THE DIRICHLET PROBLEM FOR A DOMAIN WITH A NON-LIPSCHITZ BOUNDARY

G. N. Yakovlev

Let \(G\) be an open bounded domain in the plane \((x_1,x_2)\) with boundary \(\Gamma\). Suppose that on \(\Gamma\) a certain function \(f\) is given. Denote by \(W(f)\) the class of functions \(u(x_1,x_2)\in W_2^1(G)\) for which \(u|_\Gamma=f\), i.e., the class of functions which, together with their first generalized derivatives in the sense of S. L. Sobolev [1], are square summable in the domain \(G\) and which assume on \(\Gamma\), in the mean, the prescribed values \(f\). Consider the Dirichlet problem:

\[ \sum_{i,j=1}^{2}\frac{\partial}{\partial x_i} \left(a_{ij}\frac{\partial u}{\partial x_j}\right)-bu=0 \quad \text{in } G, \tag{1} \]

\[ u=f \quad \text{on } \Gamma, \tag{2} \]

where the coefficients \(a_{ij}=a_{ji}\) and \(b\geq 0\) are bounded measurable functions in the domain \(G\). In addition, there exists a constant \(\gamma>0\) such that

\[ \sum_{i,j=1}^{2} a_{ij}\xi_i\xi_j>\gamma(\xi_1^2+\xi_2^2) \tag{3} \]

for all \((x_1,x_2)\in G\) and all \(\xi_1,\xi_2\). It is known [1], [2] that if the class \(W(f)\) is nonempty, then problem (1), (2) has a solution from this class. This solution gives the minimum of the Dirichlet integral

\[ D(u)=\iint\left(\sum_{i,j=1}^{2} a_{ij}\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j} +bu^2\right)\,dG \]

in the class of functions \(W(f)\). If the boundary \(\Gamma\) is sufficiently smooth, then the solution is unique in this class.

In the case when \(\Gamma\) is Lipschitz, it is known [3] that the class \(W(f)\) is nonempty if and only if \(f\in W_2^{1/2}(\Gamma)\). In this case problem (1), (2) has a unique solution [5]. In the present paper, necessary and sufficient conditions are obtained for the solvability of this problem when \(\Gamma\), generally speaking, is not Lipschitz. It is proved that also in this case problem (1), (2) has a unique solution in the class of functions \(W(f)\).

DEFINITIONS AND NOTATION

Necessary and sufficient conditions for the nonemptiness of the class \(W(f)\).

Any Cartesian rectangular coordinate system that is obtained from the given one by means of an orthogonal transformation will be called admissible. The coordinates of a point in any admissible coordinate system will be denoted by \(x, y\) with a superscript. The superscript changes in passing from one admissible coordinate system to another.

A point \(P\) of the boundary \(\Gamma\) is called a Lipschitz point if there exists a neighborhood \(O(P)\) of the point \(P\), a number \(\delta_p>0\), and an admissible coordinate system such that the following conditions are satisfied:

1) the equation of the piece of the curve \(\Gamma\cap O(P)\) in this coordinate system has the form
\[ y^P=\varphi_P(x^P),\quad x^P\in(-a_p,a_p),\quad a_p>0; \]

2) the function \(\varphi_P(x^P)\) satisfies the Lipschitz condition, i.e., there exists a constant \(C>0\) such that
\[ |\varphi_P(x_1^P)-\varphi_P(x_2^P)|\leq C|x_1^P-x_2^P| \]
for all \(x_1^P, x_2^P\in(-a_p,a_p)\);

3) all points of the set
\[ G_P^+=\{(x^P,y^P): -a_p<x^P<a_p,\ \varphi_P(x^P)<y^P<\varphi_P(x^P)+\delta_p\} \]
belong to the domain \(G\), while all points of the set
\[ G_P^-=\{(x^P,y^P): -a_p<x^P<a_p, \]
\[ \varphi_P(x^P)-\delta_p<y^P<\varphi_P(x^P)\} \]
do not belong to \(G\).

The boundary \(\Gamma\) is called Lipschitz if all its points are Lipschitz. One says that a function \(f\), given on \(\Gamma\), belongs to the space
\[ W_p^{1-\frac1p}(\Gamma),\quad 1<p<\infty, \]
if \(f\in L_p(\Gamma)\), i.e.,
\[ \|f\|_{L_p(\Gamma)}=\left(\int_\Gamma |f|^p\,ds\right)^{1/p}<\infty, \]
and
\[ \|f\|_{L_p^{\,1-\frac1p}(\Gamma)} = \left( \int_\Gamma\int_\Gamma \frac{|f(P)-f(Q)|^p}{|PQ|^p}\,ds_P\,ds_Q \right)^{1/p} <\infty, \]
where \(|PQ|\) is the distance between the points \(P\) and \(Q\) along the curve \(\Gamma\), and \(s_P\) and \(s_Q\) are the values of the arc length of the curve \(\Gamma\) corresponding to the points \(P\) and \(Q\). If one introduces the norm
\[ \|f\|_{W_p^{\,1-\frac1p}(\Gamma)} = \|f\|_{L_p(\Gamma)} + \|f\|_{L_p^{\,1-\frac1p}(\Gamma)}, \]
then \(W_p^{1-\frac1p}(\Gamma)\) is a Banach space. By \(W_p^1(G)\), \(1<p<\infty\), we denote the space of functions having first generalized derivatives, for which the following norm is finite:

\[ \|u\|_{W_p^1(G)}=\|u\|_{L_p(G)}+\|\operatorname{grad}u\|_{L_p(G)}. \]

Let the boundary \(\Gamma\) be Lipschitz. A function \(f\) defined on \(\Gamma\) will be called the trace of the function \(u\) on \(\Gamma\), and we shall write \(u|_\Gamma=f\), if on each \(\Gamma\cap O(P)\), in the corresponding coordinate system (see the definition of a Lipschitz point), the equality

\[ \lim_{y^P\to 0}\int_{-a_P}^{a_P} \left|u\bigl(x^P,\varphi_P(x^P)+y^P\bigr)-f(x^P)\right|^p\,dx^P=0 \tag{4} \]

holds.

Theorem. (E. Gagliardo [13]). Let the boundary \(\Gamma\) be Lipschitz. Then, if \(u\in W_p^1(G)\), then \(f=u|_\Gamma\) exists, \(f\in W_p^{1-\frac1p}(\Gamma)\), and

\[ \|f\|_{W_p^{1-\frac1p}(\Gamma)}\leq C\|u\|_{W_p^1(G)}. \]

Conversely, if \(f\in W_p^{1-\frac1p}(\Gamma)\), then there exists a function \(u\in W_p^1(G)\) such that \(u|_\Gamma=f\) and

\[ \|u\|_{W_p^1(G)}\leq C\|f\|_{W_p^{1-\frac1p}(\Gamma)}. \]

(Here and below \(C\) denotes a positive constant which does not depend on the factor standing next to it. In different formulas it may have different values.)

In particular, for \(p=2\) it follows from this that problem (1), (2) has a solution in the class of functions \(W(f)\) if and only if \(f\in W_2^{\frac12}(\Gamma)\).

In the author’s paper [6] it was shown that this theorem is also valid for some \(\Gamma\) having a finite number of non-Lipschitz points of the type of an angular point whose angle is equal to \(2\pi\). More precisely, the case was considered when the boundary \(\Gamma\) has a finite number of non-Lipschitz points \(Q\) possessing the following property: there exists a neighborhood \(O(Q)\) of the point \(Q\) such that the piece of the boundary \(\Gamma\cap O(Q)\) is divided by the point \(Q\) into two parts \(\Gamma_1\) and \(\Gamma_2\), and there exists a curve \(\Gamma_3\) such that the point \(Q\) on the curves \(\Gamma_1\cup\Gamma_3\) and \(\Gamma_2\cup\Gamma_3\) is Lipschitz. The case of a cut is not excluded.

It should be noted that if \(\Gamma\) contains non-Lipschitz points, then the previously introduced definition of the trace of a function, generally speaking, does not always make sense. In this case, by definition, we write \(u|_\Gamma=f\) if the function \(f\) is the trace of the function \(u\) in the sense of equality (4) in a neighborhood of every Lipschitz point of the boundary \(\Gamma\).

In the present paper we study the case when the boundary \(\Gamma\) has a finite number of non-Lipschitz points \(Q_1,\ldots,Q_m\) of the type of an angular point whose angle is equal to \(0\). More precisely, the point \(Q_i\), \(i=1,\ldots,m\), has the following properties:

1) there exists a neighborhood \(O(Q_i)\) of the point \(Q_i\) such that, in some admissible coordinate system with origin at the point \(Q_i\), the piece of the boundary \(\Gamma\cap O(Q_i)\) is divided by the point \(Q_i\) into two parts \(\Gamma_{ik}\), \(k=1,2\), whose equations have the form

\[ y^i=\varphi_{ik}(x^i),\quad 0\leq x^i\leq a_i,\quad k=1,2; \]

2) the functions \(\varphi_{ik}(x^i)\) are continuously differentiable on the interval \([0,a_i]\), \(\varphi_{ik}(0)=\varphi'_{ik}(0)=0\), and for all \(x^i\in(0,a_i]\)

\[ \varphi_{i1}(x)<\varphi_{i2}(x),\qquad \varphi'_{i1}(x)<\varphi'_{i2}(x); \]

3) all points of the set

\[ g_i=\{(x^i,y^i):\ 0<x^i\leq a_i,\ \varphi_{i1}(x^i)<y^i<\varphi_{i2}(x^i)\} \]

belong to the domain \(G\), and there exists a \(\delta_i>0\) such that all points of the sets

\[ \{(x^i,y^i):\ 0<x^i\leq a_i,\ \varphi_{i1}(x^i)-\delta_i<y^i<\varphi_{i1}(x^i)\}, \]

\[ \{(x^i,y^i):\ 0<x^i\leq a_i,\ \varphi_{i2}(x^i)<y^i<\varphi_{i2}(x^i)+\delta_i\}, \]

\[ \{(x^i,y^i):\ -\delta_i<x^i<0,\ -\delta_i<y^i<\delta_i\} \]

do not belong to the domain \(G\).

Theorem 1. Let \(G\) be an open bounded domain with boundary \(\Gamma\), which has a finite number of non-Lipschitz points \(Q_i,\ i=1,2,\ldots,m\). Then if \(u\in W_p^1(G)\), \(1<p<\infty\), then \(f=u|_\Gamma\) exists and, on any part \(\Gamma_0\) of the boundary \(\Gamma\) all points of which are at a positive distance from the points \(Q_i\), it belongs to the space \(W_p^{1-\frac1p}(\Gamma_0)\), and

\[ A_0=\|f\|_{W_p^{1-\frac1p}(\Gamma_0)} \leq C\|u\|_{W_p^1(G)} . \tag{5} \]

Moreover, if by \(f_{ik}\) we denote the value of the function \(f\) on \(\Gamma_{ik}\) and regard \(f_{ik}\) as a function of \(x\) in the corresponding coordinate system, then

\[ A_{ik}=\|(\varphi_{i2}-\varphi_{i1})^{\frac1p} f_{ik}\|_{L_p(0,a_i)} \leq C\|u\|_{W_p^1(G)}, \tag{6} \]

\[ B_{ik}=\left(\int_0^{a_i} d\xi \int_0^{\varphi_{i2}(\xi)-\varphi_{i1}(\xi)} \eta^{-p}|f_{ik}(\xi+\eta)-f_{ik}(\xi)|^p\,d\eta\right)^{1/p} \leq C\|u\|_{W_p^1(G)}, \tag{7} \]

where \(a_i>0\) is chosen so that

\[ a_i+\varphi_{i2}(a_i)-\varphi_{i1}(a_i)\leq a_i \]

and

\[ C_i=\|(\varphi_{i2}-\varphi_{i1})^{\frac1p-1}(f_{i1}-f_{i2})\|_{L_p(0,a_i)} \leq \]

\[ \leq C\|u\|_{W_p^1(G)},\qquad k=1,2,\quad i=1,2,\ldots,m. \tag{8} \]

Conversely, let \(\Gamma_i,\ i=0,1,\ldots,m\), form a covering of the boundary \(\Gamma\) in the sense that for any point \(P\in\Gamma\) one can specify such a \(\Gamma_i\) that \(P\in\Gamma_i\), and the distance along \(\Gamma\) from the point \(P\) to \(\Gamma\setminus\Gamma_i\) is positive. If a function \(f\), defined on \(\Gamma\), is such that \(A_0,A_{ik},B_{ik},C_i,\ i=1,\ldots,m,\ k=1,2\), are finite, then there exists a function \(u\in W_p^1(G)\) such that \(u|_\Gamma=f\) and

\[ \|u\|_{W_p^1(G)} \leq C\left\{A_0+\sum_{i=1}^{m}\sum_{k=1}^{2}(A_{ik}+B_{ik})+\sum_{i=1}^{m}C_i\right\}. \]

Corollary. The class \(W(f)\) is nonempty and problem (1), (2) has a solution in this class of functions if and only if \(A_0, A_{ik}, B_{ik}, C_i\), \(i=1,\ldots,m,\ k=1,2\), are finite.

This corollary is almost obvious, since the assertion that the class \(W(f)\) is nonempty follows from Theorem 1 for \(p=2\), while the existence of a solution of problem (1), (2) is proved, for example, by the variational method, word for word in the same way as this is done in the book of S. L. Sobolev [1]. (See also the monograph of L. D. Kudryavtsev [2], pp. 138–155.) In the following paragraph we shall show that this solution is unique in \(W(f)\).

Proof of Theorem 1. Let us first consider the first part of the theorem. The fact that the function \(f=u|_{\Gamma}\) exists, belongs to the space \(W_p^{1-\frac1p}(\Gamma_0)\), and satisfies inequality (5) follows from E. Gagliardo’s theorem. It remains to prove inequalities (6), (7), (8), and for this it is enough to show that they hold for one of the sets \(g_i\). In the proof we shall pass to the corresponding admissible system of coordinates, but for brevity we shall omit the index \(i\) everywhere.

Thus, let

\[ g=\{(x,y):\ 0<x<a,\ \varphi_1(x)<y<\varphi_2(x)\}. \]

As is usually proved (see [2], [7]), the functions \(f_1(x)\) and \(f_2(x)\) exist and for almost all \(x\) and \(y\) the equalities

\[ f_k(x)=u(x,y)-\int_{\varphi_k(x)}^{y}\frac{\partial u}{\partial t}(x,t)\,dt,\quad k=1,2. \]

hold. Integrating these equalities over \(g\), we obtain

\[ \left\|(\varphi_2-\varphi_1)^{\frac1p} f_k\right\|_{L_p(0,a)} \leq \|u\|_{L_p(g)}+ \left\| \int_{\varphi_k(x)}^{y}\frac{\partial u}{\partial t}(x,t)\,dt \right\|_{L_p(g)}. \]

In the last term we apply Hölder’s inequality to the inner integral and enlarge the limits of integration. Then we obtain inequality (6)

\[ \left\|(\varphi_2-\varphi_1)^{\frac1p} f_k\right\|_{L_p(0,a)} \leq C\|u\|_{W_p^1(g)},\quad k=1,2. \]

Next, for almost all \(x\) the equality

\[ f_2(x)-f_1(x)=\int_{\varphi_1(x)}^{\varphi_2(x)} \frac{\partial u}{\partial t}(x,t)\,dt \]

holds. We divide this equality by \(\varphi_2-\varphi_1\), integrate over \(g\), and apply Hölder’s inequality to the inner integral on the right-hand side. Then we have

\[ \left\|(\varphi_2-\varphi_1)^{\frac1p-1}(f_2-f_1)\right\|_{L_p(0,a)} \leq \left\|\frac{\partial u}{\partial y}\right\|_{L_p(g)}. \]

Inequality (8) is proved.

It remains only to prove inequality (7). To this end we perform the change of coordinates

\[ \eta=y-\varphi_1(x),\quad \xi=x . \tag{9} \]

Under this transformation the set \(g\) passes into the set \(g'\):

\[ g'=\{(\xi,\eta):\ 0<\xi<a,\ 0<\eta<\varphi_2(\xi)-\varphi_1(\xi)\}. \]

Put

\[ v(\xi,\eta)=u(\xi,\varphi_1(\xi)+\eta). \]

It is known (see, for example, [7]) that \(v(\xi,\eta)\in W_p^1(g')\), and the derivatives of the function \(v(\xi,\eta)\) are computed by the usual rule for differentiating a composite function; moreover,

\[ \|v\|_{W_p^1(g')}\leqslant C\|u\|_{W_p^1(g)} . \tag{10} \]

Then for almost all \(\xi\) and \(\eta\) the equality

\[ f_1(\xi+\eta)-f_1(\xi)= \]

\[ =\int_0^\eta \frac{\partial v}{\partial t}(\xi,t)\,dt -\int_0^\eta \frac{\partial v}{\partial t}(\xi+\eta,t)\,dt +\int_\xi^{\xi+\eta}\frac{\partial v}{\partial t}(t,\eta)\,dt . \]

Divide this identity by \(\eta\) and integrate over the set

\[ g^*=\{(\xi,\eta):\ 0<\xi<\alpha,\ 0<\eta<\varphi_2(\xi)-\varphi_1(\xi)\}, \]

where \(\alpha>0\) is chosen so that

\[ \alpha+\varphi_2(\alpha)-\varphi_1(\alpha)\leqslant a, \]

\[ \|\eta^{-1}[f_1(\xi+\eta)-f_1(\xi)]\|_{L_p(g^*)} \leqslant \left\|\frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi,t)\,dt\right\|_{L_p(g^*)} + \]

\[ + \left\|\frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi+\eta,t)\,dt\right\|_{L_p(g^*)} + \left\|\frac{1}{\eta}\int_\xi^{\xi+\eta}\frac{\partial v}{\partial t}(t,\eta)\,dt\right\|_{L_p(g^*)}. \tag{11} \]

The first term on the right-hand side of inequality (11) is estimated directly by Hardy’s inequality [8]:

\[ \left\|\frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi,t)\,dt\right\|_{L_p(g^*)} \leqslant \]

\[ \leqslant \left( \int_0^\alpha d\xi \int_0^{\varphi_2(\xi)-\varphi_1(\xi)} d\eta \left| \frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi,t)\,dt \right|^p \right)^{1/p} \leqslant \]

\[ \leqslant C\left\|\frac{\partial v}{\partial \eta}\right\|_{L_p(g')}. \tag{12} \]

We note that the function \(\eta=\varphi_2(\xi)-\varphi_1(\xi)\) increases monotonically on the interval \([0,\alpha]\) from \(0\) to \(\varphi_2(\alpha)-\varphi_1(\alpha)\). Denote by \(\xi=\varphi^{-1}(\eta)\) the inverse function. It increases monotonically on the interval \([0,\varphi_2(\alpha)-\varphi_1(\alpha)]\) from \(0\) to \(\alpha\).

Then

\[ \begin{aligned} \left\| \frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi+\eta,t)\,dt \right\|_{L_p(g^*)} &= \left( \int_0^{\varphi_2(\alpha)-\varphi_1(\alpha)} d\eta \int_{\varphi^{-1}(\eta)+\eta}^{\alpha+\eta} d\xi \left| \frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi,t)\,dt \right|^p \right)^{1/p} \leq \\ &\leq \left\| \frac{1}{\eta}\int_0^\eta \frac{\partial v}{\partial t}(\xi,t)\,dt \right\|_{L_p(g')} \leq C\left\| \frac{\partial v}{\partial \eta}\right\|_{L_p(g')}. \end{aligned} \tag{13} \]

Let us estimate the last term on the right-hand side of inequality (11). This estimate will follow from the chain of equalities and inequalities given below, in which only elementary changes of the variables of integration are made and, at the appropriate place, the generalized Minkowski inequality [8] is applied:

\[ \left\| \frac{1}{\eta}\int_\xi^{\xi+\eta}\frac{\partial v}{\partial t}(t,\eta)\,dt \right\|_{L_p(g^*)} = \left\|\int_0^1 \frac{\partial v}{\partial t}(\xi+t\eta,\eta)\,dt \right\|_{L_p(g^*)} \leq \]

\[ \leq \int_0^1 \left\| \frac{\partial v}{\partial t}(\xi+t\eta,\eta)\right\|_{L_p(g^*)}\,dt = \]

\[ = \int_0^1 \left( \int_0^{\varphi_2(\alpha)-\varphi_1(\alpha)} d\eta \int_{\varphi^1(\eta)+t\eta}^{\alpha+t\eta} \left| \frac{\partial v}{\partial \xi}(\xi,\eta) \right|^p d\xi \right)^{1/p} dt \leq \left\|\frac{\partial v}{\partial \xi}\right\|_{L_p(g')}. \tag{14} \]

From inequalities (10)—(14) there follows the inequality

\[ \left\|\eta^{-1}\bigl[f_1(\xi+\eta)-f_1(\xi)\bigr]\right\|_{L_p(g^*)} \leq C\|u\|_{W_p^1(g)}. \]

An analogous inequality for \(f_2\) is obtained in the same way; only, instead of the coordinate transformation (9), one must perform the following transformation:

\[ \eta=\varphi_2(x)-y,\qquad \xi=x. \tag{15} \]

The first part of Theorem 1 has been proved.

To prove the second part of the theorem, it is enough, for each point \(P\) of the boundary \(\Gamma\) and some neighborhood \(O(P)\) of it, to construct in the set \(O(P)\cap G\) a function that belongs to the space \(W_p^1(O(P)\cap G)\) and whose trace on \(O(P)\cap\Gamma\) is equal to \(f\). Indeed, if such functions can be constructed, then the construction of the desired function is carried out by the well-known method of Whitney and Hestenes [9]. This method, for a more general case, is set forth in detail in the work of S. M. Nikol’skii [4].

The construction of the required function on the set \(O(P)\cap G\), when the point \(P\) is Lipschitz, is well known [3], [7]. It remains for us to show how such a function is constructed for the set \(O(Q_i)\cap G\), \(i=1,2,\ldots,m\), where \(Q_i\) is a non-Lipschitz point of the type defined by us.

In what follows, in the proof we shall again, for brevity, omit the subscript \(i\).

Consider the function

\[ u_1(x,y)=\frac{1}{y-\varphi_1(x)}\int_x^{x+y-\varphi_1(x)} f_1(t)\,dt . \]

Make the change of coordinates (9) and show that the function

\[ v_1(\xi,\eta)=u_1\bigl(\xi,\eta+\varphi_1(\xi)\bigr) \]

belongs to the space \(W_p^1(g^*)\), and \(v_1(\xi,0)=f_1(\xi)\). It is easy to see that

\[ \left\|\frac{\partial v_1}{\partial \xi}\right\|_{L_p(g^*)} = \left\|\eta^{-1}\bigl[f_1(\xi+\eta)-f_1(\xi)\bigr]\right\|_{L_p(g^*)}. \]

For the function \(v_1(\xi,\eta)\) itself we have

\[ \|v_1\|_{L_p(g^*)} = \left\|\int_0^1 f_1(\xi+t\eta)\,dt\right\|_{L_p(g^*)} \le \]

\[ \le \int_0^1 \|f_1(\xi+t\eta)\|_{L_p(g^*)}\,dt \le \|(\varphi_2-\varphi_1)^{1/p}f_1\|_{L_p(0,a)}. \]

Here we have applied the generalized Minkowski inequality and extended the limits of integration. For the derivative with respect to \(\eta\) the estimate is obtained in the same way; only the exposition becomes somewhat more complicated:

\[ \left\|\frac{\partial v_1}{\partial \eta}\right\|_{L_p(g^*)} = \left\|-\frac{1}{\eta^2}\int_\xi^{\xi+\eta} f_1(t)\,dt + \frac{1}{\eta}f_1(\xi+\eta)\right\|_{L_p(g^*)} \le \]

\[ \le \left\|\frac{1}{\eta^2}\int_\xi^{\xi+\eta}\bigl[f_1(t)-f_1(\xi)\bigr]\,dt\right\|_{L_p(g^*)} + \left\|\eta^{-1}\bigl[f_1(\xi+\eta)-f_1(\xi)\bigr]\right\|_{L_p(g^*)}. \]

The first term is estimated by Hardy’s inequality [8]:

\[ \left\|\frac{1}{\eta^2}\int_\xi^{\xi+\eta}\bigl[f_1(t)-f_1(\xi)\bigr]\,dt\right\|_{L_p(g^*)} \le \left\|\frac{1}{\eta}\int_0^\eta \frac{|f_1(\xi+t)-f_1(\xi)|}{t}\,dt\right\|_{L_p(g^*)} \le \]

\[ \le C\left\|\eta^{-1}\bigl[f_1(\xi+\eta)-f_1(\xi)\bigr]\right\|_{L_p(g^*)}. \]

It remains to show that \(v_1(\xi,0)=f_1(\xi)\). Indeed, for any \(\varepsilon>0\) and \(0<\eta<\varphi_2(\varepsilon)-\varphi_1(\varepsilon)\) we have:

\[ \|v_1(\xi,\eta)-f_1(\xi)\|_{L_p(\varepsilon,a)} = \left\|\frac{1}{\eta}\int_0^\eta \bigl[f_1(\xi+t)-f_1(\xi)\bigr]\,dt\right\|_{L_p(\varepsilon,a)} \le \]

\[ \le \left\|\int_0^\eta \frac{|f_1(\xi+t)-f_1(\xi)|}{t}\,dt\right\|_{L_p(\varepsilon,a)}. \]

Apply Hölder’s inequality to the inner integral and enlarge the limits of integration. Then

\[ \left\| v_1(\xi,\eta)-f_1(\xi)\right\|_{L_p(\varepsilon,\alpha)} \le \]

\[ \le \eta^{1-\frac1p}\left\|\eta^{-1}\left[f_1(\xi+\eta)-f_1(\xi)\right]\right\|_{L_p(g^*)}, \]

which proves that \(v_1(\xi,0)=f_1(\xi)\).

Analogous considerations can also be carried out for the function

\[ u_2(x,y)=\frac{1}{\varphi_2(x)-y}\int_x^{x+\varphi_2(x)-y} f_2(t)\,dt, \]

only instead of the change of coordinates (9) one should make the change (15).

We assert that the function

\[ u(x,y)=u_1(x,y)\frac{\varphi_2(x)-y}{\varphi_2(x)-\varphi_1(x)} +u_2(x,y)\frac{y-\varphi_1(x)}{\varphi_2(x)-\varphi_1(x)} \]

is the desired one, i.e. it belongs to the space \(W_p^1(g_\alpha)\), where

\[ g_\alpha=\{(x,y):\ 0<x<\alpha,\ \varphi_1(x)<y<\varphi_2(x)\} \]

and \(u(x,\varphi_k(x))=f_k(x)\), \(k=1,2\). The last assertions are almost obvious. Let us prove, for example, that \(u(x,\varphi_1(x))=f_1(x)\). Indeed, for any \(\varepsilon>0\) and sufficiently small \(h\), \(0<h<\frac12[\varphi_2(\varepsilon)-\varphi_1(\varepsilon)]\), the inequality

\[ \left\|u(x,\varphi_1(x)+h)-f_1(x)\right\|_{L_p(\varepsilon,\alpha)} \le \]

\[ \le \left\|u_1(x,\varphi_1(x)+h)-f_1(x)\right\|_{L_p(\varepsilon,\alpha)} + \]

\[ +hC\sum_{k=1}^{2}\left\|(\varphi_2-\varphi_1)^{\frac1p}f_k\right\|_{L_p(0,a)} \]

holds, where the positive constant \(C\) does not depend on \(h\). (The constant \(C\) depends on \(\varepsilon\), and \(C\to\infty\) as \(\varepsilon\to0\).) Hence the equality

\[ \lim_{h\to0}\left\|u(x,\varphi_1(x)+h)-f_1(x)\right\|_{L_p(\varepsilon,\alpha)}=0 \]

follows.

It is easy to see that

\[ \|u\|_{L_p(g_\alpha)} \le \sum_{k=1}^{2}\|u_k\|_{L_p(g_\alpha)} \le \sum_{k=1}^{2}\left\|(\varphi_2-\varphi_1)^{\frac1p}f_k\right\|_{L_p(0,a)}. \]

Now let us consider the derivatives of the function \(u(x,y)\):

\[ \frac{\partial u}{\partial x} = \frac{\partial u_1}{\partial x}\frac{\varphi_2-y}{\varphi_2-\varphi_1} + \frac{\partial u_2}{\partial x}\frac{y-\varphi_1}{\varphi_2-\varphi_1} + \]

\[ +(u_1-u_2)\, \frac{\varphi_2'(\varphi_2-\varphi_1)-(\varphi_2-y)(\varphi_2'-\varphi_1')} {(\varphi_2-\varphi_1)^2}, \]

\[ \frac{\partial u}{\partial y} = \frac{\partial u_1}{\partial y}\frac{\varphi_2-y}{\varphi_2-\varphi_1} + \frac{\partial u_2}{\partial y}\frac{y-\varphi_1}{\varphi_2-\varphi_1} + \frac{u_2-u_1}{\varphi_2-\varphi_1}. \]

For them we obtain

\[ \left\|\frac{\partial u}{\partial x}\right\|_{L_p(g_\alpha)} \leq \sum_{k=1}^{2}\left\|\frac{\partial u_k}{\partial x}\right\|_{L_p(g_\alpha)} + C\left\|\frac{u_1-u_2}{\varphi_2-\varphi_1}\right\|_{L_p(g_\alpha)}, \]

\[ \left\|\frac{\partial u}{\partial y}\right\|_{L_p(g_\alpha)} \leq \sum_{k=1}^{2}\left\|\frac{\partial u_k}{\partial y}\right\|_{L_p(g_\alpha)} + \left\|\frac{u_1-u_2}{\varphi_2-\varphi_1}\right\|_{L_p(g_\alpha)}. \]

Finally,

\[ \begin{aligned} \left\|\frac{u_1-u_2}{\varphi_2-\varphi_1}\right\|_{L_p(g_\alpha)} &\leq \sum_{k=1}^{2}\left\|\frac{u_k-f_k}{\varphi_2-\varphi_1}\right\|_{L_p(g_\alpha)} + \left\|\frac{f_2-f_1}{\varphi_2-\varphi_1}\right\|_{L_p(g_\alpha)} = \\[4pt] &= \left\| \frac{1}{\varphi_2-\varphi_1}\, \frac{1}{y-\varphi_1} \int_{x}^{x+y-\varphi_1}\bigl[f_1(t)-f_1(x)\bigr]\,dt \right\|_{L_p(g_\alpha)} + \\[4pt] &\quad+ \left\| \frac{1}{\varphi_2-\varphi_1}\, \frac{1}{\varphi_2-y} \int_{x}^{x+\varphi_2-y}\bigl[f_2(t)-f_2(x)\bigr]\,dt \right\|_{L_p(g_\alpha)} + \\[4pt] &\quad+ \left\|(\varphi_2-\varphi_1)^{\frac1p-1}(f_2-f_1)\right\|_{L_p(0,\alpha)} \leq \\[4pt] &\leq \sum_{k=1}^{2} \left\| \frac1\eta\int_{0}^{\eta} \left|\frac{f_k(\xi+t)-f_k(\xi)}{t}\right|\,dt \right\|_{L_p(q^*)} + \\[4pt] &\quad+ \left\|(\varphi_2-\varphi_1)^{\frac1p-1}(f_2-f_1)\right\|_{L_p(0,\alpha)} \leq \\[4pt] &\leq C\sum_{k=1}^{2} \left\|\eta^{-1}\bigl[f_k(\xi+\eta)-f_k(\xi)\bigr]\right\|_{L_p(q^*)} + \\[4pt] &\quad+ \left\|(\varphi_2-\varphi_1)^{\frac1p-1}(f_2-f_1)\right\|_{L_p(0,\alpha)}. \end{aligned} \]

Consequently, the function \(u(x,y)\) is the desired one. Theorem 1 is proved.

EXISTENCE AND UNIQUENESS OF THE SOLUTION OF THE DIRICHLET PROBLEM

Let the domain \(G\) and its boundary \(\Gamma\) be the same as in the preceding subsection. Denote by \(\overset{\circ}{W}{}^1_p(G)\) the subspace of the space \(W^1_p(G)\) consisting of functions whose traces on \(\Gamma\) are equal to zero. By \(\overset{\circ}{C}{}^\infty(G)\) we shall denote the set of infinitely differentiable functions each of which is equal to zero outside some closed domain contained in the domain \(G\). Further, by \(\overline{C}{}^\infty(G)\) we shall denote the closure of the set \(\overset{\circ}{C}{}^\infty(G)\) in the metric of the space \(W^1_p(G)\).

Theorem 2. \(\overset{\circ}{G}{}^\infty(G)=\overset{\circ}{W}{}^{1}_{p}(G)\).

Proof. The inclusion \(\overline{\overset{\circ}{C}{}^\infty(G)}\subset \overset{\circ}{W}{}^{1}_{p}(G)\) in the case of a Lipschitz boundary is well known. The presence of non-Lipschitz points \(Q_i\), \(i=1,2,\ldots,m\), introduces no essential changes into the proof; therefore we shall not dwell on this. We shall prove the inclusion that is of greater interest to us,
\(\overset{\circ}{W}{}^{1}_{p}(G)\subset \overset{\circ}{C}{}^\infty(G)\). This inclusion is also known for Lipschitz boundaries. It is proved, for example, in the work of J. Nečas [5]. We shall give another proof, which, as it seems to us, is of interest also for the case of domains with Lipschitz boundaries, since it is easily carried over also to the so-called weighted spaces \(W^{1}_{p}\) [2].

For each Lipschitz point \(P\) of the boundary \(\Gamma\) we construct the set

\[ G_P(a_P)=G_P^{+}\cup G_P^{-}. \]

(See the definition of a Lipschitz point.) For each point \(Q_i\), \(i=1,\ldots,m\), construct two sets \(G_{i1}(a_i)\) and \(G_{i2}(a_i)\). To this end we first define functions \(\varphi_{ik}(x^i)\), \(i=1,2,\ldots,m\), \(k=1,2\), also for negative \(x^i\), setting them there equal to zero. Then

\[ G_{ik}(a_i)=\{x^i,y^i):\ -\hat{\delta}_i<x^i<a_i,\ \varphi_{ik}(x^i)-\hat{\delta}_i<y^i<\varphi_{ik}(x^i)+\hat{\delta}_i\}, \]

\(k=1,2\). The sets \(G_P(a_P/2)\) and \(G_{ik}(a_i/2)\) form a covering of the boundary \(\Gamma\). From this covering choose a finite subcovering so that it contains the sets \(G_{ik}(a_i/2)\), \(i=1,2,\ldots,m\), \(k=1,2\). The sets \(G_P(a_P/2)\) entering this finite covering will be denoted by \(G_j(a_j/2)\), \(j=1,2,\ldots,N\). Obviously, the sets \(G_{ik}\left(\frac{3}{4}a_i\right)\), \(i=1,\ldots,m\), \(k=1,2\), and \(G_j\left(\frac{3}{4}a_j\right)\), \(j=1,2,\ldots,N\), also form a covering of the boundary \(\Gamma\).

In what follows, the function \(\varphi\) and the coordinates \(x,y\) corresponding to the set \(G_j\) will be denoted by \(\varphi_j\) and \(x^j,y^j\). For each function \(\varphi_j\) construct the function \(\psi_j\):

\[ \psi_j(x^j)= \begin{cases} \varphi_j(x^j), & \text{if } |x^j|\le \dfrac{3}{4}a_j,\\[6pt] \varphi_j(x^j)-|x^j|+\dfrac{3}{4}a_j, & \text{if } \dfrac{3}{4}a_j<|x^j|<a_j. \end{cases} \]

Analogously, construct the functions \(\psi_{i1}\) and \(\psi_{i2}\).

\[ \psi_{i1}(x^i)= \begin{cases} \varphi_{i1}(x^i), & \text{if } -\hat{\delta}_i<x^i\le \dfrac{3}{4}a_i,\\[6pt] \varphi_{i1}(x^i)-x^i+\dfrac{3}{4}a_i, & \text{if } \dfrac{3}{4}a_i<x^i<a_i. \end{cases} \]

\[ \psi_{i2}(x^i)= \begin{cases} \psi_{i2}(x^i), & \text{if } -\hat{\delta}_i<x^i\le \dfrac{3}{4}a_i,\\[6pt] \varphi_{i2}(x^i)+x^i-\dfrac{3}{4}a_i, & \text{if } \dfrac{3}{4}a_i<x^i<a_i. \end{cases} \]

It is easy to see that the functions \(\psi_j\) and \(\psi_{ik}\) satisfy the Lipschitz condition. From the functions \(\psi_j\) and \(\psi_{ik}\) we construct the sets \(G_j^*\) and \(G_{ik}^*\):

\[ G_j^*=\{(x^j,y^j): -a_j<x^j<a_j;\ \psi_j(x^j)-\delta_j<y^j<\psi_j(x^j)+\delta_j\}, \]

\[ G_{ik}^*=\{(x^i,y^i): -\delta_i<x^i<a_i,\ \psi_{ik}(x^i)-\delta_i<y^i<\psi_{ik}(x^i)+\delta_i\}. \]

Let us construct the functions \(\eta_j,\ j=1,\ldots,N\), in the following way: outside the set \(G_j^*\cap G_j(a_j)\) we put \(\eta_j=1\), and at the remaining points we define it by the formula:

\[ \eta_j= \begin{cases} 0, & \text{if } y^j\leqslant \psi_j(x^j)+h,\\[6pt] \dfrac{y^j-\psi_j(x^j)-h}{h}, & \text{if } \psi_j(x^j)+h\leqslant y^j\leqslant \psi_j(x^j)+2h,\\[8pt] 1, & \text{if } y^j\geqslant \psi_j(x^j)+2h. \end{cases} \]

Similarly we define the functions \(\eta_{ik}\). Outside the set \(G_{ik}^*\cap G_{ik}(a_i)\) we put \(\eta_{ik}=1\), and at the remaining points we define them by the formulas:

\[ \eta_{i1}= \begin{cases} 0, & \text{if } y^i\leqslant \psi_{i1}(x^i)+h,\\[6pt] \dfrac{y^i-\psi_{i1}(x^i)-h}{h}, & \text{if } \psi_{i1}(x^i)+h\leqslant y^i\leqslant \psi_{i1}(x^i)+2h,\\[8pt] 1, & \text{if } y^i\geqslant \psi_{i1}(x^i)+2h, \end{cases} \]

\[ \eta_{i2}= \begin{cases} 0, & \text{if } y^i\geqslant \psi_{i2}(x^i)-h,\\[6pt] \dfrac{y^i-\psi_{i2}(x^i)+h}{-h}, & \text{if } \psi_{i2}(x^i)-h\geqslant y^i\geqslant \psi_{i2}-2h,\\[8pt] 1, & \text{if } y^i\leqslant \psi_{i2}(x^i)-2h. \end{cases} \]

The functions \(\eta_j\) and \(\eta_{ik}\) are defined on the entire plane. They are discontinuous functions, but, for sufficiently small \(h>0\), as is easy to see, they are continuous in \(G\) and have bounded generalized derivatives there.

Let \(u\in \overset{\circ}{W}{}^1_p(G)\), \(1<p<\infty\). Denote by \(\eta\) the product of all functions \(\eta_j,\ j=1,\ldots,N\), and \(\eta_{ik},\ i=1,\ldots,m,\ k=1,2\). Obviously, the function \(u_h=u\eta\) is equal to zero in some strip along the boundary \(\Gamma\) and, for sufficiently small \(h\), belongs to the space \(W^1_p(G)\). We shall show that

\[ \lim_{h\to 0}\|u-u_h\|_{W^1_p(G)}=0. \]

Indeed,

\[ \|u-u_h\|^p_{L_p(G)} \leqslant \sum_{j=1}^{N}\|u-u_h\|^p_{L_p(G_{jh})} + \sum_{k=1}^{2}\sum_{i=1}^{m}\|u-u_h\|^p_{L_p(G_{ikh})}, \]

where

\[ G_{jh}=\{(x^j,y^j): -a_j<x^j<a_j,\ \varphi_j(x^j)<y^j<\varphi_j(x^j)+2h\}, \]

\[ G_{i1h}=G\cap \{(x^i,y^i): 0<x^i<a_i,\ \varphi_{i1}(x^i)<y^i<\varphi_{i1}(x^i)+2h\}, \]

\[ G_{i2h}=G\cap \{(x^i,y^i): 0<x^i<a_i,\ \varphi_{i2}(x^i)-2h<y^i<\varphi_{i2}(x^i)\}. \]

And since \(0\leqslant 1-\eta\leqslant 1\), then

\[ \|u-u_h\|_{L_p(G)} \leq C \sum_{j=1}^{N}\|u\|_{L_p(G_{jh})} + C \sum_{k=1}^{2}\sum_{i=1}^{m}\|u\|_{L_p(G_{ikh})}. \tag{16} \]

Further,

\[ \|\operatorname{grad}(u-u_h)\|_{L_p(G)} \leq \]

\[ \leq \|(1-\eta)|\operatorname{grad}u|\|_{L_p(G)} + \|u|\operatorname{grad}\eta|\|_{L_p(G)}. \tag{17} \]

The first term is estimated in the same way as the norm of the difference \(u-u_h\):

\[ \|(1-\eta)|\operatorname{grad}u|\|_{L_p(G)}^{p} \leq \sum_{j=1}^{N}\|\operatorname{grad}u\|_{L_p(G_{jh})}^{p} + \]

\[ + \sum_{k=1}^{2}\sum_{i=1}^{m}\|\operatorname{grad}u\|_{L_p(G_{ikh})}^{p}. \tag{18} \]

It is easy to see that

\[ |\operatorname{grad}\eta| < \frac{C}{h}, \quad C>0. \]

Moreover, \(|\operatorname{grad}\eta_j|\ne 0\) in \(G\) only on a set that is part of the set \(G_{jh}\), and \(|\operatorname{grad}\eta_{ik}|\ne 0\) in \(G\) only on a set that is part of the set \(G_{ikh}\). Therefore

\[ \|u|\operatorname{grad}\eta|\|_{L_p(G)} \leq \frac{C}{h}\sum_{j=1}^{N}\|u\|_{L_p(G_{jh})} + \frac{C}{h}\sum_{k=1}^{2}\sum_{i=1}^{m}\|u\|_{L_p(G_{ikh})}. \]

All terms on the right-hand side are estimated in the same way. Let us estimate, for example, one of the terms of the first sum. In doing so, all arguments will be carried out in the corresponding admissible coordinate system. Then, applying Hölder’s inequality and taking into account that in \(G_{jh}\), \(0<y^i-\varphi_j<h\), we obtain

\[ \frac{1}{h}\|u\|_{L_p(G_{jh})} = \frac{1}{h}\left\| \int_{\varphi_{i1}}^{y^i} \frac{\partial u}{\partial t}(x^i,t)\,dt \right\|_{L_p(G_{jh})} \leq C \left\| \frac{\partial u}{\partial y^i} \right\|_{L_p(G_{jh})}. \]

Thus,

\[ \|u|\operatorname{grad}\eta|\|_{L_p(G)} \leq C \sum_{j=1}^{N}\|\operatorname{grad}u\|_{L_p(G_{jh})} + \]

\[ + C \sum_{k=1}^{2}\sum_{i=1}^{m}\|\operatorname{grad}u\|_{L_p(G_{ikh})}. \tag{19} \]

From inequalities (16), (17), (18), and (19) it follows that

\[ \lim_{h\to 0}\|u-u_h\|_{W_p^1(G)}=0. \]

Extend the function \(u_h\) to the whole plane by setting it equal to zero outside \(G\). Construct the sequence of mean functions of S. L. Sobolev [1] for the function \(u_h\). This sequence converges to \(u_h\) in the norm

space \(W_p^1(G)\). For a sufficiently small radius of averaging, its terms belong to \(\overset{\circ}{C}{}^\infty(G)\). Thus, for any function \(u \in \overset{\circ}{W}{}_p^1(G)\) one can construct a sequence of functions from \(\overset{\circ}{C}{}^\infty(G)\) converging to the function \(u\) in the norm of the space \(W_p^1(G)\). Theorem 2 is proved.

Now let us consider the Dirichlet problem (1), (2). Let the function \(f\) be such that the class \(W(f)\) is nonempty. A function \(u \in W(f)\) will be called a generalized solution of problem (1), (2) if, for any function \(v \in \overset{\circ}{C}{}^\infty(G)\), the following equality holds:

\[ \iint \left( \sum_{i,j=1}^{2} a_{ij}\,\frac{\partial u}{\partial x_i}\,\frac{\partial v}{\partial x_j} + buv \right)\,dG = 0. \]

Theorem 3. Let the domain \(G\) and its boundary \(\Gamma\) be the same as in Theorem 1, and let \(f\) be such that the class \(W(f)\) is nonempty. Then there exists a unique generalized solution of the Dirichlet problem (1), (2).

Proof. We have already discussed the existence of a solution. (See the corollary after Theorem 1.) Let us show that this solution is unique in \(W(f)\). Suppose there are two generalized solutions \(u_1\) and \(u_2\) of problem (1), (2). Then \(v_0 = u_1 - u_2\) is a solution of problem (1), (2) with \(f \equiv 0\). Consider the linear functional

\[ J(v)=\iint \left( \sum_{i,j=1}^{2} a_{ij}\,\frac{\partial v_0}{\partial x_i}\,\frac{\partial v}{\partial x_j} + bv_0v \right)\,dG, \]

where \(v \in \overset{\circ}{W}{}_2^1(G)\). This functional is bounded and vanishes on the set \(\overset{\circ}{C}{}^\infty(G)\), which by Theorem 2 is dense in \(\overset{\circ}{W}{}_2^1(G)\). Therefore \(J(v)=0\) for all \(v \in \overset{\circ}{W}{}_2^1(G)\), and, in particular, for \(v_0\). Then from inequality (3) it follows that

\[ \|\operatorname{grad} v_0\|_{L_p(G)}=0. \]

And since \(v_0 \in \overset{\circ}{W}{}_2^1(G)\), it follows that \(v_0 \equiv 0\). The theorem is proved.

References

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Received by the editors
March 1, 1965

Moscow Institute of Physics and Technology