Abstract
Full Text
L. I. Kamynin, V. N. Maslennikova
Boundary Estimates for the Solution of the Problem with an Oblique Derivative for a Parabolic Equation in a Noncylindrical Domain
(Presented by Academician S. L. Sobolev on 30 VI 1964)
In our work ((^5)), a priori estimates were obtained in Hölder norms for the solution of the III boundary-value problem (with a conormal derivative) for a general parabolic equation of the 2nd order in a noncylindrical domain. The method of these papers makes it possible to obtain a priori estimates in Hölder norms for the solution of the problem with an oblique derivative for a parabolic equation of the 2nd order and for the solution of general boundary-value problems for parabolic systems in the sense of I. G. Petrovskii, satisfying a solvability condition of the Lopatinskii type.
In the present paper we consider the parabolic equation of the 2nd order
[
\sum_{i,j=1}^{n} a_{ij}(x,t)\frac{\partial^2 u}{\partial x_i \partial x_j}
+
\sum_{i=1}^{n} b_i(x,t)\frac{\partial u}{\partial x_i}
+
c(x,t)u
-
\frac{\partial u}{\partial t}
=
f(x,t),\quad (x,t)\in Q,
\tag{1}
]
with the initial condition
[
u(x,0)=\psi(x),\quad x\in \Omega=\overline{Q}\cap{t=0}
\tag{2}
]
and the boundary condition
[
\sum_{i=1}^{n} \nu_i(x,t)\frac{\partial u(x,t)}{\partial x_i}
+
\mu(x,t)u(x,t)
=
\varphi(x,t),\quad (x,t)\in \Gamma,
\tag{3}
]
where (Q) is a domain (possibly unbounded in (x_i)) in the space of variables ((x,t)=(x_1,x_2,\ldots,x_n;t)), lying between the hyperplanes (t=0) and (t=T>0); (\Gamma) is the lateral surface of the domain (Q). If (\gamma_i(x,t)) ((i=1,2,\ldots,n)) are the cosines of the angles formed by the inward normal to (\Gamma_t) (at the point ((x,t))) with the axes (ox_i), where (\Gamma_t) is the section of the surface (\Gamma) by the hyperplane (t=\text{const}), then
[
\sum_{k=1}^{n} \nu_k(x,t)\gamma_k(x,t)\geq \nu_0>0,\quad \nu_0=\text{const}.
]
(A_1.) The lateral surface (\Gamma) has at each point a tangent plane nowhere orthogonal to the axis (ot). For each point (P(x,t)\in\Gamma) there exists an ((n+1))-dimensional sphere (S_\delta(P)) with center at the point (P) and radius (\delta>0) ((\delta) does not depend on the choice of the point (P) on (\Gamma)) such that the part of (\Gamma) lying in the sphere (S_\delta(P)) can, for some (i) ((1\leq i\leq n)), be represented in the form
[
x_i=h(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n;t),
]
where the function (h) has a first derivative with respect to (t) and second derivatives with respect to (x_k), satisfying Hölder conditions in (x) and (t) with exponents (\alpha) and (\alpha/2), respectively. Here and below (0<\alpha<1). In this case we shall say that the function (h(x,t)) belongs to the class (C_{x,t}^{2+\alpha,\,1+\alpha/2}) (has bounded norm (|h|_{2+\alpha}), cf. ((^5))).
A₂. Equation (1) is uniformly parabolic in (Q), i.e., for any real vector (\xi) and for all ((x,t)\in \overline Q)
[
M_1\sum_{i=1}^n \xi_i^2 \geq \sum_{i,j=1}^n a_{ij}(x,t)\xi_i\xi_j \geq M_0\sum_{i=1}^n \xi_i^2,
]
where (M_i>0) ((i=0,1)) are constants.
A₃.
[
|a_{ij}|\alpha^Q+|b_i|\alpha^Q+|c|\alpha^Q\leq M_2,\qquad |f|\alpha^Q<+\infty .
]
A₄.
[
|\nu_i|{1+\alpha}^{\Gamma}+|\mu|\leq M_2,\qquad}^{\Gamma
|\varphi|{1+\alpha}^{\Gamma}<+\infty,\qquad
|\psi|<+\infty .}^{\Omega
]
A₅. The functions (f,\psi,\varphi,\mu), and (\nu_i) are compatible, by virtue of equation (1), on the edge (\Gamma\cap\Omega).
Consider in the strip
[
D_T={(x,t),\ |x_i|<+\infty,\ i=1,2,\ldots,n-1;\ 0<x_n<+\infty,\ 0<t<T<+\infty}
]
the boundary-value problem for the heat-conduction equation
[
\sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}-\frac{\partial u}{\partial t}=f(x,t),\qquad (x,t)\in D_T;
\tag{4}
]
[
u(x,0)=\psi(x),\qquad x\in \Omega_0=\overline{D_T}\cap{t=0};
\tag{5}
]
[
\frac{\partial u(x,t)}{\partial l_0}=\varphi(x,t),\qquad (x,t)\in \Gamma_0=\overline{D_T}\cap{x_n=0},
\tag{6}
]
where (l_0) is a constant direction lying in the plane passing through the point ((x,t)) orthogonally to the (ot) axis and forming an acute angle with the axis (ox_n).
Lemma 1. Let the function (f(x,t)) be defined in (D_T) and (|f|^{D_T}<+\infty), let the function (\psi(x)) be defined in (\Omega_0) and (|\psi|{2+\alpha}^{\Omega_0}<+\infty), and let the function (\varphi(x_1,\ldots,x)) solution (v(x,t)) of problem (4)—(6), for which the inequalities},t)) be defined on (\Gamma_0) and (|\varphi|_{1+\alpha}^{\Gamma_0}<+\infty), with (f,\psi), and (\varphi) compatible on the edge (\Gamma_0\cap\Omega_0). Then there exists a unique (in the class of bounded functions continuous on (\overline{D_T
[
[v]{2+\alpha}^{D_T}\leq C(M_0,M_1,l_0)\bigl(|f|\alpha^{D_T}+[\psi]{2+\alpha}^{\Omega_0}+[\varphi]\bigr),}^{\Gamma_0
]
[
[v]{0}^{D_T}\leq C(M_0,M_1,T,l_0)\bigl([f]+[\psi]}^{D_T{0}^{\Omega_0}+[\varphi]\bigr).}^{\Gamma_0
]
It is sufficient to prove Lemma 1 for the one-dimensional heat-conduction equation with zero initial condition (see ((^1,^5,^6))). In this case the solution of the problem under consideration in (D_T) can be represented explicitly in the form (for brevity of notation we put (n=2))
[
v(x,t)=-\frac{1}{2\pi}\int_0^t d\tau \int_{-\infty}^{+\infty}
H(x_1-\xi_1,x_2,t-\tau)\varphi(\xi_1,\tau)\,d\xi_1,
\tag{7}
]
where the half-space kernel (H) has the form
[
\begin{aligned}
H(x_1-\xi_1,x_2,t-\tau)
&=\frac{a_2}{t-\tau}\exp\left{-\frac{(x_1-\xi_1)^2+x_2^2}{4(t-\tau)}\right}\
&\quad-\frac{a_1[a_1x_2-a_2(x_1-\xi_1)]}{(t-\tau)^{3/2}}
\exp\left{-\frac{[a_1x_2-a_2(x_1-\xi_1)]^2}{4(t-\tau)}\right}\
&\quad\times\left[
\Phi\left(\frac{a_1(x_1-\xi_1)+a_2x_2}{2(t-\tau)^{1/2}}\right)-\frac{\sqrt{\pi}}{2}
\right].
\end{aligned}
\tag{8}
]
Here ((a_1,a_2)) are the direction cosines of the direction (l_0),
[
\Phi(x)=\int_0^x \exp{-y^2}\,dy .
]
The solution (7) was obtained by us by applying the Fourier transform with respect to (x_i) ((i=1,2,\ldots,n-1)) and the Laplace transform with respect to (t). A kernel of the type (8) is also given in the paper ((^2)) for arbitrary cylindrical domains. By estimating the solution (7) directly, we obtain the estimates of Lemma 1. The uniqueness of the representation (7) is established with the aid of the paper ((^3)).
Bearing in mind the interior a priori estimates and the a priori estimates near the base (\Omega), the estimates in the whole noncylindrical domain (Q) of the solution of (1)—(3) are established with the aid of Lemma 1 by the method of localization.
Theorem 1. Let (Q_1 \subseteq Q) be a subdomain of (Q) (in particular, it may coincide with (Q)), with (\Omega_1=\overline{Q}_1\cap\Omega), (\Gamma_1=\overline{Q}_1\cap\Gamma). Suppose that conditions (A_1)—(A_5) are fulfilled in the domains (Q,\Omega_1,\Gamma_1), respectively, and that (u(x,t)) is a solution, bounded in (\overline{Q}), of problem (1)—(3), where (\Omega) in (2) and (\Gamma) in (3) are replaced respectively by (\Omega_1) and (\Gamma_1). Suppose, finally, that
[
|u|_{2+\alpha}^{Q\cup\Gamma_1\cup\Omega_1}<+\infty .
]
Then
[
|u|{2+\alpha}^{Q_1}\leq C\bigl(Q_1,Q,M_0,M_1,M_2,\nu_0,\delta\bigr)
\bigl(|f|+|\varphi|}^{Q{1+\alpha}^{\Gamma}+|\psi|\bigr),}^{\Omega}+[u]_0^{Q
]
where ([u]_0^Q=\sup_Q |u|).
Remark. Increasing the requirements on the smoothness of the surface (\Gamma), the coefficients of equation (1), the functions appearing in (2), (3), and the order of compatibility leads to a corresponding increase in the smoothness of the solution of problem (1)—(3).
Theorem 2. In the case of a bounded domain (Q), the estimate
[
|u|{2+\alpha}^{Q}\leq C\bigl(|f|+|\varphi|}^{Q{1+\alpha}^{\Gamma}+|\psi|\bigr)}^{\Omega
]
is valid, where the constant (C) depends on equation (1) and on the domain (Q), but does not depend on the function (u(x,t)).
Using refined investigations of the properties of heat potentials (see ((^4))), by the method of continuation in a parameter, with the aid of Theorem 2 one can prove the following existence theorem.
Theorem 3. Let the surface (\Gamma) be of type (J_{1,1,(1+\beta)/2}^{1,\beta,\beta/2}) ((0<\alpha<\beta<1)), where (\beta>0) is arbitrary (see ((^4))). Suppose that conditions (A_2)—(A_5) are fulfilled.
Then there exists a solution (u(x,t)) of problem (1)—(3), having in (\overline{Q}_T) a first derivative with respect to (t) and second derivatives with respect to (x_i), satisfying Hölder conditions in (x) and (t) with exponents (\alpha) and (\alpha/2), respectively.
Moscow State University
named after M. V. Lomonosov
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
23 VI 1964
CITED LITERATURE
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- R. Výborný, Czechoslovak Math. Journal, 8 (83) (1958).
- L. I. Kamynin, Dokl. Akad. Nauk SSSR, 160, No. 2 (1964).
- L. I. Kamynin, V. N. Maslennikova, Dokl. Akad. Nauk SSSR, 153, No. 3 (1963).