L. N. Slobodetskii

Estimates in \(L_p\) of Solutions of Elliptic Systems

(Presented by Academician V. I. Smirnov on 4 VII 1958)

1. In the present note a generalization is given, in one direction, of the results set forth in the notes \((^{1-3})\) and in the article of Gagliardo \((^4)\). In doing so we adhere to the terminology adopted in \((^2,{}^3)\).

2. Theorem 1. Let \(l\) be a natural number, \(1<p<+\infty\), and let \(\Omega\) be a bounded domain of the \(n\)-dimensional space \(E_n\), bounded by a surface \(S\) that is continuously differentiable a finite number \(l+1\) of times. Let
   \(v=v(x)\in W_p^{(l)}(\Omega)\).

Then, for \(k=0,1,\ldots,l-1\), the normal derivatives \(\partial^k v/\partial \nu^k\), as functions of a point of the surface \(S\), belong to the spaces \(W_p^{(l-k-1/p)}(S)\). Moreover,

\[ \left\|\frac{\partial^k v}{\partial \nu^k}\right\|_{W_p^{(l-k-1/p)}(S)} \leq C_1 \|v\|_{W_p^{(l)}(\Omega)}, \tag{1} \]

where \(C_1\) depends only on \(\Omega\).

Conversely, if functions \(\varphi_k(x')\in W_p^{(l-k-1/p)}(S)\) \((k=0,1,\ldots,l-1)\) are given, then there exists a function \(\bar v\in W_p^{(l)}(\Omega)\) satisfying the boundary conditions

\[ \left.\frac{\partial^k \bar v}{\partial \nu^k}\right|_S=\varphi_k \quad (k=0,1,\ldots,l-1). \tag{2} \]

Moreover,

\[ \|\bar v\|_{W_p^{(l)}(\Omega)} \leq C_2 \sum_{k=0}^{l-1}\|\varphi_k\|_{W_p^{(l-k-1/p)}(S)}, \tag{3} \]

where \(C_2\) depends only on \(\Omega\).

For \(p=2\), a more general result (obtained with the aid of the Fourier transform and Parseval’s equality) is given in note \((^2)\). For \(l=1\), Theorem 1 was proved by Gagliardo \((^4)\).

The first part of our theorem follows directly from Gagliardo’s theorem. We shall therefore dwell in more detail on the proof of its second part.

It suffices to assume that \(\Omega\) is the half-space \(x_n>0\). Assuming that
\(\psi_s(x')\in W_p^{(l-s-1/p)}(E_{n-1})\) \((0\leq s\leq l-1)\), consider, for \(x_n>0\), the function

\[ v_s(x)=K_n e^{-x_n}\frac{x_n^{s+1}}{s!} \int_{E_{n-1}} \frac{\psi_s(x'+t')\,dt'}{(|t'|^2+x_n^2)^{n/2}}, \]

where

\[ K_n=\left(\int_{E_{n-1}}\frac{dt'}{(|t'|^2+1)^{n/2}}\right)^{-1}. \]

It is easy to see that

\[ \left.\frac{\partial^k v_s}{\partial x_n^k}\right|_{x_n=0} =0 \quad (0\leq k\leq s-1); \qquad \left.\frac{\partial^s v_s}{\partial x_n^s}\right|_{x_n=0} =\psi_s(x'). \]

Let

\[ \frac{n-2}{p}+1<\varepsilon<\frac{n-1}{p}+1. \]

Applying Hölder’s inequality, we obtain:

\[ \begin{aligned} |v_s(x)| &\leq C_1 e^{-x_n}x_n^{s-n+1} \int_{E_{n-1}} \frac{|\psi_s(x'+t')|}{(1+|t'|^2/x_n^2)^{n/2}}\,dt' \\ &\leq C_1 e^{-x_n}x_n^{s-n+1} \left[ \int_{E_{n-1}} \frac{|\psi_s(x'+t')|^p}{(1+|t'|^2/x_n^2)^{\varepsilon p/2}}\,dt' \right]^{1/p} \times \\ &\qquad\qquad\times \left[ \int_{E_{n-1}} \frac{dt'}{(1+|t'|^2/x_n^2)^{\frac{n-\varepsilon}{2}p'}} \right]^{1/p'} \\ &= C_2 e^{-x_n}x_n^{s-\frac{n-1}{p}} \left[ \int_{E_{n-1}} \frac{|\psi_s(x'+t')|^p\,dt'}{(1+|t'|^2/x_n^2)^{\varepsilon p/2}} \right]^{1/p}. \end{aligned} \]

Hence

\[ \begin{aligned} \int_0^{+\infty} dx_n \int_{E_{n-1}} |v_s|^p\,dx' &\leq C_3 \int_0^{+\infty} e^{-p x_n}x_n^{ps-n+1}\,dx_n \int_{E_{n-1}} \frac{dt'}{(1+|t'|^2/x_n^2)^{\varepsilon p/2}} \times \\ &\qquad\qquad\times \int_{E_{n-1}}|\psi_s(x'+t')|^p\,dx' \\ &= C_3 \int_0^{+\infty} e^{-p x_n}x_n^{ps}\,dx_n \times \\ &\qquad\qquad\times \int_{E_{n-1}} \frac{dt'}{(|t'|^2+1)^{\varepsilon p/2}} \int_{E_{n-1}}|\psi_s(x')|^p\,dx' \\ &= C_4\int_{E_{n-1}}|\psi_s(x')|^p\,dx'. \end{aligned} \tag{4} \]

Next we have

\[ \begin{aligned} D_{x'}^{\,l}v_s &= K_n e^{-x_n}\frac{x_n^{s+1}}{s!} \int_{E_{n-1}} D^{\,l-s-1}\psi_s(x'+t')\, D_{t'}^{\,s+1}\left[\frac{1}{(|t'|^2+x_n^2)^{n/2}}\right]\,dt' \\ &= K_n e^{-x_n}\frac{x_n^{s+1}}{s!} \int_{E_{n-1}} \bigl[D^{\,l-s-1}\psi_s(x'+t')-D^{\,l-s-1}\psi_s(x')\bigr]\, D_{t'}^{\,s+1} \left(\frac{1}{(|t'|^2+x_n^2)^{n/2}}\right)\,dt'. \end{aligned} \]

Hence, with the aid of Hölder’s inequality, we obtain

\[ \begin{aligned} \int_0^{+\infty} dx_n \int_{E_{n-1}} |D_{x'}^{\,l}v_s|^p\,dx' &\leq C_1 \int_0^{+\infty} dx_n \int_{E_{n-1}} \left[ \int_{E_{n-1}} \frac{|D^{\,l-s-1}\psi_s(x'+t')-D^{\,l-s-1}\psi_s(x')|} {(|t'|^2+x_n^2)^{n/2}}\,dt' \right]^p dx' \\ &\leq C_2 \int_{E_{n-1}} dx' \int_0^{+\infty} x_n^{-(n-1)-p}\,dx_n \int_{E_{n-1}} \frac{|D^{\,l-s-1}\psi_s(x'+t')-D^{\,l-s-1}\psi_s(x')|^p} {(1+|t'|^2/x_n^2)^{\varepsilon p/2}}\,dt' \\ &\leq C_3 \int_{E_{n-1}} dx' \int_{E_{n-1}} \frac{|D^{\,l-s-1}\psi_s(x'+t')-D^{\,l-s-1}\psi_s(x')|^p} {|t'|^{\,n-2+p}}\,dt'. \end{aligned} \tag{5} \]

In a somewhat more complicated way one obtains (5) for \(D_{x_n}^{l}v_s\). From (4) and (5) it follows that

\[ \|v_s\|_{W_p^{(l)}(x_n>0)}\leq C_5\|\psi_s\|_{W_p^{(l-s-1/p)}(E_{n-1})}. \]

By the method described above we construct the function \(v_0(x)\) from \(\psi_0=\varphi_0\), and \(v_s\) from

\[ \psi_s=\varphi_s-\left.\sum_{k=0}^{s-1}\frac{\partial^s v_k}{\partial x_n^s}\right|_{x_n=0} \qquad (s=1,2,\ldots,l-1). \]

Then, from the first part of Theorem 1 and the estimates obtained for \(v_s\), it is easy to obtain that the function

\[ \bar v=\sum_{s=0}^{l-1}v_s \]

satisfies conditions (2) and (3).

1. From Theorem 1 of the present work and Theorem 3 of note (3) there follows the following proposition.

Theorem 2. Let \(L=L(x,\partial/\partial x)\) be an elliptic differential operator of order \(2k\), defined in \(\overline{\Omega}=\Omega+S\); let \(R_\mu=R_\mu(x',\partial/\partial x)\) be differential operators of orders \(m_\mu\) \((\mu=1,2,\ldots,k)\), defined on \(S\) and connected with \(L\) by condition (L) (see (3)). Let \(l\) be a natural number \(\geq 2k\) and \(m_\mu\leq l-1\) \((\mu=1,2,\ldots,k)\).

If the coefficients of the differential operators \(L\) and \(R_\mu\) have \(l-2k\) bounded derivatives and their leading coefficients are continuous in the domains of their definition, and if the surface is \(l+1\) times continuously differentiable, then for any function \(u=u(x)\in W_p^{(l)}(\Omega)\) the inequality

\[ C_1\left[\|Lu\|_{W_p^{(l-2k)}(\Omega)} +\sum_{\mu=1}^{k}\|R_\mu u\|_{W_p^{(l-m_\mu-1/p)}(S)}\right]\leq \]

\[ \leq \|u\|_{W_p^{(l)}(\Omega)} \leq C_2\left[\|Lu\|_{W_p^{(l-2p)}(\Omega)} +\sum_{\mu=1}^{k}\|R_\mu u\|_{W_p^{(l-m_\mu-1/p)}(S)} +\|u\|_{L_p(\Omega)}\right], \tag{6} \]

holds, where \(C_1\) and \(C_2\) are positive constants independent of \(u(x)\).

1. Both theorems admit a generalization to the case when \(\Omega\) is an unbounded domain and \(S\) is an unbounded sufficiently smooth surface without boundary.

Let \(\overline{\Omega}\) be a closed, but, generally speaking, unbounded domain of the space \(E_n\). A function \(f(x)\), defined in \(\overline{\Omega}\), will be called continuous in this domain if it is continuous at each of its points and if there exists a finite limit \(f(x)\) as \(|x|\to+\infty\).

Let \(\sigma\) be a finite or infinite \((n-1)\)-dimensional surface given by the equation \(x=x(\gamma')\) \((\gamma'=(\gamma_1,\ldots,\gamma_{n-1}))\), where the vector function \(x(\gamma')\) is defined in some domain \(d(\sigma)\) of the \((n-1)\)-dimensional Euclidean space of the points \(\gamma'\). Denote by \(\nu=\nu(\gamma')\) the unit vector of the normal to \(\sigma\). We shall say that \(\sigma\in K^{(l)}\) if, for some \(b>0\), the domain \(\Delta(\sigma)\), defined by the relations \(\gamma'\in d(\sigma)\), \(|\gamma_n|<b\), is mapped one-to-one onto some \(n\)-dimensional neighborhood \(D(\sigma)\) of the surface \(\sigma\) by means of

\[ x=x(\gamma')+\nu(\gamma')\gamma_n=x(\gamma),\qquad \gamma=(\gamma',\gamma_n), \]

and, moreover, so that \(x=x(\gamma)\) has in \(\Delta(\sigma)\), and \(\gamma=\gamma(x)\) has in \(D(\sigma)\), bounded derivatives up to order \(l\), while these vector functions themselves are continuous in the above-indicated sense.

Let \(\delta>0\). Denote by \(\Omega_\delta\) the subdomain of the domain \(\Omega\) consisting of those points of the domain \(\Omega\) whose distances to its boundary are greater than \(\delta\).

We shall say that an \((n-1)\)-dimensional surface without boundary \(S\) of the space \(E_n\) is a surface of class \(R^{(l)}\) if it can be covered by a finite number of surfaces \(\sigma_i^{(0)}\) \((i=1,2,\ldots,q)\) of class \(K^{(l)}\) with the following properties: a) each point of \(S\) belongs to at least one of the \(\sigma_i^{(0)}\); b) each point of any one of the \(\sigma_i^{(0)}\) belongs to \(S\); c) for some

for \(\delta>0\), the parts \(\sigma_i^{(1)}\) of the surfaces \(\sigma_i^{(0)}\) lying in \(D_\delta(\sigma_i^{(0)})\) also possess properties a) and b).

The definition of the spaces \(W_p^{(l)}\) by means of parametrization is easily extended to surfaces \(S\in R^{(l)}\).

We can now say that Theorem 1 is valid for the case where \(\Omega\) is an infinite domain bounded by a surface of class \(R^{(l)}\). In the same case Theorem 2 will also be valid if it is further added that the continuity of the leading coefficients of the operators \(L\) and \(R_\mu\) is understood in the sense indicated above.

Leningrad State
Pedagogical Institute
named after A. I. Herzen

Received
2 VII 1958

CITED LITERATURE

1. L. N. Slobodetskii, V. M. Babich, DAN, 106, No. 4 (1956).
2. L. N. Slobodetskii, DAN, 118, No. 2 (1958).
3. L. N. Slobodetskii, DAN, 120, No. 2 (1958).
4. E. Gagliardo, Rend. Sem. Mat. di Padova, 27 (1957).