

If \(\bar\rho=mj\) (\(m\) an integer), then, in addition, the inequality
\[ \left(\int_0^h \left| f^{(mj)}(u)-(-1)^m f^{(mj)}(-u)\right|^p\,du\right)^{1/p}\leq c h^\beta \]
holds.

Remark. Property 1) follows from the general embedding theorem of S. M. Nikol’skii \((1^3)\); property 2) reflects the harmonicity of the function \(U\) in a neighborhood of the point \(P_0\). It turns out that Theorem 1 is completely invertible only in the case when \(\omega=\pi/j,\ j=1,2,\ldots\), as is seen from the following theorem.

Theorem 2. Let \(\omega=\pi/j,\ j=1,2,\ldots;\ r\) and \(r-1/p\) be noninteger positive numbers; \(r-2/p>0\). Let \(f(s)\) satisfy the conditions:
\[ \text{1) } f(s)\in H_p^{r-1/p}(M,[0,l]); \]
\[ \text{2) } f^{(kj)}(+0)=(-1)^k f^{(kj)}(-0) \]
for all \(k=0,1,\ldots\) for which \(kj<\bar\rho\), where \(\rho=r-1/p=\bar\rho+\beta\), \(\bar\rho\) is an integer, \(0<\beta<1\), and in the case \(\bar\rho=mj\) (\(m\) an integer) the inequality
\[ \left(\int_0^h \left| f^{mj}(u)-(-1)^m f^{mj}(-u)\right|^p\,du\right)^{1/p}\leq c |h|^\beta \]
also holds.

Then the function \(U(x,y)\), harmonic in \(\Omega\), with boundary function \(f(s)\), belongs to the class \(W_p^r H^\alpha(M_1,\Omega)\), where \(r+\alpha=r\), and
\[ M_1\leq c_1\left(M+c+\|f(s)\|_{L(p)}\right). \]

For \(\omega=\pi/2\), Theorems 1 and 2 were proved by S. M. Nikol’skii \((^3)\).

Theorem 3. If \(\omega\) is not equal to one of the numbers \(\pi/j,\ j=1,2,\ldots\), then the converse of Theorem 2 holds only for \(r-2/p<\pi/\omega\). In the case \(r-2/p>\pi/\omega\), one can construct a boundary function \(f(s)\), having arbitrarily many derivatives for \(0\leq s\leq l\) and equal to zero on the segments \(0\leq s\leq\delta,\ l-\delta\leq s\leq l\), with the property that the harmonic function \(U\), for which \(U|_\Gamma=f\), does not belong to the class \(W_p^r H^\alpha(\Omega)\), \(r+\alpha=r\).

The proofs of the theorems were carried out by means of methods of the theory of double-layer potentials, as well as methods of the theory of conformal mappings. In the course of the proofs we needed the following lemmas.

Lemma 1. Let the contour \(\Gamma\), bounding a two-dimensional domain \(\Omega\), be smooth everywhere, and let
\[ \Gamma\in H_\infty^{r+2+1/p}(M_1),\qquad r=r+\alpha. \]
If \(\mu(s)\) is the density of a double-layer potential and \(\mu(s)\in H_p^r(M,\Gamma)\), then the corresponding value of the double-layer potential on the contour \(\Gamma\), \(\bar\mu(s)\), belongs to the class \(H_p^{r+1}(\bar M,\Gamma)\), where
\[ \bar M<c\{M+\|\mu\|_{L_p}(\Gamma)\}. \]

Lemma 2. Let the contour \(\Gamma\), bounding a two-dimensional domain \(\Omega\), be smooth everywhere, and let
\[ \Gamma\in H_\infty^{r+2+1/p}(M_1). \]

If \(\mu(s)\) is the density of the double-layer potential and \(\mu(s)\in H_p^r(M,\Gamma)\), then \(U(x,y)\), the double-layer potential with density \(\mu(s)\), belongs to the class \(W_p^{\bar \rho}H^\alpha(\bar M,\Omega)\), where \(r+1/p=\rho=\bar\rho+\alpha\),

\[ \bar M \leq c\{M+\|\mu\|_{L_p}\}. \]

Lemmas 1 and 2 in the three-dimensional case were proved by N. P. Mozherova \((^2)\); the proof for two-dimensional domains is analogous.

Lemma 3. Let \(G\) be a two-dimensional domain bounded by a piecewise smooth contour \(\Gamma\) with angles \(\omega_l\), \(l=0,1,\ldots,N\), at the points \(P_l\), where \(0<\omega_l<2\pi\); let each smooth piece

\[ \Gamma\in H_\infty^{r+2+1/p}\bigl([s_k,s_{k+1}]\bigr), \]

where \(s_0,s_1,\ldots,s_N\) are the values of the arc \(s\) corresponding to the corner points of \(\Gamma\). If

\[ f(s)\in H_p^r\bigl(M,[s_k,s_{k+1}]\bigr),\qquad k=0,1,\ldots,N-1, \]

then for any domain \(\Omega'\in\Omega\) such that

\[ \rho(\Omega',P_k)>\varepsilon>0,\qquad k=0,1,\ldots,N, \]

\[ U\in W_p^{\bar\rho}H^\alpha(\bar M,\Omega'),\qquad \rho=\bar\rho+\alpha=r+\frac1p, \]

where

\[ \bar M \leq c(\varepsilon)\{M+\|f\|_{L_p}\}. \]

Remark. Theorems 1, 2, and 3 are formulated in the corresponding way (and, with the aid of Lemma 3, proved) in the case of domains with an infinite number of corner points.

Received
9 XI 1959

REFERENCES

1. S. M. Nikol’skii, Matem. sborn., 33 (75), No. 2, 261 (1953).
2. N. P. Mozherova, DAN, 118, No. 4, 636 (1958).
3. S. M. Nikol’skii, Matem. sborn., 43 (85), 1 (1957).
4. Ya. S. Bugrov, DAN, 115, No. 4, 639 (1957).
5. Lichtenstein, Enzykl. d. Math. Wissensch., 2, 3, Leipzig, 1899—1916.
6. N. M. Günter, Theory of the Potential and Its Application to the Basic Problems of Mathematical Physics, Moscow, 1953.
7. J. L. Walsh, D. Young, J. Math. and Phys., 36, 2 (1957).