UDC 517.946.9

A NONLINEAR DIFFERENTIAL EQUATION WITH A POLYNOMIAL FUNCTIONAL GENERATING IT

Yu. D. Kashchenko

This article is devoted to the study of a boundary value problem of the first kind for one nonlinear second-order differential equation. Many works have been devoted to the theory of nonlinear equations; among them we mention the studies of M. I. Vishik [4] and O. A. Ladyzhenskaya [10]. A characteristic feature of the present article is that, for an equation with a generating polynomial non-quadratic functional, not only is the existence of a generalized solution established, but it is also proved that it is an ordinary solution in the sense of the generalized derivatives of S. L. Sobolev.

Consider the interior boundary value problem

\[ \sum_{i=1}^{n} \frac{\partial}{\partial x_i} \left\{ a_i(x)\left[\frac{\partial u}{\partial x_i}(x)\right]^{2p-1}\right\} = f(x), \tag{1} \]

\[ u|_{\Gamma}=\Phi(x), \tag{2} \]

where \(\Gamma\) is the boundary of the cube \(\Delta=\{x:0<x_i<1\}\), \(x=(x_1,\ldots,x_n)\), \(2\le n<+\infty\), \(p\ge 1\) is an integer.

Let us denote

\[ \mathscr{D}(F)=\int_{\Delta}\sum_{i=1}^{n} a_i \left(\frac{\partial F}{\partial x_i}\right)^{2p}\,dx, \]

\[ dx=dx_1\cdots dx_n,\qquad (F,f)=\int_{\Delta}Ff\,dx, \]

\[ K(F)=\mathscr{D}(F)+2p(F,f),\qquad \mathscr{D}(u,v)= \]

\[ =\int_{\Delta}\sum_{i=1}^{n} a_i \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \frac{\partial v}{\partial x_i}\,dx, \]

\[ K(u,v)=\mathscr{D}(u,v)+(f,v). \]

Derivatives are understood in the sense of S. L. Sobolev [1, 2]; measure and integral—in the sense of Lebesgue.

The coefficients \(a_i\) and the function \(f\) are such that \(a_i(x)\ge 0\);

\[ \sum_{i=1}^{n}\int_{\Delta} \left(\frac{\partial F}{\partial x_i}\right)^{2p} dx \le \tilde{c}\,\mathscr{D}(F) \le \tilde{\tilde{c}}\sum_{i=1}^{n}\int_{\Delta} \left(\frac{dF}{dx_i}\right)^{2p} dx; \]

\[ a_i,\quad \frac{\partial a_i}{\partial x_k}\le C_i'=\mathrm{const};\qquad f,\ \frac{\partial f}{\partial x_i}\in L_{\frac{2p}{2p-1}}(\Delta). \]

Denote by \(\mathfrak M=\mathfrak M(\Phi)\) the class of functions for which the integral \(\mathscr D(F)\) is finite and the boundary conditions (2) are satisfied; denote by \(\mathfrak M_0\) the class of functions with the same properties, but with zero values on the boundary.

The boundary function \(\Phi\) is required to be such that the class \(\mathfrak M\) is nonempty\(^*\).

Theorem. In the class \(\mathfrak M\) there exists a unique function \(u\) satisfying simultaneously the following three conditions:

1)
\[ K(u)=\min_{F\in\mathfrak M} K(F); \tag{3} \]

2)
\[ K(u,v)=0 \qquad (v\in\mathfrak M_0); \tag{4} \]

3) \(u\) satisfies the differential equation (1).

In the class \(\mathfrak M\), each of the conditions 1)—3) implies the other two. Thus, for each of the conditions 1)—3), in the class \(\mathfrak M\) there exists only one function satisfying that condition.

The proof of the theorem will be obtained by means of the following assertions.

Lemma 1. There exists a unique function \(u\in\mathfrak M\) for which (3) holds.

We carry out the proof according to the usual scheme [5], using Clarkson’s inequality [1].

Lemma 2. If \(u\in\mathfrak M\) satisfies (3), then (4) is satisfied.

Proof.

\[ \frac{\partial}{\partial\lambda}K(u+\lambda v) = 2p\int_\Delta \sum_{i=1}^{n} a_i \left( \frac{\partial u}{\partial x_i} + \lambda\frac{\partial v}{\partial x_i} \right)^{2p-1} \frac{\partial v}{\partial x_i}\,dx + 2p\int_\Delta f v\,dx, \]

\[ \left. \frac{\partial}{\partial\lambda}K(u+\lambda v) \right|_{\lambda=0} = 2pK(u,v), \]

\[ \left. \frac{\partial}{\partial\lambda}K(u+\lambda v) \right|_{\lambda=0} = 0, \qquad K(u,v)=0. \]

Lemma 3. The function \(u\in\mathfrak M\), realizing \(\min_{F\in\mathfrak M}K(F)\), is the only function in the class \(\mathfrak M\) satisfying the equation in variations (3).

Proof. Let \(K(u^*,v)=0\).

Put
\[ v=u-u^*,\qquad \mu(\lambda)=K(u^*+\lambda v). \]

We have
\[ \mu'(\lambda) = 2p\int_\Delta \sum_{i=1}^{n} a_i \left( \frac{\partial u^*}{\partial x_i} + \lambda\frac{\partial v}{\partial x_i} \right)^{2p-1} \frac{\partial v}{\partial x_i}\,dx + 2p\int_\Delta f v\,dx, \]

\[ \mu''(\lambda) = 2p(p-1)\int_\Delta \sum_{i=1}^{n} a_i \left( \frac{\partial u^*}{\partial x_i} + \lambda\frac{\partial v}{\partial x_i} \right)^{2p-2} \left( \frac{\partial v}{\partial x_i} \right)^2 dx \ge 0, \]

\[ \mu'(0)=2pK(u^*,v)=0,\qquad \mu'(1)=0,\quad \text{since } \mu(1)=\min \mu(\lambda). \]

Consequently, \(\mu'(\lambda)=0\) on \([0,1]\), \(\mu(\lambda)\equiv\mathrm{const}\) on \([0,1]\), \(\mu(0)=\mu(1)\), \(K(u^*)=K(u)\), \(u^*=u\).

\[ \text{----------------} \]

\(^*\) In [3] it is shown: if a function has summable derivatives, then it possesses boundary values in the sense of limiting values almost everywhere.

The lemma is proved.

Denote

\[ x_\tau=(x_1+\tau,\ x_2,\ \ldots,\ x_n),\qquad \Delta\Psi=\Psi(x_h)-\Psi(x), \]

\[ \omega(\Psi)=\frac{\Delta\Psi}{h},\qquad A(\Psi)=\frac{\Delta(\Psi^{2p-1})}{\Delta\Psi}. \]

These notations remain valid for \(n=1\). By the oddness of \(2p-1\), one always has \(A(\Psi)>0\).

Let \(\sigma\) denote an arbitrary open ball in \(\Delta\) with positive distance \(\rho\) from the boundary \(\Gamma\). Put \(\delta=\dfrac{\rho}{3}\).

Lemma 4. Let \(u\in\mathfrak M\) satisfy (4). Then

\[ \int_\sigma \frac{\Delta\left[\left(\dfrac{\partial u}{\partial x_i}\right)^{2p-1}\right]}{h}\, \frac{\Delta\left(\dfrac{\partial u}{\partial x_i}\right)}{h}\,dx<C \qquad (i=1,\ldots,n), \tag{5} \]

where \(C\) does not depend on \(h\).

Proof. Denote by \(\Delta_\lambda\) the set of points of \(\Delta\) whose distance from \(\Gamma\) is greater than \(\lambda\).

Let \(\theta\) be a function such that \(\theta\equiv1\) on \(\Delta_{3\delta}\), \(\theta\equiv0\) outside \(\Delta_{2\delta}\), and

\[ \left|\frac{\partial\theta}{\partial x_i}\right|<C_1\sqrt{\theta} \qquad (i=1,\ldots,n)^* . \tag{6} \]

Put \(v=\omega(u)\theta\). Then from \(K(u,v)=0\) it is easy to obtain

\[ \int_{\Delta_{2\delta}} \sum_{i=1}^{n} \omega\left[ a_i\left(\frac{\partial u}{\partial x}\right)^{2p-1} \right] \frac{\partial v}{\partial x_i}\,dx + \int_{\Delta_{2\delta}} \omega(f)v\,dx=0 . \tag{7} \]

Remark. If there exists

\[ \int_D \left(\frac{\partial\Psi}{\partial x_1}\right)^m dx<+\infty, \]

then

\[ \int_D [\omega(\Psi)]^m dx \leq \int_D \left(\frac{\partial\Psi}{\partial x_1}\right)^m dx \quad [7]. \]

Denote \(g=g_\delta=\Delta_{2\delta}\). We have

\[ \left|\int_g \omega(f)v\,dx\right| = \left\{ \int_g [\omega(f)]^{\frac{2p}{2p-1}}\,dx \right\}^{\frac{2p-1}{2p}} \left(\int_g v^{2p}\,dx\right)^{\frac{1}{2p}} \leq \]

\[ \leq \left[ \int_g \left(\frac{\partial f}{\partial x_1}\right)^{\frac{2p}{2p-1}} dx \right]^{\frac{2p-1}{2p}} \left( \int_g \left(\frac{\partial u}{\partial x_1}\right)^{2p} dx \right)^{\frac{1}{2p}} <C_2 . \]

Next,

\[ \omega\left\{ a_i(x)\left[\frac{\partial u}{\partial x_i}(x)\right]^{2p-1} \right\} = a_i(x)\omega\left\{ \left[\frac{\partial u}{\partial x_i}(x)\right]^{2p-1} \right\} + \omega[a_i(x)] \left[\frac{\partial u}{\partial x_i}(x_h)\right]^{2p-1}, \]

\[ ^*)\ \text{See, for example, [6].} \]

\[ \frac{\partial v}{\partial x_i} = \omega\!\left(\frac{\partial u}{\partial x_i}\right)\theta + \omega(u)\frac{\partial\theta}{\partial x_i}. \]

Relation (7) can be rewritten as follows:

\[ \int_g \sum a_i\,\omega\!\left[\left(\frac{\partial u}{\partial x_i}\right)^{2p-1}\right] \left[ \omega\!\left(\frac{\partial u}{\partial x_i}\right)\theta + \omega(u)\frac{\partial\theta}{\partial x_i} \right]\,dx \le \]

\[ \le -\int_g \sum \omega(a_i) \left[ \frac{\partial u}{\partial x_i}(x_h) \right]^{2p-1} \frac{\partial v}{\partial x_i}\,dx + C_3, \]

i.e.

\[ \int_g \sum a_i A\!\left(\frac{\partial u}{\partial x_i}\right) \left[ \omega\!\left(\frac{\partial u}{\partial x_i}\right) \right]^2 \theta\,dx \le -\int_g \sum a_i A\!\left(\frac{\partial u}{\partial x_i}\right) \omega\!\left(\frac{\partial u}{\partial x_i}\right) \omega(u)\frac{\partial\theta}{\partial x_i}\,dx - \]

\[ - \int_g \sum \omega(a_i) \left[ \frac{\partial u}{\partial x_i}(x_h) \right]^{2p-1} \frac{\partial v}{\partial x_i}\,dx + C_3. \]

Or, denoting the last three integrals by \(J^2, J_1, J_2\), respectively, \(J^2\le -J_1-J_2+C_3\). (The integral on the left-hand side is nonnegative.) Obviously, \(|J_2|<C_4\).

We shall show that \(|J_1|<C_5J\). By Schwarz’s inequality,

\[ \left| \int_g a_i A\!\left(\frac{\partial u}{\partial x_i}\right) \omega\!\left(\frac{\partial u}{\partial x_i}\right) \omega(u)\frac{\partial\theta}{\partial x_i}\,dx \right| \le C_i \left\{ \int_g a_i A\!\left(\frac{\partial u}{\partial x_i}\right) \left[ \omega\!\left(\frac{\partial u}{\partial x_i}\right) \right]^2 \theta\,dx \right\}^{\frac12} \times \]

\[ \times \left\{ \int_g a_i A\!\left(\frac{\partial u}{\partial x_i}\right) [\omega(u)]^2\,dx \right\}^{\frac12}. \]

Applying Hölder’s inequality, we have

\[ \int_g a_i A\!\left(\frac{\partial u}{\partial x_i}\right) [\omega(u)]^2\,dx \le C_i' \left\{ \int_g \left[ A\!\left(\frac{\partial u}{\partial x_i}\right) \right]^{\frac{p}{p-1}}\,dx \right\}^{\frac{p-1}{p}} \times \]

\[ \times \left\{ \int_g [\omega(u)]^2\,dx \right\}^{\frac1p} \le C_5^{(i)}. \]

Thus, \(|J_1|<C_5J\). It is now clear that \(J^2\le C_5J+C_6\), whence \(J^2<C<+\infty\). Passing to the domain \(\sigma\), we thereby eliminate the factor \(\theta\):

\[ \int_\sigma \sum A\!\left(\frac{\partial u}{\partial x_i}\right) \left[ \omega\!\left(\frac{\partial u}{\partial x_i}\right) \right]^2\,dx < C. \]

Since \(A\ge 0\), it follows that

\[ \int_\sigma A\!\left(\frac{\partial u}{\partial x_i}\right) \left[ \omega\!\left(\frac{\partial u}{\partial x_i}\right) \right]^2\,dx < C \qquad (i=1,\ldots,n). \]

Lemma 5. If

\[ \varphi\in L_{2p}(0,1),\qquad \sup_{\substack{vrai\\0<x<1}}|\varphi(x)|=+\infty, \]

then

\[ \lim_{h\to +0}\int_0^{1-h}\frac{\Delta(\varphi^{2p-1})}{h}\,\frac{\Delta\varphi}{h}\,dx=+\infty . \]

Proof. Without loss of generality in the reasoning, one may assume that \(\varphi\geqslant 0\). Indeed,

\[ A(\varphi)[\omega(\varphi)]^2 =|\omega(\varphi^{2p-1})|\,|\omega(\varphi^{2p-1})|\geqslant \]

\[ \geqslant |\omega(|\varphi|^{2p-1})|\,|\omega(|\varphi|)| = A(|\varphi|)\,[\omega(|\varphi|)]^2, \]

i.e.

\[ A(\varphi)[\omega(\varphi)]^2\geqslant A(|\varphi|)\,[\omega(|\varphi|)]^2 . \]

Consequently, if the lemma is true for nonnegative functions and, in particular, for \(|\varphi|\), then it is true also without the requirement that the function \(\varphi\) be nonnegative.

Thus, let \(\varphi\geqslant 0\). Then it is enough to show that

\[ X=\int_0^{1-h}|\omega(\varphi)|\,\varphi^{p-1}\,dx\to 0 \qquad (h\to +0)^{*}. \]

Let

\[ \operatorname{supvrai}_{0<x<\frac12}\varphi(x)=+\infty^{**}, \]

and let \(N\) be arbitrary, but so large that

\[ \operatorname{mes}\left\{x:\frac12<x<1;\ \varphi(x)<N\right\}> \frac45\,\operatorname{mes}\left(\frac12,1\right). \]

Denote

\[ E=\left\{x:0<x<\frac12;\ \varphi(x)>8N\right\}. \]

\[ {}^{*})\ \text{In fact, the integral }X\text{ is taken over the set }(0,1-h), \text{ contained in the set of finite measure }(0,1). \]
Therefore, if \(X\to+\infty\), then also

\[ \int_0^{1-h}\Psi\,dx\to+\infty \]

\[ (h\to+0), \]
where

\[ \Psi= \begin{cases} |\omega(\varphi)|\varphi^{p-1}, & \text{if } |\omega(\varphi)|\varphi^{2p-1}\geqslant 1,\\ 0, & \text{if } |\omega(\varphi)|\varphi^{2p-1}<1. \end{cases} \]

We have

\[ \Psi\leqslant \Psi^2\leqslant \varphi^{2p-2}[\omega(\varphi)]^2 \leqslant A(\varphi)[\omega(\varphi)]^2. \]

\[ {}^{**})\ \text{If}\qquad \operatorname{supvrai}_{0<x<\frac12}\varphi(x)<+\infty, \]

then

\[ \operatorname{supvrai}_{\frac12<x<1}\varphi(x)=+\infty \]

and, instead of \(\varphi\), one may consider the function \(\Psi(x)=\varphi(1-x)\), for which

\[ \operatorname{supvrai}_{0<x<\frac12}\Psi(x)=+\infty \]

and the value of the integral \(X\) is the same as for \(\varphi\).

Take an open set \(\Omega\) such that \(\left(0,\frac{1}{2}\right)\supset \Omega \supset E\) and \(\operatorname{mes} E > \frac{4}{5}\operatorname{mes}\Omega\).

Partition \(\Omega\) into pairwise disjoint intervals \(\Omega_k\). Among the intervals \(\Omega_k\) there is an \(\Omega_{k_0}\) such that

\[ \operatorname{mes}(\Omega_{k_0}\cap E)>\frac{4}{5}\operatorname{mes}\Omega_{k_0}. \]

Fix an arbitrary \(h\) such that \(0<h<\frac{\operatorname{mes}\Omega_{k_0}}{6}\); we shall show that for it \(X>N\).

Among the intervals \((ih,(i+1)h)\) (\(i\geq 0\) an integer) there will be\(^*\) intervals

\[ (\alpha,\beta)\subset\left(0,\frac{1}{2}+h\right), \]

\[ (\gamma,\delta)\subset\left(\frac{1}{2}-h,1+h\right)^{**}, \]

such that

\[ \alpha<\beta\leq\gamma<\delta,\qquad \operatorname{mes}[\alpha,\beta)\cap E]>\frac{3}{5}h, \]

\[ \operatorname{mes}\{x:\gamma\leq x<\delta,\ \varphi(x)<N\}>\frac{3}{5}h. \]

Put

\[ \varphi_1(x)= \begin{cases} 8N, & \varphi(x)>8N,\\ \varphi(x), & 1\leq \varphi(x)\leq 8N,\\ 1, & \varphi(x)<1. \end{cases} \]

We have

\[ X\geq \int_{\alpha}^{\gamma}\frac{\varphi_1(x)-\varphi_1(x+h)}{h}\,dx^{***}; \]

\(^*\) Indeed. Denote \(m_0=\operatorname{mes}\Omega_{k_0}\). Let \(g_k\) be those of the \((ih,(i+1)h)\) which intersect \(\Omega_{k_0}\).

Denote \(G=\bigcup_k g_k\). We have

\[ \frac{\operatorname{mes}(G\cap E)}{\operatorname{mes}G} \geq \frac{\operatorname{mes}(\Omega_{k_0}\cap E)}{\operatorname{mes}G} > \frac{\frac{4}{5}m_0}{m_0+2\frac{m_0}{6}} = \frac{3}{5}. \]

Consequently, the indicated \((\alpha,\beta)\) exists. The existence of the indicated \((\gamma,\delta)\) is proved similarly.

\(^ {**}\) If \((\gamma,\delta)\) goes beyond \((0,1)\), extend \(\varphi\), setting at points of \((\gamma,\delta)\setminus(0,1)\)
\(\varphi\equiv 8N\).

\(^ {***}\) Indeed. Denote

\[ Y=|\varphi(x)-\varphi(x+h)|\,|\varphi(x)|^{p-1},\qquad Z=\varphi_1(x)-\varphi_1(x+h). \]

We shall show that for \(x\in(0,1-h)\), \(Z\leq Y\). We have

\[ Y\geq0,\qquad |\varphi(x)-\varphi(x+h)|\geq |\varphi_1(x)-\varphi_1(x+h)|. \]

Let \(\varphi(x)\leq \varphi(x+h)\); then \(Z\leq0\).

\[ \int_\alpha^\gamma \frac{\varphi_1(x)-\varphi_1(x+h)}{h}\,dx = \frac1h\left[ \int_\alpha^\beta \varphi_1(x)\,dx - \int_\gamma^\delta \varphi_1(x)\,dx \right]\ge \]

\[ \ge \frac1h\left\{ h\frac35\,8N - h\left(\frac35\,N+\frac25\,8N\right) \right\}=N,\qquad X\ge N. \]

The lemma is proved.

Denote
\[ y^{(j)}=(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n),\quad \sigma_j=\sigma_{x_j}(y^{(j)}),\quad y=y^{(1)}. \]

Lemma 6. Let \(u\in\mathfrak M\) satisfy (4). Then for almost all \(y^{(j)}\in\sigma_{y^{(j)}}\)

\[ \operatorname{supvrai}_{x_j\in\sigma_j}\left|\frac{\partial u}{\partial x_i}\right|<+\infty \qquad (i,j=1,\ldots,n). \tag{8} \]

Proof. Choose \(j=1\), fix some \(i\), and carry out the proof by contradiction.

Let \(\mathcal E\subset\sigma_y\) be the set of those \(y\)’s for which (8) is not satisfied, and let \(m=\operatorname{mes}\mathcal E>0\). By Lemma 5,

\[ \Psi_n=\int_{\sigma_1} A\left(\frac{\partial u}{\partial x_1}\right) \omega\left[\left(\frac{\partial u}{\partial x_i}\right)^2\right]\,dx_1\to+\infty \]

for \(y\in\mathcal E\), \(h=\frac1n\), \(n\to+\infty\), and, consequently, \(\Psi_n\to+\infty\) at least on \(\mathcal E\).

Thus, for any \(0<\mathcal K<+\infty\) there exists \(n_0\) such that for \(n>n_0\), \(\Psi_n(y)>\mathcal K\) on some set \(\widetilde{\mathcal E}\subset\mathcal E\), \(\operatorname{mes}\widetilde{\mathcal E}>\frac m2\); whence

\[ \int_\sigma A\left(\frac{\partial u}{\partial x_1}\right) \left[ \omega\left(\frac{\partial u}{\partial x_i}\right)^2 \right]\,dx > \frac{\mathcal K\cdot m}{2} \]

for \(h=\frac1n\), \(n>n_0\), which contradicts (5).

Lemma 7. Let \(u\in\mathfrak M\) satisfy the variational equation (4). Then for almost all \(y^{(j)}\in\sigma_y^{(j)}\) there exist derivatives

\[ \frac{\partial}{\partial x_j} \left[\left(\frac{\partial u}{\partial x_i}\right)^{2p-1}\right] \]

\((x_j\in\sigma_j)\), and

\[ \int_{\sigma_j} \left\{ \frac{\partial}{\partial x_j} \left[\left(\frac{\partial u}{\partial x_i}\right)^{2p-1}\right] \right\}^2 dx_j<+\infty \qquad (i,j=1,\ldots,n). \]

Proof. Choose \(j=1\), fix \(i\in[1,n]\). Fix \(0<M<+\infty\). Inequality (5) remains valid if, instead of \(\dfrac{\partial u}{\partial x_i}\), one substitutes into it

Let \(\varphi(x)>\varphi(x+h)\) and \(\varphi(x)<1\); then \(Z=0\).

Let \(\varphi\gg1\); then

\[ Y\ge |\varphi(x)-\varphi(x+h)|\ge |\varphi_1(x)-\varphi_1(x+h)| = \varphi_1(x)-\varphi_1(x+h)=Z, \]

i.e. \(Z\le Y\). If \(\gamma>1-h\), \(\delta>1\), then for \(1-h<x<\gamma\) we have \(x+h>1\), \(\varphi(x+h)=8N\), \(\varphi_1(x)\le \varphi_1(x+h)=8N\), \(z\le0\).

\[ \Psi = \begin{cases} \dfrac{\partial u}{\partial x_i}(x), & \left|\dfrac{\partial u}{\partial x_i}(x)\right| \le M, \\[1.2ex] M \cdot \operatorname{sign}\dfrac{\partial u}{\partial x_i}(x), & \dfrac{\partial u}{\partial x_i}(x) > M. \end{cases} \tag{*} \]

After multiplication by the bounded nonnegative function \(A(\Psi)\) we obtain

\[ \int_{\sigma}\left[\omega\left(\Psi^{2p-1}\right)\right]^2\,dx \le C^* \]

(\(C^* = C(2p-2)M^{2p-2}\) does not depend on \(h\)). Hence it follows that the derivative with respect to \(x_1\) of the function \(\Psi^{2p-1}\) exists and is square-summable:

\[ \int_{\sigma}\left[\frac{\partial}{\partial x_1}\left(\Psi^{2p-1}\right)\right]^2\,dx < +\infty \quad [8,9]. \]

For any \(y \in \sigma_y\) (possibly except for a set of \((n-1)\)-dimensional measure zero) there is an \(M\) such that, almost everywhere on \(x_1 \in \sigma_1\),

\[ \frac{\partial u}{\partial x_i}(x_1,y) \equiv \Psi(x_1,y). \]

Hence, and by Lebesgue’s theorem on repeated integration, it follows that for almost all \(y \in \sigma_y\) there exists

\[ \int_{\sigma_1} \left\{ \frac{\partial}{\partial x_1} \left[ \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \right] \right\}^2 \,dx_1 < +\infty . \]

Lemma 8. The function \(u \in \mathfrak M\), satisfying (4), satisfies (1).

Proof. Let \(v \in \mathfrak M_0\) and \(v \equiv 0\) outside the domain \(\sigma\). Since an arbitrary ball lying strictly in \(\Delta\) is taken, Lemma 7 remains valid if \(\Delta_{2\delta} \supset \sigma\) is taken instead of \(\sigma\); consequently, over the domain \(\sigma\) one may integrate by parts the expressions containing

\[ \left(\frac{\partial u}{\partial x_i}\right)^{2p-1}. \]

Fix \(i \in [1,n]\). Denote

\[ \Psi = \Psi_i = a_i\left(\frac{\partial u}{\partial x_i}\right)^{2p-1}. \]

We have

\[ \int_{\sigma}\Psi\,\frac{\partial v}{\partial x_i}\,dx = \int_{\sigma_y^{(i)}}dy^{(i)} \int_{\sigma_i}\Psi\,\frac{\partial v}{\partial x_i}\,dx_i, \]

\[ {}^{*}\ \text{Indeed,} \]

\[ A\left(\frac{\partial u}{\partial x_i}\right) \left[ \omega\left(\frac{\partial u}{\partial x_i}\right) \right]^2 = \left| \omega\left[ \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \right] \right| \cdot \left| \omega\left(\frac{\partial u}{\partial x_i}\right) \right| \ge \]

\[ \ge \left| \omega\left(\Psi^{2p-1}\right) \right| \cdot \left| \omega(\Psi) \right| = A(\Psi)\,[\omega(\Psi)]^2 . \]

\[ \int_{\sigma_i} \Psi \frac{\partial v}{\partial x_i}\, dx_i = \int_{\sigma_i} \frac{\partial \Psi}{\partial x_i} v\,dx_i + [\Psi v]_{x_i=a}^{x_i=b}, \]

where \(a=a(y^{(i)})<b=b(y^{(i)})\) are the values of the coordinate \(x_i\) for the points of intersection of the line parallel to the axis \(Ox_i\) with the boundary of the domain \(\sigma\). Thus,

\[ \int_{\sigma} \Psi \frac{\partial v}{\partial x_i}\, dx = - \int_{\sigma} \frac{\partial \Psi}{\partial x_i} v\,dx . \]

We have

\[ \int_{\sigma} \sum \frac{\partial}{\partial x_i} \left[ a_i \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \right] v\,dx = - \int_{\sigma} \sum a_i \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \frac{\partial v}{\partial x_i}\,dx, \]

\[ - \int_{\sigma} \sum a_i \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \frac{\partial v}{\partial x_i}\,dx = \int_{\sigma} f v\,dx, \]

\[ \int_{\sigma} \sum \frac{\partial}{\partial x_i} \left[ a_i \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \right] v\,dx = \int_{\sigma} f v\,dx. \]

By the fundamental lemma of the calculus of variations,

\[ \sum \frac{\partial}{\partial x_i} \left[ a_i \left(\frac{\partial u}{\partial x_i}\right)^{2p-1} \right] = f \]

(for \(x\in\sigma\), and consequently also for \(x\in\Delta\), since the ball \(\sigma\) is arbitrary).

Lemma 9. Let \(u\in\mathfrak{M}\) satisfy (1). Then \(u\) satisfies (4).

The proof is carried out according to the usual scheme [6].

Now the theorem can be obtained by a simple combination of Lemmas 1—3, 8, 9.

Remark. Our results will apparently remain valid (without essential changes in the proofs) if, instead of the parallelepiped \(\Delta\), one takes an arbitrary bounded domain with piecewise smooth boundary.

The author expresses deep gratitude to S. M. Nikol’skii for his attention to this work.

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Received by the editors
25 July 1966

Moscow Institute of Physics and Technology