UDC 517.945.9

MATHEMATICS

M. D. IVANOVICH

ESTIMATES OF SOLUTIONS OF GENERAL BOUNDARY-VALUE PROBLEMS FOR A HIGHER-ORDER ELLIPTIC EQUATION IN THE SPACES \(C_{l+\alpha(r)}\)

(Presented by Academician I. G. Petrovskii on 28 XII 1966)

In [1] sharp estimates were obtained for the derivatives of solutions of the Dirichlet problem for a second-order elliptic equation, under the condition that the coefficients and the right-hand side of the equation, as well as the second derivatives of the boundary function, had a modulus of continuity not exceeding a certain function \(\omega(r)\) satisfying Dini’s condition. It was proved that the second derivatives of the solution have a modulus of continuity not exceeding

\[ C\int_0^r \frac{\omega(t)}{t}\,dt . \]

On fundamental solutions see [4, 5].

Theorems 1 and 4 contain sharp interior estimates for solutions of elliptic and parabolic systems.

In Theorem 2, estimates are obtained for solutions of general boundary-value problems in a half-ball, on the flat part of whose boundary general boundary conditions are prescribed. With the aid of Theorems 1 and 2 we shall obtain estimates for solutions of general boundary-value problems for an arbitrary domain with sufficiently smooth boundary. In Hölder norms these estimates were obtained in [2, 3].

Let us introduce the following definitions: we shall say that a function \(f(x)\) satisfies a uniform Hölder condition in the domain \(D\) with refined exponent \(\alpha(r)\) if, for any points \(x,x'\in D\),

\[ |f(x)-f(x')|\le C|x-x'|^{\alpha(|x-x'|)}; \]

the function \(f(x)\) satisfies a local Hölder condition with refined exponent \(\alpha(r)\) if it satisfies a uniform Hölder condition with refined exponent \(\alpha(r)\) in every subdomain \(D'\) of the domain \(D\) that is strictly interior with respect to \(D\), where the function \(\alpha(r)\) is defined and continuous for \(0<r<\infty\) and satisfies the following conditions: \(\alpha(r)\to \lambda\), \(\lambda\in[0,1)\), \(\alpha'(r)r\ln r\to 0\) both as \(r\to0\) and as \(r\to\infty\), and, if \(\lambda=0\), then \(\alpha(r)\ln r\to -\infty\) as \(r\to0\), and \(\alpha(r)+r\ln r\cdot \alpha'(r)>0\) for \(r\in R_0=(0,r_0)+(1/r_0,\infty)\), where \(r_0\) is some sufficiently small number. (It is assumed that \(\alpha'(r)\) exists and is continuous, at least for sufficiently small and sufficiently large \(r\).) From these conditions it follows that the function \(r^{\alpha(r)}\) increases monotonically for \(r\in R_0\).

The function \(r^{\alpha(r)-\lambda}\) is slowly varying, i.e.

\[ (kr)^{\alpha(kr)-\lambda}/r^{\alpha(r)-\lambda}\to 1 \]

as \(r\to0\) and as \(r\to\infty\), uniformly with respect to \(k\), \(0<a\le k\le b<\infty\).

We denote by \(C^{\alpha(r)}(D)\) the class of functions satisfying a local Hölder condition with refined exponent \(\alpha(r)\) in the domain \(D\). By \(C^{l+\alpha(r)}(D)\) we denote the class of functions for which \(D^l f(x)\in C^{\alpha(r)}(D)\).

By \(C_{l+\alpha(r)}(D)\) we denote the space of functions for which the norm is finite:

\[ |f|_{l+\alpha(r)}^{D} = \sum_{k=0}^{l} \sup_{\substack{x\in D\\ |j|=k}} |D^j f(x)| + \sup_{\substack{x,x'\in D\\ |j|=l}} \frac{|D^j f(x)-D^j f(x')|} {|x-x'|^{\alpha(|x-x'|)}} \equiv \sum_{k=0}^{l} [f]_k + [f]_{l+\alpha(r)} . \]

By \(C_{p+\beta(r),\,l+\alpha(r)}(D)\) we mean the space of functions for which the norm

\[ |f|_{p+\beta(r),\,l+\alpha(r)}^{D} = \sum_{k=0}^{l}\sup_{\substack{x\in D\\ |j|=k}} d_x^{p+k}|D^j f(x)| + \]

\[ + \sup_{\substack{x,x'\in D\\ |j|=l}} d_{xx'}^{p+l+\beta(d_{xx'})+\alpha(d_{xx'})} \frac{|D^j f(x)-D^j f(x')|} {|x-x'|^{\alpha(|x-x'|)}} \equiv \sum_{k=0}^{l}[f]_{p,k} + [f]_{p+\beta(r),\,l+\alpha(r)}, \]

is finite, where \(d_x\) is the distance from the point \(x\) to the boundary of the domain \(D\) and \(d_{xx'}=\min(d_x,d_{x'})\), \(p+l\ge 0\), \(p\) is an integer, \(\beta(r)\) is some function, \(\alpha(r)+\beta(r)\ge 0\).

By \(\mathcal A_l\) we denote the class of refined exponents \(\alpha(r)\) for which \(A^l\alpha(r)<\infty\), where

\[ A\alpha(r)=\int_0^r r^{\alpha(r)-1}\,dr,\qquad l=1,2,\ldots \]

Let \(\alpha(r)\in \mathcal A_1\) and let

\[ B\alpha(r)=\frac{1}{\ln r}\ln\frac{A\alpha(r)}{A\alpha(1)}. \]

Theorem 1. Let the functions \(u_j(x)\in C^t_j(D)\) satisfy the elliptic system in the sense of Douglis--Nirenberg

\[ \sum_{j=1}^{N} l_{ij}(x,D)u_j(x)=f_i(x),\qquad i=1,2,\ldots,N, \tag{1} \]

\[ l_{ij}(x,D)=\sum_{|p|\le s_i+t_j} a_{ij,p}(x)D^p,\qquad s_i,\ t_i\text{ are integers},\quad \max s_i=0. \]

Suppose that

\[ |a_{ij,p}|_{-\alpha(r),\,-s_i+l+\alpha(r)},\qquad |p|=s_i+t_j, \]

\[ |a_{ij,p}|_{s_i+t_j-|p|+B\alpha(r)-\alpha(r),\,-s_i+l+\alpha(r)},\qquad |p|<s_i+t_j, \]

are bounded by the constant \(K_1\); \(\alpha(r)\in\mathcal A_1\), \(l\ge 0\) is an integer. Then

\[ |u_j|_{t-t_j,\,t_j+l+B\alpha(r)} \le \]

\[ \le K_2\left( \sup_i |f_i|_{s_i+t+B\alpha(r)-\alpha(r),\,-s_i+l+\alpha(r)} + \sum_{t_j>0}|u_j|_{t-t_j,\,0} \right), \qquad t=\max_i t_i , \tag{2} \]

where \(K_2\) depends on \(K_1\), \(n\), the ellipticity constant and on the function \(A\alpha(r)\), the domain \(D\), \(s_i\) and \(t_i\).

In the proof the following lemma is used.

Lemma 1. Let

\[ u(x)=\int_S D_y^{2m}\Gamma(x-y,a(x))(g(y)-g(x))h(y)\,dy, \]

where \(S=\{|x_0-x|\le d,\ d\le 1\}\), \(\Gamma(x,a(\xi))\) is the fundamental solution of the elliptic equation with constant coefficients

\[ \sum_{|k|=2m} a_k(\xi)D_x^k u(x)=0. \]

Let \(g(x)\in C_{\alpha(r)}(S)\), \(h(x)\in C_{\gamma(r)}(S)\), where \(\alpha(r)\in\mathcal A_1\),

\[ \gamma(r)=\frac{1}{\ln r}\ln\left(r^{B\alpha(r)}+r^{\alpha(r)}\ln\frac{1}{r}\right) \]

and let

\[ [a]_{\alpha(r)}^{S}=\sum_{|k|=2m}[a_k]_{\alpha(r)}^{S}<\infty. \]

Then the estimate holds

\[ |u(x_0)-u(x)| \le C[g]_{\alpha(r)}^{S}\bigl(1+[a]_{\alpha(r)}^{S}\bigr)|h|_{\gamma(r)}^{S} |x_0-x|^{B\alpha(|x_0-x|)} \]

where the constant \(C\) depends on \(n\), \(A\alpha(r)\), and the ellipticity constant.

Let now \(\Sigma_R\) be a half-ball in the space \(E_{n+1}\), \(\Sigma_R=\{|x|<R,\ x_{n+1}\geqslant 0\}\), \(R\leqslant 1\). Consider in \(\Sigma_R\) the problem

\[ L(x,D)u(x)=f(x) \tag{3} \]

\[ B_j(x',D)u(x)\big|_{x_{n+1}=0}=\varphi_j(x),\qquad x'=(x_1,x_2,\ldots,x_n),\qquad j=1,\ldots,m, \]

and suppose: 1) the operators \(L\) and \(B_j\) satisfy conditions i) and ii) of [3], p. 70 (the leading coefficients of \(L\) do not depend on \(x_{n+1}\) if \(\lambda=0\)); 2) the coefficients of the operators \(L\) and \(B_j\) are bounded by the constant \(K_3\) in the norms \(|\cdot|_{l-2m+\alpha(r)}\) and \(|\cdot|_{l-m_j+B\alpha(r)}\), \(l\geqslant l_0=\max(2m,m_j)\), respectively, \(\alpha(r)\in A_4\).

Theorem 2. Let the function \(u(x)\in C^{l_0+B^2\alpha(r)}(\Sigma_R)\) be a solution of problem (3). Let assumptions 1), 2) be fulfilled. Suppose that the norms
\[ |\bar f|_{2m+B^2\alpha(r)-\alpha(r),\,l-2m+\alpha(r)},\quad |\varphi_j|_{m_j+B^2\alpha(r)-B\alpha(r),\,l-m_j+B\alpha(r)} \]
and \(|u|_0\) are finite, \(\alpha(r)\in A_4\). Then the estimate holds

\[ |\bar u|_{0,\,l+B^2\alpha(r)} \leqslant K_4\left( |\bar f|_{2m+B^2\alpha(r)-\alpha(r),\,l-2m+\alpha(r)} +\sum_{j=1}^{m}|\varphi_j|_{m_j+B^2\alpha(r)-B\alpha(r),\,l-m_j+B\alpha(r)} +|u|_0\right), \tag{4} \]

where \(K_4\) depends on \(n\), on the function \(A^4\alpha(r)\), \(K_3\), and on the constants entering condition 1). (The bar over the norm means that the distance entering its definition is measured only up to the curvilinear boundary \(\Sigma_R\).)

Estimate (4) is proved from a certain representation for the derivatives \(D^l u(x)\) and with the aid of Lemma 2.

Lemma 2. Let
\[ u(x)=\int K(x'-y',x_{n+1})f(y')\,dy',\qquad x_{n+1}>0, \]
where
\[ K(x)=W(x/|x|)|x|^{-n}, \]
\(W(Q)\) is a continuously differentiable function on the half-sphere \(|Q|=1,\ x_{n+1}\geqslant 0\), and
\[ \int_{|\xi|=1} W(\xi',0)\,d\Omega_\xi=0. \]
Let \(f(x')\in C_{\alpha(r)}(E_n)\), \(\alpha(r)\in A_1\), and let \(f(x')\in L_p\) for some finite \(p\geqslant 1\). Then
\[ [u]_{B\alpha(r)}\leqslant C[f]_{\alpha(r)}, \]
where \(C\) depends on \(n\), \(\max |W|\), \(\max |DW|\), and \(A\alpha(r)\).

Lemma 3. Let
\[ u(x)=\int K(x'-y',x_{n+1}) \int D^{2m}\Gamma(y'-\eta',-\eta_{n+1})(g(\eta)-g(x_0',0))h(\eta)\,d\eta\,dy', \]
\[ x_{n+1}>0, \]
where the kernel \(K(x)\) satisfies the conditions of Lemma 2, while \(g(x)\in C_{\alpha(r)}(E_{n+1})\), \(h(x)\in C_{B^2\alpha(r)}(E_{n+1})\) and is finite, and the function \(\Gamma(x)\) is the fundamental solution of an elliptic equation with constant coefficients
\[ \sum_{|k|=2m} a_k D^k u(x)=0,\qquad \alpha(r)\in A_4. \]
Then
\[ |u(x_0)-u(x)| \leqslant C[g]_{\alpha(r)} \left(|h|_0+|x_0-x|^{B^4\alpha(|x_0-x|)}[h]_{B^2\alpha(r)}\right) |x_0-x|^{B^2\alpha(|x_0-x|)}, \]
where the constant \(C\) depends on \(n\), \(\max |W|\), \(\max |DW|\), and on the function \(A^4\alpha(r)\) and the constant of ellipticity.

Let now \(D\) be an arbitrary domain, which may be unbounded, in the space \(E_{n+1}\). We shall denote its boundary by \(\partial D\). Consider in the domain \(D\) the problem
\[ Lu(x)=f(x),\qquad B_j u(x)\big|_\Gamma=\varphi_j(x),\qquad j=1,\ldots,m, \tag{5} \]
where \(\Gamma\) is a part of the boundary \(\partial D\), which may coincide with the entire boundary of the domain \(D\); \(L\) and \(B_j\) are differential operators of orders \(2m\) and \(m_j\). Put \(l_0=\max(2m,m_j)\) and let \(l\geqslant l_0\) be some integer and \(\alpha(r)\in A_4\).

Consider a subdomain (possibly unbounded) \(U\) of the domain \(D\), such that \(\partial U\cap\partial D\subset \Gamma\). We shall assume that \(\Gamma\in C^{l+\alpha(r)}\). Suppose that for \(U\) and \(D\) the conditions of [3], p. 76, are fulfilled, and it is assumed that each component of the mapping \(T_P\) (see [3], p. 76) and its inverse has norm \(|\cdot|_{l+\alpha(r)}\) bounded by the constant \(K_5\), independent of \(P\). With respect to the equation and the boundary conditions we shall assume that, under any such mapping \(T_P\), they pass into system (3) in the half-ball \(\Sigma_{R(P)}\), satisfying conditions 1), 2). Пред-

assume that the coefficients of the operators \(L\) and \(B_j\) have, respectively in the domain \(D\), finite \(K_6\)-norms \(|\cdot|_{l-2m+\alpha(r)}\) and \(|\cdot|_{l-m_j+B\alpha(r)}\), and that the operator \(L\) is uniformly elliptic in this domain.

Let \(|f|^{D}_{l-2m+\alpha(r)}\), \(|\varphi_j|^{\Gamma}_{l-m_j+B\alpha(r)}\), and \(|u|^{D}_{0}\) be finite, \(\alpha(r)\in\mathcal A_4\).

Theorem 3. Let the function \(u(x)\in C^{l+B^{2}\alpha(r)}\) in \(D+\Gamma\) be a solution of problem (5). Then

\[ u(x)\in C^{l+B^{2}\alpha(r)} \quad \text{in } \overline U, \]

\[ |u|^{U}_{l+B^{2}\alpha(r)} \le K_7\left( |f|^{D}_{l-2m+\alpha(r)} + \sum |\varphi_j|^{\Gamma}_{l-m_j+B\alpha(r)} + |u|^{D}_{0} \right), \]

where the constant \(K_7\) depends on \(d\), \(K_5\), \(m\), \(m_j\), \(K_6\), on the function \(A^{4}\alpha(r)\), and on the constants entering condition 1).

If the domain \(D\) is bounded, then in Theorem 3 one may take \(U=D\) and \(\Gamma=\partial D\). In this case the normal solvability of problem (5) follows from Theorem 3.

For \(\alpha(r)\in\mathcal A_1\), interior estimates have been obtained for parabolic systems in the sense of Petrovsky (for second order, see \((^{6})\)). For brevity we formulate the result for a single equation. Let \(D\) be a bounded domain in \((n+1)\)-dimensional space \((x_1,x_2,\ldots,x_n,t)\). Introduce the distance

\[ d(P,Q)=\sqrt{|x-\xi|^{2}+|t-t'|^{1/m}}. \tag{*} \]

Let

\[ |f|_{p+3(r),\alpha(r)} = \sup_{P\in D} d_P^{p}|f(P)| + \sup_{P,Q\in D} d_{PQ}^{p+\alpha(d_{PQ})+\ell(d_{PQ})} \frac{|f(P)-f(Q)|}{d(P,Q)^{\alpha(d(P,Q))}}, \]

where \(p\ge 0\) is an integer, \(d_P\) is the distance in the sense of \((*)\) from the point \(P(x,t)\) to the boundary of \(D\) lying in the half-space \(t<\tau\); \(d_{PQ}=\min(d_P,d_Q)\).

Theorem 4. Let \(u(x,t)\) have continuous derivatives with respect to \(x\) up to order \(2m\) and a continuous derivative with respect to \(t\), and satisfy the parabolic equation

\[ \sum_{|k|\le 2m} a_k(x,t)D_x^k u(x,t)-D_tu(x,t)=f(x,t). \]

Suppose that \(|a_k|_{-\alpha(r),\alpha(r)}\), \(|k|=2m\), and \(|a_k|_{2m-|k|+B\alpha(r)-\alpha(r),\alpha(r)}\), \(|k|<2m\), are bounded by the constant \(\widetilde K\). Suppose that \(|f|_{2m+B\alpha(r)-\alpha(r),\alpha(r)}<\infty\), \(\alpha(r)\in\mathcal A_1\). Then

\[ |u|_{0,\,2m+B\alpha(r)} \le K\left( |f|_{2m+B\alpha(r)-\alpha(r),\alpha(r)} + |u|_{0} \right), \]

where

\[ |u|_{0,\,2m+B\alpha(r)} = \sum_{|k|<2m}\sup_{P\in D} d_P^{|k|}\,|D_x^k u(P)| + \sum_{|k|=2m}|D_x^k u|_{2m,B\alpha(r)} + |D_tu|_{2m,B\alpha(r)} \]

and the constant \(K\) depends on the constant of parabolicity, \(\widetilde K\), \(n\), the diameter of the domain \(D\), and the function \(\alpha(r)\).

For \(\alpha(r)\in\mathcal A_2\), Schauder-type estimates have been obtained for solutions of the first boundary-value problem for parabolic systems in the sense of Petrovsky. For \(\alpha(r)=\lambda\), \(\lambda\in(0,1)\), these results were obtained in papers \((^{7,8})\).

I take this opportunity to express my sincere gratitude to my scientific adviser T. D. Venttsel.

Moscow State University
named after M. V. Lomonosov

Received
21 XII 1966

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