Abstract
Full Text
MATHEMATICS
V. A. SOLONNIKOV
ON A PRIORI ESTIMATES FOR SOME BOUNDARY-VALUE PROBLEMS
(Presented by Academician V. I. Smirnov, January 28, 1961)
No. 1. Let (\Omega) be a rectangular parallelepiped in the space (E_n: a_i < x_i < b_i) (the case of infinite (a_i) and (b_i) is not excluded). In (1) the concept of the spaces (\mathfrak{L}{p}^{l_1\ldots l_n}(\Omega)) and (\mathfrak{W}(\Omega)). Let}^{l_1\ldots l_n}(\Omega)) was introduced and their properties were investigated. We shall now introduce the spaces (L_{p}^{l_1\ldots l_n}(\Omega)) and (W_{p}^{l_1\ldots l_n
[
D_i^k=\partial^k/\partial x_i^k,\qquad \lambda_i=l_i-[l_i]
]
and
[
|f|{L^p =}^{l_i}(\Omega)
\begin{cases}
\displaystyle
\int_{\Omega} dx \int_{0}^{b_i-x_i}
\left|D_i^{[l_i]}f(x_1\ldots x_i+h\ldots x_n)-D_i^{[l_i]}f(x)\right|^p
\frac{dh}{h^{1+p\lambda_i}},
& \text{for } \lambda_i>0,\[2.2ex]
\displaystyle
\int_{\Omega} |D_i^{l_i}f(x)|^p\,dx,
& \text{for } \lambda_i=0.
\end{cases}
]
We define the spaces (L_{p}^{l_1\ldots l_n}(\Omega)) and (W_{p}^{l_1\ldots l_n}(\Omega)) as the closure of the set of smooth functions equal to zero for large (|x|), in the norms
[
|f|{L}^{l_1\ldots l_n}(\Omega)
=
\sum_{i=1}^{n}|f|{L,\qquad}^{l_i}(\Omega)
|f|{W}^{l_1\ldots l_n}(\Omega)
=
|f|{L_p(\Omega)}+|f|.}^{l_1\ldots l_n}(\Omega)
]
For nonintegral (l_i) these spaces coincide with (\mathfrak{L}{p}^{l_1\ldots l_n}(\Omega)) and (\mathfrak{W}(\Omega)).}^{l_1\ldots l_n
Theorem 1. Let (f\in W_{p}^{l_1\ldots l_n}(\Omega)), (0<s0.
]
Then
[
D_1^{\nu_1}\cdots D_n^{\nu_n}f\in
\mathfrak{W}{p}^{\,l_1\mu\ldots l_s\mu}(\Omega_s),
\qquad
\left|D_1^{\nu_1}\cdots D_n^{\nu_n}f\right|{p}^{\,l_1\mu\ldots l_s\mu}(\Omega_s)}
\le
C_1|f|,}^{l_1\ldots l_n}(\Omega)
]
where (\Omega_s) is the section of (\Omega) by the hyperplane (x_{s+1}=\mathrm{const},\ x_n=\mathrm{const}).
This theorem is a generalization of the well-known results of Aronszajn, Babich, Slobodetskii, and Gagliardo on the boundary properties of functions from (W_p^l) for integral (l). It admits a converse.
Theorem 2. Let (\varphi(x)) be an infinitely differentiable function of one variable, defined on the positive half-axis and vanishing together with all its derivatives at (x=0), and also for sufficiently large (x), and let (\varphi^(x)) be its extension to the whole axis, with (\varphi^(x)=0) for (x<0). Then, for all (l\ge 0), except (l=k+1/p), where (k\ge 0) is an integer, the inequality
[
|\varphi^*|{\mathfrak{L}}^{l}(E_1)
\le
C_2|\varphi|{\mathfrak{L}}^{l}(0,\infty)
]
holds.
For (l=k+1/p) this inequality is false.
No. 2. Consider, in the half-space (x_n \ge 0), the boundary-value problems:
[
\Delta u=f(x),\qquad u\big|{x_n=0}=\varphi(x_1\ldots x),
\tag{1}
]
[
\Delta v=f(x),\qquad
\left.\frac{\partial v}{\partial x_n}\right|{x_n=0}
=\psi(x_1\ldots x);
]
[
u_t-\Delta u=f(x,t),\qquad
u\big|{t=0}=u_0(x),\qquad
u\big|,t),}=\varphi(x_1\ldots x_{n-1
\tag{2}
]
[
v_t-\Delta v=f(x,t),\qquad
v\big|{t=0}=v_0(x),\qquad
\left.\frac{\partial v}{\partial x_n}\right|
=\psi(x_1\ldots x_{n-1},t),
]
and in the half-space (x_3 \ge 0) of three-dimensional space (E_3) the problem
[
\Delta \mathbf u=\operatorname{grad} p+\mathbf f,\qquad
\operatorname{div}\mathbf u=\rho(x),\qquad
\mathbf u\big|_{x_3=0}=\mathbf a(x_1,x_2).
\tag{3}
]
Introduce the following notation: (D_n) is the half-space (x_n \ge 0) of the space (E_n); (\widetilde D_n) is the half-space (t \ge 0) of the space (E_n) of the variables (x_1,\ldots,x_{n-1},t); (A_{n+1}) is the domain (x_n \ge 0,\ t \ge 0) of the space (E_{n+1}). Thus, the function (f(x)) is given in (D_n), (\varphi(x_1\ldots x_{n-1})) and (\psi(x_1\ldots x_{n-1})) in (E_{n-1}), (f(x,t)) in (A_{n+1}), (u_0) and (v_0) in (D_n), (\varphi(x_1\ldots x_{n-1},t)) and (\psi(x_1\ldots x_{n-1},t)) in (\widetilde D_n), etc. It is well known that, by extending in the proper way the functions (f(x)), (f(x,t)), (u_0(x)), (v_0(x)), and others to the whole space (E_n) or (E_{n+1}), one can write the solutions of problems (1) and (2) explicitly in terms of electrostatic or heat potentials.
As for problem (3), it is a generalization of the first boundary-value problem for the stationary linear Navier–Stokes system and is easily reduced to the latter by introducing the new unknown vector
[
\mathbf v=\mathbf u-\operatorname{grad} q(x),
]
where (q(x)) is the solution of the problem
[
\Delta q(x)=\rho,\qquad
\left.\frac{\partial q}{\partial x_3}\right|_{x_3=0}=a_3(x_1,x_2).
]
Since (\mathbf v) can be expressed in terms of Odqvist hydrodynamic potentials (2), the vector (\mathbf u) is expressed as a sum of hydrodynamic and electrostatic potentials.
All these potentials can be estimated in the norms (\mathfrak L_p^l) and (L_p^l) (the heat potentials in the norms (\mathfrak L_{px,t}^{2l,l}) and (L_{px,t}^{2l,l}))* in terms of the corresponding norms of their densities. With the help of these estimates (which we do not write out), as well as Theorems 1 and 2 and some results of the work (1), the following propositions are proved:
Theorem 3. For the solutions of problems (1) the following inequalities hold:
[
|u|{\mathfrak L_p^{l+2}(D_n)}
\le
C_3\bigl(|f|}(D_n)
+|\varphi|{\mathfrak L_p^{\,l+2-1/p}(E\bigr),})
]
[
|u|{L_p^{l+2}(D_n)}
\le
C_4\bigl(|f|}(D_n)
+|\varphi|{\mathfrak L_p^{\,l+2-1/p}(E\bigr),})
]
[
|v|{\mathfrak L_p^{l+2}(D_n)}
\le
C_5\bigl(|f|}(D_n)
+|\varphi|{\mathfrak L_p^{\,l+1-1/p}(E\bigr),})
]
[
|v|{L_p^{l+2}(D_n)}
\le
C_6\bigl(|f|}(D_n)
+|\psi|{\mathfrak L_p^{\,l+1-1/p}(E\bigr).})
]
Theorem 4. For the solution of problem (3) the following inequalities hold:
[
\sum_{i=1}^{3}|u_i|{\mathfrak L_p^{l+2}(D_3)}
+|p|}(D_3)
\le
]
* By (\mathfrak L_{px,t}^{2l,l}) one should understand the space (\mathfrak L_p^{2l\ldots 2l,l}), regarding (t) as the last coordinate, and by (\mathfrak L_p^l), (\mathfrak L_p^{l,\ldots,l}).
[
\leq C_7\left(\sum_{i=1}^{3}|f_i|{L_p^l(D_3)}+|\rho|}(D_3)
+\sum_{i=1}^{3}|a_i|_{\mathfrak L_p^{\,l+2-1/p}(E_2)}\right),
]
[
\sum_{i=1}^{3}|u_i|{L_p^{l+2}(D_3)}+|p|\leq}(D_3)
]
[
\leq C_8\left(\sum_{i=1}^{3}|f_i|{L_p^l(D_3)}+|\rho|}(D_3)
+\sum_{i=1}^{3}|a_i|_{\mathfrak L_p^{\,l+2-1/p}(E_2)}\right).
]
Theorem 5. For the solutions of problem (2) the following inequalities hold
[
|u|{\mathfrak L}^{\,2l+2,\,l+1}(A_{n+1})
\leq C_9\left(|f|{\mathfrak L}^{\,2l,l}(A_{n+1})
+|u_0|{\mathfrak L_p^{\,2l+2-2/p}(D_n)}
+|\varphi|\right),}^{\,2l+2-1/p,\,l+1-1/2p}(\widetilde D_n)
]
[
|u|{L}^{\,2l+2,\,l+1}(A_{n+1})
\leq C_{10}\left(|f|{L}^{\,2l,l}(A_{n+1})
+|u_0|{\mathfrak L_p^{\,2l+2-2/p}(D_n)}
+|\varphi|\right),}^{\,2l+2-1/p,\,l+1-1/2p}(\widetilde D_n)
]
provided only that (l+1-1/2p\ne m+1/p), where (m\geq 0) is an integer, and
[
|v|{\mathfrak L}^{\,2l+2,\,l+1}(A_{n+1})
\leq C_{11}\left(|f|{\mathfrak L}^{\,2l,l}(A_{n+1})
+|v_0|{\mathfrak L_p^{\,2l+2-2/p}(D_n)}
+|\psi|\right),}^{\,2l+1-1/p,\,l+1/2-1/2p}(\widetilde D_n)
]
[
|v|{L}^{\,2l+2,\,l+1}(A_{n+1})
\leq C_{12}\left(|f|{L}^{\,2l,l}(A_{n+1})
+|v_0|{\mathfrak L_p^{\,2l+2-2/p}(D_n)}
+|\psi|\right),}^{\,2l+1-1/p,\,l+1/2-1/2p}(\widetilde D_n)
]
provided only that (l+1/2-1/2p\ne m+1/p). If (l+1-1/2p=m+1/p) or (l+1/2-1/2p=m+1/p), the corresponding inequalities do not hold.
The indicated exceptional cases disappear if, instead of (2), one considers problems without initial data for (-\infty<t<\infty).
No. 3. Let (\Omega) be a bounded domain with sufficiently smooth boundary. Introduce the following notation:
[
N_l(f,\Omega)=\left(\int_\Omega dx\int_\Omega
\frac{|f(x)-f(y)|^p}{|x-y|^{\,n+pl}}\,dy\right)^{1/p}
\qquad \text{for } 0<l<1;
]
[
N_1(f,\Omega)=\left(\int_\Omega dx\int_{A(x)}
\frac{\left|f(x)-2f\left(\frac{x+y}{2}\right)+f(y)\right|^p}
{|x-y|^{\,n+p}}\,dy\right)^{1/p},
]
where (A(x)) is the set of those (y\in\Omega) such that (\dfrac{x+y}{2}\in\Omega), and
[
N_l(f,\Omega)=\sum_{\sum\nu_i=\bar l}
N_\lambda\left(D_1^{\nu_1}\cdots D_n^{\nu_n}f(x),\Omega\right)
\qquad \text{for } l=\bar l+\lambda,
]
where (\bar l\geq 0) is an integer and (0<\lambda\leq 1).
Now one can define the spaces (\mathfrak W_p^l(\Omega)) and (\widetilde W_p^l(\Omega)) as the closures of the set of smooth functions in the norms
[
|f|{\mathfrak W_p^l(\Omega)}=|f|+N_l(f,\Omega),
]
[
|f|{\widetilde W_p^l(\Omega)}=
\begin{cases}
|f| l,\[6pt]}, & \text{for nonintegral
|f|{L_p(\Omega)}+\displaystyle\sum
|D_1^{\nu_1}\cdots D_n^{\nu_n}f|_{L_p(\Omega)}, & \text{for integral } l.
\end{cases}
]
Theorem 6. If (\Omega) is a rectangular parallelepiped, then the norms
(|f|{\mathfrak W_p^l(\Omega)}) and (|f|) are equivalent to the norms
(|f|{\mathfrak W_p^{\,l\ldots l}(\Omega)}) and (|f|), introduced above.}(\Omega)
Theorem 7. Every function in (\widetilde{\mathfrak W}^{\,l}{p}(\Omega)) or (\widetilde W^{\,l}(\Omega)) can be extended to the whole space (E_n) with preservation of equivalence of norms.
For the spaces (W^{\,l}{p}(\Omega)) this theorem has been proved (see ((^{3,4}))), so that what is essentially new here is the case when (f\in \widetilde{\mathfrak W}^{\,l}(\Omega)) with integral (l).
In the usual way (by means of parametrization) the spaces (\mathfrak W^{\,l}{p}) and (\widetilde W^{\,l}}) are defined if the functions are given on some smooth manifold, for example the boundary (S) of a domain. The preceding theorems generalize to the spaces of functions (\mathfrak W^{\,l_1\ldots l_m{p(x_1)\ldots(x_m)}(Q)) and (\widetilde W^{\,l_1\ldots l_m}(Q)) is the closure of the set of smooth functions in the norm}(Q)), whose elements are given in the cylinder (Q=\Omega_1\times\cdots\times\Omega_m), where the (\Omega_i) are smooth manifolds and (x_i\in\Omega_i). For example, if the (\Omega_i) are domains, then (\mathfrak W^{\,l_1\ldots l_m}_{p(x_1)\ldots(x_m)
[
|f|^{p}{\mathfrak W^{\,l_1\ldots l_m}}(Q)
=
\sum_{i=1}^{m}
\int_{\Omega_1} dx_1\cdots
\int_{\Omega_{i-1}} dx_{i-1}
\int_{\Omega_{i+1}} dx_{i+1}\cdots
\int_{\Omega_m}
|f|^{p}{\mathfrak W^{\,l_i}\,dx_m .}(\Omega_i)
]
No. 4. Consider in a bounded (n)-dimensional domain (\Omega) the following boundary-value problems:
[
\mathcal L(x)u=f,\qquad u|_S=\varphi,
]
[
\mathcal L(x)v=f,\qquad \left.\frac{\partial v}{\partial N}\right|_S=\psi;
\tag{4}
]
[
u_t-\mathcal L(x,t)u=f,\qquad u|_{t=0}=u_0,\qquad u|_S=\varphi,
]
[
v_t-\mathcal L(x,t)v=f,\qquad v|_{t=0}=v_0,\qquad
\left.\frac{\partial v}{\partial N}\right|_S=\psi,\qquad
0\leq t\leq T,
\tag{5}
]
where (\mathcal L(x)) and (\mathcal L(x,t)) are second-order elliptic operators with coefficients depending respectively on (x) and on ((x,t)). Also consider, for (n=3), the problem
[
\Delta \mathbf u=\operatorname{grad}p+\mathbf f,\qquad
\operatorname{div}\mathbf u=\rho,\qquad
\mathbf u|_S=\mathbf a(x),
\tag{6}
]
assuming that the necessary condition is satisfied
[
\int_{\Omega}\rho\,dx=\int_{S}a_n\,dS .
]
Following the arguments of Schauder ((^{5})), one can, with the aid of Theorems 3–5, obtain estimates for solutions of problems (4)–(6). Their difference from the inequalities of Theorems 3–5 consists in the fact that instead of the norms (L^{\,l}{p}) and (L'^{\,l}}), the norms (\mathfrak W^{\,l{p}) and (\widetilde W^{\,l}) occur everywhere. In addition, the right-hand side of the estimates for the solutions of problem (4) will contain the norms of these solutions in the space (L_p(\Omega)).
It is necessary to note that some of the results presented above are known. Estimates for elliptic equations in the norms (\widetilde W^{\,l}{p}) for integral (l) and in (\widetilde W^{\,l}}) for arbitrary (l) have been obtained by a number of authors ((^{6-9})). In the works ((^{10,8})) estimates were also obtained in (\widetilde W^{\,2l,l{2x,t}) for solutions of parabolic equations. An analogous theorem for problems with zero initial and boundary conditions was announced by L. N. Slobodetskii also for the case (p\ne2) ((^{8})); however, as the author has communicated, his proof is inaccurate. As for problem (6), an estimate of its solutions for (\rho=0) in the space (\widetilde W^{\,2})).}(\Omega)) was published by the author of the present article in ((^{11
Received
19 I 1961
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