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On the Classical Solution of Mixed Problems for Nonstationary Equations
N. I. Brish
1. Consider, in the cylinder \(Q=\Omega\times(0,T)\), where \(\Omega\) is a bounded domain of the space \(x=(x_1,\ldots,x_n)\) with boundary \(S\), the following mixed problem:
\[ -\frac{\partial^2 u}{\partial t^2}+Au=f(x,t), \tag{1} \]
\[ u\big|_{t=0}=\varphi(x),\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=\psi(x), \tag{2} \]
\[ \frac{\partial^k u}{\partial \nu^k}\bigg|_{\Gamma}=0,\qquad k=0,1,\ldots,m-1,\quad \Gamma=S\times[0,T]. \tag{3} \]
Here \(Au\) is a formally self-adjoint, positive elliptic differential expression in \(\overline{\Omega}\) of the form
\[ Au=\sum_{|\alpha|=|\beta|\le m}(-1)^{|\alpha|}D^\alpha\bigl(a_{\alpha\beta}(x)D^\beta u\bigr), \tag{4} \]
where \(\alpha=(\alpha_1,\ldots,\alpha_n)\), \(|\alpha|=\alpha_1+\cdots+\alpha_n\), \(D^\alpha=\partial^{|\alpha|}/\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}\), and the \(a_{\alpha\beta}\) are real functions in \(\overline{\Omega}\), symmetric with respect to permutation of the indices; \(\nu\) is the outward normal to \(S\).
In the paper [1] the solvability of problem (1)—(3) was proved by the Fourier method in the Sobolev spaces \(W_{x,t,2}^{km,k}(Q)\) of functions \(u(x,t)\) with norm
\[ \|u\|_{km,k}^{2} = \sum_{|\alpha|+ms\le km} \left\|D^\alpha \partial^s u/\partial t^s\right\|_{L_2(Q)}^{2}. \tag{5} \]
From this, with the aid of embedding theorems, one obtains as a special case, for
\[
k=\left[\frac{n}{2m}\right]+r+1,
\]
a solution of the mixed problem (1)—(3) in the spaces \(C_{x,t}^{mr,r}(\overline{Q})\) of functions \(u(x,t)\) with norm
\[ |u|_{mr,r} = \sum_{|\alpha|+ms\le mr} \left|D^\alpha \partial^s u/\partial t^s\right|_{C(\overline{Q})} \tag{6} \]
and, consequently, a classical solution, i.e. a solution in the space \(C_{x,t}^{2m,2}(\overline{Q})\). However, such a path for proving the existence of a classical ...
…ical solution requires high smoothness of the boundary surface \(S\). For this it is in fact necessary that the surface \(S\) be \(\left(\left[\dfrac{n}{2m}\right]+3\right)m+1\) times continuously differentiable [1]*.
In the present article we shall show that such severe restrictions on the smoothness of \(S\) are not imposed by the essence of the matter. We shall give a justification of the Fourier method for problem (1)—(3) directly from the point of view of classical solutions under the condition that the surface \(S\) is \(2m+1\) times continuously differentiable. We note that the results of V. A. Il’in [2], concerning the case \(m=1\), are stronger. In the general case analogous results have not yet been established.
- We pass to the solution of the problem posed. In doing so we shall use a number of notions and assertions, connected with the question under study, which are contained in [1].
Suppose that for any function \(u\in C_0^\infty(\Omega)\) the condition
\[ (Au,u)\geq C_1\|u\|^2, \tag{7} \]
is fulfilled, where \(C_1>0\) and is independent of \(u\). This condition will be fulfilled if the coefficient \(a_{00}>0\) is sufficiently large. The latter condition can always be satisfied by making in problem (1)—(3) the substitution of the sought function \(u\) according to the formula \(u=v\exp(ct)\), where \(c\) is a sufficiently large positive constant. In this case the appearance of the term \(\dfrac{\partial v}{\partial t}\) will not cause any complications.
Let \(\{v_i\}\) be an orthonormal system of generalized eigenfunctions [1] of the equation
\[ Av=\lambda v \tag{8} \]
under the boundary conditions
\[ \left.\frac{\partial^k v}{\partial \nu^k}\right|_S=0,\qquad k=0,1,\ldots,m-1, \tag{9} \]
corresponding to the system of eigenvalues \(\{\lambda_i\}\), numbered in increasing order. Applying the method of separation of variables, we obtain the formal solution of the mixed problem (1)—(3) in the following form:
\[ u(x,t)=\sum_{i=1}^{\infty} v_i(x)\left(\varphi_i\cos\sqrt{\lambda_i}\,t+\frac{\psi_i}{\sqrt{\lambda_i}}\sin\sqrt{\lambda_i}\,t\right)+ \]
\[ +\sum_{i=1}^{\infty}\frac{v_i(x)}{\sqrt{\lambda_i}}\int_0^t f_i(\tau)\sin\sqrt{\lambda_i}(t-\tau)\,d\tau, \tag{10} \]
where \(\varphi_i\), \(\psi_i\), and \(f_i(t)\) are the Fourier coefficients of \(\varphi\), \(\psi\), and \(f\) with respect to the system \(\{v_i\}\).
We shall determine those conditions under which the series (10) gives a solution of the mixed problem (1)—(3) in the space \(C_{x,t}^{2m,2}(Q)\). We begin the study of this question with the smoothness of the generalized eigenfunctions of problem (8), (9).
* In [1], on p. 1250, obvious misprints were made: instead of \(S\in R^{2m+[n/2]+1}\) it should be \(S\in R^{([n/2m]+3)m}\), and instead of \(a_{\alpha\beta}\in W_2^{|\alpha|+[n/2]+1}(\Omega)\) it should be \(a_{\alpha\beta}\in W_2^{|\alpha|+m([n/2m]+1)}(\Omega)\).
Denote by \(C_h^l(\Omega)\) the space of all functions \(u\) from \(C^l(\overline{\Omega})\) whose derivatives of order \(l\) uniformly satisfy a Hölder condition with exponent \(h\), \(0<h<1\). For functions \(u\in C_h^l(\Omega)\) define the norm
\[ |u|_{l,h}=|u|_{C^l(\overline{\Omega})} +\sum_{|\alpha|=l}\sup_{x,y\in\Omega} \frac{|D^\alpha u(x)-D^\alpha u(y)|}{|x-y|^h}. \tag{11} \]
Lemma 1. If the surface \(S\) is \(2m+1\) times continuously differentiable, \(a_{\alpha\beta}\in C_h^{|\alpha|}(\Omega)\), then the generalized eigenfunctions of problem (8), (9) belong to the space \(C_h^{2m}(\Omega)\).
Proof. From [1] it follows that \(v_i\in W_2^{2m}(\Omega)\). Hence, by the embedding theorems, \(v_i\in L_p(\Omega)\), where \(p=2n/(n-4m)\) if \(n-4m>0\), and \(p\) is arbitrary if \(n-4m\le 0\). Now to the solutions of equation (8) satisfying conditions (9) we apply the a priori estimates in \(L_p(\Omega)\) [3], [4]. We obtain that \(v_i\in W_p^{2m}(\Omega)\). Repeating this argument a sufficient number of times, we see that \(v_i\in W_p^{2m}(\Omega)\) for every \(p\ge 2\). Choose now \(p>n\); then, by the embedding theorems, we conclude that \(v_i\in C_h^{2m-1}(\Omega)\), where \(h=1-n/p\). Having obtained such information about the smoothness of \(v_i\), we apply to the solutions of equation (8) satisfying conditions (9) the Schauder a priori estimates [3], [4]. We obtain the assertion of Lemma 1.
Theorem. Let the boundary \(S\) of the domain \(\Omega\) be \(2m+1\) times continuously differentiable and let, for \(k=\left[\dfrac{n}{2m}\right]+1\), the following conditions be fulfilled:
\[ \begin{aligned} &1)\quad a_{\alpha\beta}\in W_2^{|\alpha|+mk}(\Omega); \\ &2)\quad \varphi\in W_2^{m(k+2)}(\Omega),\quad \psi\in W_2^{m(k+1)}(\Omega) \end{aligned} \]
and, moreover, the functions
\[
\varphi,\ A\varphi,\ \ldots,\ A^{\left[\frac{k+1}{2}\right]}\varphi
\quad\text{and}\quad
\psi,\ A\psi,\ \ldots,\ A^{\left[\frac{k}{2}\right]}\psi
\]
belong to \(\overset{\circ}{W}{}_{2}^{m}(\Omega)\);
\[
3)\quad f\in W_{x,t}^{m(k+1),\,k+1}(Q)
\]
and, moreover, the functions
\[
f,\ Af,\ \ldots,\ A^{\left[\frac{k}{2}\right]}f
\]
belong to \(\overset{\circ}{W}{}_{2}^{m,\,1}(Q)\).
Then the series (10) converges in the norm \(C_{x,t}^{2m,\,2}(\overline{Q})\), and its sum \(u(x,t)\) is a classical solution of the mixed problem (1), (3).
The lemmas below are used in the proof of the theorem. The first of them is of independent interest.
In what follows, by \(A\) we shall denote the operator generated by the differential expression (4), with domain of definition \(D(A)\) coinciding with the subspace of all functions from \(W_2^{2m}(\Omega)\) which, in the sense of the embedding theorems, satisfy the boundary conditions (9). The operator \(A\) is self-adjoint in \(L_2(\Omega)\).
Lemma 2. If \(S\) is \(2m+1\) times continuously differentiable, \(a_{\alpha\beta}\) \((0\le |\alpha|\le m)\) has bounded derivatives up to order \(|\alpha|\), and \(k=\left[\dfrac{n}{2m}\right]+1\), then the operators \(A^{-k/2}\) and \(A^{-(k+1)/2}\) are bounded from \(L_2(\Omega)\) into \(C_{1/2}(\Omega)\) and \(C_{1/2}^{m}(\Omega)\), respectively. If, moreover, \(a_{\alpha\beta}\in C_{1/2}^{|\alpha|}(\Omega)\), \((|\alpha|\le m)\), then the operator \(A^{-(k+2)/2}\) is bounded from \(L_2(\Omega)\) into \(C_{1/2}^{2m}(\Omega)\). In this case there exists a symmetric kernel \(G_{s/2}(x,y)\) \((s=k,\ k+1,\ k+2)\) such that
\[ \left(A^{-\frac{s}{2}} f\right)(x)=\int\limits_{\Omega} G_{\frac{s}{2}}(x,y) f(y)\,dy \tag{12} \]
for any function \(f\in L_2(\Omega)\), and the integrals
\[ \int\limits_{\Omega} G_{\frac{k}{2}}^2\,dy,\quad \int\limits_{\Omega} \left(D_x^\alpha G_{\frac{k+1}{2}}\right)^2 dy,\ |\alpha|\leq m,\quad \int\limits_{\Omega} \left(D_x^\alpha G_{\frac{k+2}{2}}\right)^2 dy,\ |\alpha|\leq 2m \tag{13} \]
exist and are functions belonging to \(C_{\frac12}(\Omega)\).
Proof. First of all let us note that \(A^{-1}\) is a bounded operator from \(L_2(\Omega)\) into \(W_2^{2m}(\Omega)\). Hence, from a priori estimates in \(L_p\) [3], [4], it follows that \(A^{-1}\) is also a bounded operator from \(L_p(\Omega)\) into \(W_p^{2m}(\Omega)\) for any \(p\geq 2\).
Consider the case \(s=k\). Let first \(k=2l+1\). The operator \(A^{-\frac12}\) is bounded from \(L_2(\Omega)\) into \(W_2^m(\Omega)\). By the embedding theorems, \(A^{-\frac12}\) is bounded from \(L_2(\Omega)\) into \(L_p(\Omega)\), where \(1/p=1/2-m/n\). Consequently,
\[
A^{-\frac32}=A^{-1}A^{-\frac12}
\]
is a bounded operator from \(L_2(\Omega)\) into \(W_p^{2m}(\Omega)\), i.e. into \(L_q(\Omega)\) for \(1/q=1/2-3m/n\). Since \(1/p-2m(l-1)/n>0\), and \(1/p-2ml/n<0\), this process can be continued. We obtain that \(A^{-\frac{k}{2}}\) is a bounded operator from \(L_2(\Omega)\) into \(W_r^{2m}(\Omega)\), where \(1/r=1/2-m(k-2)/n\). Hence it follows, by the embedding theorems, that \(A^{-\frac{k}{2}}\) is a bounded operator from \(L_2(\Omega)\) into \(C_{\frac12}(\Omega)\). The cases \(k=2l\) and \(s=k+1\) are considered similarly.
The assertion of Lemma 2 for the case \(s=k+2\) follows from the corresponding assertion in the case \(s=k\) and from Schauder a priori estimates [3], [4].
Since
\[
\left|\left(A^{-\frac{s}{2}}f\right)(x)\right|\leq \left\|A^{-\frac{s}{2}}\right\|\cdot \|f\|,
\]
for each fixed \(x\) the value \(\left(A^{-\frac{s}{2}}f\right)(x)\) is a bounded linear functional on \(L_2(\Omega)\). Therefore there exists an element \(g_{x,s/2}(y)\in L_2(\Omega)\) such that
\[
\left(A^{-\frac{s}{2}}f\right)(x)=(f,g_{x,s/2}).
\]
Putting \(G_{\frac{s}{2}}(x,y)=g_{x,s/2}(y)\), we obtain (12).
The properties of the kernel \(G_{\frac{s}{2}}(x,y)\) follow from the first part of Lemma 2 proved above.
Corollary. If the conditions of Lemma 2 are satisfied, then for
\[
k=\left[\frac{n}{2m}\right]+1
\]
the following bilinear series converge uniformly in \(\overline{\Omega}\):
\[ \sum_{i=1}^{\infty}\lambda_i^{-k}v_i^2,\quad \sum_{i=1}^{\infty}\lambda_i^{-(k+1)}\left(D^\alpha v_i\right)^2,\ |\alpha|\leq m,\quad \sum_{i=1}^{\infty}\lambda_i^{-(k+2)}\left(D^\alpha v_i\right)^2,\ |\alpha|\leq 2m. \tag{14} \]
Lemma 3. If all the conditions of the theorem are satisfied (with the exception of condition 2), then, for
\(k=\left[\dfrac{n}{2m}\right]+1\), the series
\[ \sum_{i=1}^{\infty}\lambda_i^{k+1}\int_0^T f_i^2(t)\,dt \]
converges, and the series
\[ \sum_{i=1}^{\infty}\lambda_i^k f_i^2(t) \]
converges uniformly on \([0,T]\).
The first part of Lemma 3 follows from Lemma 2 of [1], which remains valid also in the case under consideration, when \(S\) is continuously differentiable \(2m+1\) times. The second part follows from the first on the basis of the inequality \(f_i^2(t)\le C_2(\|f_i\|^2+\|\dot f_i\|^2)\). For the proof of the theorem, in fact, only the first part of Lemma 2 is needed. On the basis of the conditions of the theorem we conclude that
\[ a_{\alpha\beta}\in C_{1/2}^{|\alpha|}(\Omega),\quad \varphi\in D\left(A^{\frac{k}{2}+1}\right),\quad \psi\in D\left(A^{\frac{k+1}{2}}\right) \quad\text{and}\quad f\in D\left(A^{\frac{k+1}{2}}\right). \]
The assertion of the theorem follows from Lemmas 2 and 3 on the basis of Theorem 1 of [5]. One may also, instead of Lemma 2, use the corollary to this lemma. Then the theorem is proved by a direct investigation of the uniform convergence in \(\overline Q\) of series (10) and of the series obtained by termwise differentiation of it the appropriate number of times with respect to \(x_i\) and \(t\), using this corollary, Lemma 2 of [1], and Lemma 3.
Let us note that in the course of the proof of the theorem the following estimate for the solution is obtained:
\[ |u|_{2m,2}\le C_3\bigl(\|\varphi\|_{m(k+2)}+\|\psi\|_{m(k+1)}+\|f\|_{m(k+1),\,k+1}\bigr), \tag{15} \]
where the constant \(C_3\) depends on the domain \(Q\) and on the coefficients \(a_{\alpha\beta}\).
Using as an example the mixed problem with Dirichlet boundary conditions, we have presented a method for proving the uniform convergence of the series representing the generalized solution of this problem. This method is based on Lemmas 1 and 2, which are proved with the aid of a priori estimates for solutions of elliptic equations in the norms of the spaces \(W_p^l(\Omega)\) and \(C_h^l(\Omega)\). As is known [3], [4], such estimates are also valid in the case of general boundary conditions. Therefore the method set forth above carries over without change to mixed problems for equation (1) and for nonstationary equations with first derivative with respect to time of the form
\[ \frac{\partial u}{\partial t}+Au=f(x,t), \tag{16} \]
\[ \frac{1}{i}\frac{\partial u}{\partial t}+Au=f(x,t) \tag{17} \]
with general boundary conditions
\[ B_j u|_{\Gamma} = \sum_{|\beta|\le k_j} b_{\beta j}(x)D^\beta u|_{\Gamma} =0, \tag{18} \]
where \(k_j\le 2m-1\), \(j=1,\ldots,m\). Generalized solutions of such problems were obtained by the Fourier method in [6].
References
- Brish N. I., Valeshkevich I. N. DAN SSSR, 146, No. 6, 1962, pp. 1247–1250.
- Il’in V. A. UMN, 15, issue 2, 1960, pp. 97–154.
- Browder F. E. Proc. Nat. Acad. Sci. USA, 15, No. 3, 1959, 365–372.
- Agmon S., Douglis A., Nirenberg L. Estimates of solutions of elliptic equations near the boundary. IL, Moscow, 1962.
- Krasnosel’skii M. A., Pustyl’nik E. I. DAN SSSR, 122, No. 6, 1958, pp. 978–981.
- Brish N. I., Valeshkevich I. N. Differential Equations, 1, No. 3, 1965, pp. 393–399.
Received by the editors
21 December 1964
Belorussian State University named after V. I. Lenin