Abstract
Full Text
UDC 517.946
MATHEMATICS
M. I. KLYUCHANTSEV
ESTIMATES OF SOLUTIONS OF BOUNDARY-VALUE PROBLEMS WITH THE BESSEL OPERATOR
(Presented by Academician I. N. Vekua on March 6, 1969)
In the modern theory of boundary-value problems, an important place is occupied by the question of limiting sharp estimates of the norms of solutions of boundary-value problems in terms of the norms of the right-hand sides and boundary conditions. The present note is devoted mainly to estimates of solutions of boundary-value problems for certain singular differential equations. These estimates can be obtained rather simply, once explicit formulas are known for the solutions of equations with constant coefficients.
In the metrics (L_{2,k}), estimates of solutions were obtained earlier by I. A. Kipriyanov in the paper ((^{1})).
- Let (E_{n+2}^{++}) be the Euclidean space of points ((x,y,t)), (x=(x_1,\ldots,x_n)), (y\ge 0), (t\ge 0). In this domain we shall consider a solution (u(x,y,t)) of the (B)-elliptic ((^{1})) equation of order (2m) with constant (complex) coefficients
[
\mathscr{L}(D,B_y)u=\sum_{i+2r=2m} a_{i,r}D^iB_y^r u=f(x,y,t)
\quad \text{for } t>0,
\tag{1}
]
satisfying (m) boundary conditions
[
H_j(D,B_y)u=\sum b_{i,r}^j D^iB_y^r u=g_j(x,y)
\quad \text{for } t=0.
\tag{2}
]
These conditions are expressed by differential operators (H_j) with constant coefficients. The order of the operator (H_j) is equal to (m_j). The numbers (m_j) are nonnegative and may exceed the number (2m)—the order of the operator (\mathscr{L}). In addition, (\mathscr{L}) and all (H_j) are homogeneous. Here
[
B_y=-\left(\frac{\partial^2}{\partial y^2}+\frac{k}{y}\frac{\partial}{\partial y}\right),\quad k>0,
]
[
D=(D_x,D_t),\quad \text{where } \quad D_t=\frac{1}{i}\frac{\partial}{\partial t}.
]
The functions (f) and (g_j) for (t\ge 0) and (y\ge 0) are infinitely differentiable and have compact support. In what follows we shall assume that the boundary operators are connected with the operator (\mathscr{L}) by the Ya. B. Lopatinskii condition (the complementing condition ((^{2}))).
For sufficiently large (N), let (C_B^N) denote the class of functions continuously differentiable (N) times with respect to (x) and (t), and continuously admitting (N/2) applications of the Bessel operator with respect to the variable (y) in the case of even (N), and (\frac{\partial}{\partial y}B_y^{(N-1)/2}) in the case of odd (N). Extend the function (f) to the entire half-space (y\ge 0) so that (f_N\in C_B^N). The solution of equation (1) with right-hand side (f_N) has ((^{3})) the form
[
v=v_N(x,y,t)=\mathscr{E}*f_N
=\int T_y^z\mathscr{E}(x-s,y,t-\tau) f_N(s,z,\tau) z^k\,ds\,dz\,d\tau,
\tag{3}
]
where (\mathscr{E}) is the fundamental solution of the (B)-elliptic equation, determined by the formula ((^{3}))
[
\mathscr{E}(P)=|P|^{2m-n-k-2}\Omega!\left(\frac{P}{|P|}\right)+q(P)\ln|P|,
\quad P=(x,y,t),
]
and (T_y^z) is the generalized shift operator
[
T_y^z f(y)=
\frac{\Gamma((k+1)/2)}{\Gamma(1/2)\Gamma(k/2)}
\int_0^\pi
f!\left(\sqrt{y^2+z^2-2yz\cos\alpha}\right)\sin^{k-1}\alpha\,d\alpha.
\tag{4}
]
Integration in formula (3) is carried out over the whole half-space (z>0).
Let (u=v_N+w). To determine the function (w) we obtain the following problem: (\mathscr{L}w=0) for (t>0) and (H_jw=\varphi_j(x,y)) for (t=0), where the function
[
\varphi_j(x,y)=g_j(x,y)-H_jv_N\big|_{t=0}
\tag{5}
]
is no longer finite. Using the results of paper ((^4)), for the solution of the latter problem we obtain the representation
[
D^iB_y^r w
=
\sum_{j=1}^{m}
\int_{E_{n+1}^{+}}
D^iB_y^rT_y^zK_j(x-s,y,t)\varphi_j(s,z)z^k\,ds\,dz,
\tag{6}
]
where (i+2r=l_0=\max(2m,m_j)), (K_j) is the Poisson kernel of the (B)-elliptic problem constructed in ((^4)), so that for the finite function (u) we obtain the representation
[
D^iB_y^r u=D^iB_y^r v_N+D^iB_y^r w.
\tag{7}
]
The uniqueness of the function (u) defined by (7), (6), can be proved according to the scheme of the proof of Theorem 4.1 of paper ((^2)).
- Let (l) be an arbitrary integer (\ge l_0). For any function (v) of the class (C_B^l) put
[
[v]l=\sup |D^iB_y^r v|,\qquad
|v|_l=\sum[v]_j,}^{l
]
where (i+2r=l). The subclass of all functions from (C_B^l) for which the expressions (D^iB^r v) uniformly satisfy the Hölder condition with exponent (\alpha) ((0<\alpha<1)) will be denoted by (C_B^{l+\alpha}). For these functions we define the seminorm by the formula
[
[v]_{l+\alpha}
=
\sup
\frac{|D^iB^r v(P)-D^iB^r v(Q)|}{|P-Q|^\alpha},
]
where the supremum is taken over all (i+2r=l) and (P\ne Q).
Theorem 1. Let (u(x,y,t)) be a solution of problem (1)—(2), belonging to the class (C_B^{l_0+\alpha}), and let it have compact support. If (f\in C_B^{l-2m+\alpha}) in the quarter-space (y>0,\ t>0), and (g_j\in C_B^{l-m_j+\alpha}) on the half-plane (t=0).
Then (u\in C_B^{l+\alpha}),
[
[u]{l+\alpha}\le C\left([f]\right),}+\sum [g_j]_{l-m_j+\alpha
\tag{8}
]
where (C) depends only on (l,\alpha) and on the (B)-ellipticity constant of our problem.
Let (\Sigma^+=\Sigma_R^+) denote the quarter-ball:
[
|x|^2+y^2+t^2<R^2,\qquad y\ge 0,\quad t\ge 0
]
in ((x,y,t))-space. Denote by (\sigma_R^+) the flat part of the boundary: (t=0,\ y\ge 0); (d_P) is the distance from the point (P) in (\Sigma^+) to the spherical part of the boundary (\Sigma^+), and (d_{P,Q}=\min(d_P,d_Q)). By (\widehat{C}_B^l(\Sigma^+)) denote the class of functions (u) for which the norm is finite
[
|\widehat{u}|{l+\alpha}
=
\sum}^{l}\widehat{[u]j+\widehat{[u]},
]
where
[
\widehat{[u]}_j=\sup d_P^j|D^iB_y^r u|,
]
[
\widehat{[u]}{l+\alpha}
=
\sup d}^{\,l+\alpha
\frac{|D^iB^r u(P)-D^iB^r u(Q)|}{|P-Q|^\alpha}.
]
Theorem 2. Let (u) be a bounded solution of problem (1)—(2), belonging to the class (C_B^{l_0+\alpha}). Suppose, moreover, that (f \in \widehat C_B^{\,l-2m+\alpha}(\Sigma^+)) and (g_j \in \widehat C_B^{\,l-m_j+\alpha}(\sigma^+)) for fixed (l \ge l_0).
Then the function (u \in C_B^{l+\alpha}) and the inequality holds
[
[\widehat u]{l+\alpha}\le
C\left(d[\widehat f]}^{2m{l-2m+\alpha}
+\sum d}^{m_j}[\widehat g_j]_{l-m_j+\alpha
+[\widehat u]_0\right),
\tag{9}
]
where (C) does not depend on (u, f, g_j), and (R).
If growth of the solution at infinity with some rate is allowed, then Theorem 2 implies a generalization of Theorem 1.
Theorem 3. Suppose the quantity
[
M_0=\lim_{R\to\infty} R^{-(l+\alpha)}\max_{\Sigma^+}|u|
]
is finite.
Then (u \in C_B^{l+\alpha}) and the inequality holds
[
[u]{l+\alpha}\le
C\left([f]+M_0\right).}+\sum [g_j]_{l-m_j+\alpha
\tag{10}
]
3. The theorems given above make it possible, by means of the usual procedure of “freezing” the coefficients, to obtain analogues of Theorems 1 and 2 for equations with variable coefficients. In this case our equations have the form
[
\mathcal L(P,D,B_y)u(P)\equiv
\sum_{i+2r\le 2m} a_{i,r}(P)D^iB_y^r u(P)=F(P),\qquad t>0,
\tag{11}
]
[
H_j(P',D,B_y)u(P)=
\sum_{i+2r\le m_j} b_{i,r}^{\,j}(P')D^iB_y^r u(P)=G_j(P'),\qquad t=0,
\tag{12}
]
where (P=(x,y,t)), (P'=(x,y,0)).
We also assume that the operators (\mathcal L, H_j) and their coefficients satisfy certain conditions of type i)—iii) of work ({}^{(2)}) (p. 70). We shall not formulate them, but note that they are essential.
Theorem 4. Let the function (u) belong to the class (C_B^{l+\alpha}) and be a solution of the boundary-value problem (11)—(12) in the domain (E_{n+2}^{++}).
Then
[
|u|{l+\alpha}\le
C\left([F]+|u|_0\right),}+\sum [G_j]_{l-m_j+\alpha
\tag{13}
]
where (C) does not depend on (u, F), and (G_j).
Theorem 5. Let (u) be a solution of problem (11)—(12), belonging to the class (C_B^{l_0+\alpha}) in (\Sigma^+) for (R<1).
Then (u\in C_B^{l+\alpha}(\Sigma^+)) and
[
|\widehat u|{l+\alpha}\le
C\left(d[\widehat F]}^{2m{l-2m+\alpha}
+\sum d}^{m_j}[\widehat G_j]_{l-m_j+\alpha
+|\widehat u|_0\right),
\tag{14}
]
where (C) does not depend on (R).
4. In conclusion we give (L_{p,k})-estimates for solutions of problems (1)—(2) and (11)—(12). Put, for (p>1) ((t=x_{n+2})),
[
[u]{j,L=}
\left(
\sum_{i=1}^n \int_{E_{n+2}^{++}} |D_{x_i}^j u|^p y^k\,dx\,dy
+\int_{E_{n+2}^{++}} |D_t^j u|^p y^k\,dx\,dy
+[B_y^{[j/2]}u]{y,j-2[j/2],L^p}
\right)^{1/p},
]
[
|u|{l,L=}
\left(\sum_{j=0}^{l}[u]{j,L.}}^p\right)^{1/p
\tag{15}
]
where
[
\bigl[B_y^{[j/2]}u\bigr]{y,\,j-2[j/2],\,L_p,\,k}
=
\left(
\int}^{++}
\left|B_y^{j/2}u\right|^p y^k\,dx\,dy
\right)^{1/p}
]
in the case of even (j). If (j) is odd ((j=2r+1)), then
[
\bigl[B_y^{[j/2]}u\bigr]{y,\,1,\,L_p,\,k}
=
\left(
\int}^{++}
y^k\,dx\,dy
\int_0^\infty
\frac{\left|T_y^{2h}B^r u-2T_y^hB^r u+B^r u\right|^p}{h^{1+p}}\,dh
\right)^{1/p}.
]
For functions (\varphi(x,y)) that are boundary values of functions (u(x,y,t)), and for positive integers (l), define the norms
[
|\varphi|{l-1/p,\,L_p,\,k}
=
\inf |u|,
]
where the infimum is taken over all such functions (u) that (u(x,y,0)=\varphi(x,y)).
Theorem 6. Let (u) be a solution of problem (1)—(2) and let it be finite. Suppose, moreover, that (|u|_{l,\,L_p,\,k}<\infty), (l\ge l_0+1).
Then
[
|u|{l,\,L_p,\,k}
\le
C\left(
|f|
+
\sum |g_j|_{l-m_j-1/p,\,L_p,\,k}
\right),
\tag{16}
]
where the constant (C) depends only on (l), (p), and the constant of (B)-ellipticity of our problem.
Theorem 7. Let (u) be a solution of problem (11)—(12) and tend to zero as (|P|>\rho), where (\rho) is sufficiently small. Suppose that the norm (|u|_{l_0+1,\,L_p,\,k}) is finite.
Then the norm (|u|_{l,\,L_p,\,k}) for (l\ge l_0+1) is also finite and
[
|u|{l,\,L_p,\,k}
\le
C\left(
|F|
+
\sum |G_j|{l-m_j-1/p,\,L_p,\,k}
+
|u|
\right),
\tag{17}
]
where (C) does not depend on (u,F,G_j).
Voronezh State University
Received
3 III 1969
References Cited
- I. A. Kipriyanov, DAN, 158, No. 2 (1964).
- S. Agmon, A. Douglis, L. Nirenberg, Estimates of Solutions of Elliptic Equations Near the Boundary, IL, 1962.
- V. I. Kononenko, DAN, 172, No. 2 (1967).
- I. A. Kipriyanov, M. I. Klyuchantsev, DAN, 183, No. 5 (1968).