CC BY
32
УДК 517.955
On Carleman-type Formulas for Solutions to the Heat Equation
Ilya A. Kurilenko* Alexander A. ShlapunoV
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 28.02.2019, received in revised form 11.03.2019, accepted 20.04.2019 We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equation. More precisely we recover a function, satisfying the heat equation in a cylindrical domain, via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural (anisotropic) spaces (Sobolev and Holder spaces, etc). Finally, we obtain a uniqueness theorem for the problem and a criterion of its solvability and a Carleman-type formula for its solution.
Keywords: the heat equation, ill-posed problems, integral representation method, Carleman formulas. DOI: 10.17516/1997-1397-2019-12-4-421-433.
The integral representation method is the core of investigations of the ill-posed problems for partial differential equations, see [1-3]. Lev Aizenberg [4,5] noted that the Cauchy problem for the Cauchy-Riemann equations is closely related to the problem of analytic continuation even if its entries are not analytic. He found principal ingredients, leading to the construction of integral formulas for its solution (Carleman formulas): a proper integral formula recovering the function via the data on the whole boundary, the uniqueness theorem and an effective tool, providing the analytic continuation from a domain to a larger one. This method was successfully used in the framework of Hilbert space methods to investigate the Cauchy problem for general elliptic systems of partial differential equations, see [6-8], and even to elliptic complexes of differential operators, see [9]. It provided both a solvability criterion and formulas for exact and approximate solutions. Recently, the scheme of using the concept of analytic continuation was adopted to obtain a solvability criterion for the ill-posed Cauchy problem in Holder spaces for some parabolic operators as well, see [10,11].
In this paper we concentrated our efforts in the construction of Carleman-type formulas for solutions to the heat equation. Namely, we recover a function satisfying the heat equation in a cylindrical domain via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. The principal difference with [10,11] is that we use anisotropic Holder and Sobolev spaces and we succeed to involve Hilbert space methods, too.
1. Preliminaries
Let Q be a bounded domain in n-dimensional linear space 1" with the coordinates x = = (xi,... ,xn). As usual we denote by Q the closure of Q, and we denote by dQ its boundary. In the sequel we assume that dQ is piece-wise smooth. We denote by QT the bounded
1 [email protected] © Siberian Federal University. All rights reserved
open cylinder Q x (0,T) in Rn+1 with a positive altitude T. Let also r c dQ be a non empty connected relatively open subset of dQ. Then = r x (0, T) and = r x [0, T].
We consider functions on subsets in Rn and Rn+1. As usual, for s G Z+ we denote by Cs(Q) the space of all s times continuously differentiable functions in Q. Next, for a (relatively open) set S c dQ denote by Cs(Q U S) the set of such functions from the space Cs(Q) that all their derivatives up to order s can be extended continuously onto Q U S. The standard topology of these metrizable spaces induces the uniform convergence on compact subsets in Q U S together with all the partial derivatives up to order s (the case S = 0 is included). We will also use the so-called Holder spaces (cf., [12, Ch. 1, Sec. 1]. Recall that a function u, defined on a set M G Rn, is called Holder continuous with a power 0 < A < 1 on M, if there is such a constant C > 0 that
\u(x) — u(y)\
< U >x,M = sup ^-^ < C.
x,yeM, x=y \x — y\
Let Cs'x(Q U S) stand for the set of such Holder continuous functions that all their partial derivatives up to order s are also Holder continuous functions with a power A over each compact subset of Q U S. Clearly, Cs,x(Q U dQ) = Cs,x(Q) is a Banach space with the norm,
IMIc^(n) = Y^ (ma- \d"u(x)\ + <d°U>A,n)
\a\^s
see, for instance, [12-14]. In general, the space Cs'x(Q U S) again can be treated as a metrizable space, generated by a system of seminorms p:mix(u) = ||u||cs,*(Qm) with a suitable exhaustion {Qm}meN of the set Q U S. By the definition, the space Cs'x(Q U S) is continuously embedded into Cs'x'(Q U S) if s + A > s' + A', A, A' G [0,1).
To investigate the heat equation we need also the anisotropic (parabolic) spaces, see [12, Ch. 1], [14, Ch. 8]. For this purpose, let C2s,s(QT) stand for the set of all continuous functions u in QT, having in QT the continuous partial derivatives dj d^u with all multi-indexes
n
(a,j) G Z+ x Z+ satisfying \a\ + 2j < 2s where, as usual, \a\ = ^ a.j. Similarly, we de-
j=i
note by C2s+k's(QT) the set of continuous functions in QT, such that all partial derivatives d@u belong to C2s's(QT) if ¡3 G Z+ satisfies \ft\ < k, k G Z+. Of course, it is natural to agree that C2s+0's(Qt) = C2s's(Qt), C0'0(Qt) = C(Qt) and C0(Q) = C(Q). We also denote by C2s+k,s((Q u s)t) the set of such functions u from the space C2s+k,s(QT) that their partial derivatives dju, 2j + \a\ < 2s, \fl\ < k, can be extended continuously onto (Q U S)T. The standard topology of these metrizable space induces the uniform convergence on compact subsets of (QUS)T together with all partial derivatives used in its definition (the case S = 0 is included).
We will also use the so-called anisotropic Holder spaces (cf., [12, Ch. 1], [14, Ch. 8]). Recall that a function u(x,t), defined on a set MT = M x [0,T] G Rn+1, is called anisotropic Holder continuous with a power 0 < A < 1 on MT, if there is such a constant C > 0 that
<u>x,Mt = sup <u(^,t) >x,M + sup <u(x, •) >x/2,[0,T]< C.
te[0,T] xeM
Let C2s+k's'x'x/2((Q U S)T) stand for the set of anisotropic Holder continuous functions with a power A over each compact subset of (Q U S)T together with all partial derivatives da+i3dju where \3\ < k, \a\ + 2j < 2s. Clearly, C2s+k,s,x,x/2(QT) is a Banach space with the norm,
||u||c№,A,A/2^) = £ £ ( \dZ+i3 dj u(x,t)\+ <da+f} dj u>x^)
\?Kk M+2j<2s ^ (x't)enT /
see, for instance, [14, Ch. 8]. In general, the space C2s+k's'x'x/2((Q U S)T) can be treated again as a metrizable space, generated by a system of seminorms p^f (u) = ||u||C2s+fc,s,*,v2(Qm T) with
a suitable exhaustion {Qm}meN of the set Q U S. Obviously, the space C2s+k's'x,x/2((Q u S)T) is continuously embedded into C2s'+k' ,s',x',x'/2((Q u S)T) if s + A > s' + A', A, A' e [0,1) and k > k'.
In order to invoke the Hilbert space approach, we need anisotropic (parabolic) Sobolev spaces H2s's(QT), s e Z+, see, [12,15], i.e. the set of all measurable functions u in QT such that all generalized partial derivatives djd^u with all multi-indexes (a,j) e Z+ x Z+ satisfying |a| + 2j < 2s, belong to the Lebesgue class L2(QT). This is a Hilbert space with the inner product
(u,v)H2s,s(QT) = / djd^v(x, t) djdXu(x, t)dxdt. (1)
We also may define H2s's(QT) as the completion of the space C2s's(QT) with respect to the norm
|| • ||H2s,s(QT) generated by the inner product (1). For s = 0 we have H0'0(QT) = L2(QT). "
Let now An dl. x. be the Laplace operator in 1" and let L = dt — stand for the
j=i " 3
heat operator in 1"+i. This operator plays essential role in the contemporary natural science,
"
see, for instance, [17]. Now let dv = ^ Vjdx denote the derivative at the direction of the exterior
j=i
unit normal vector v = (vi,..., vn) to the surface dQ. As dQ is piece-wise smooth, the normal vector v is defined almost everywhere on dQ. It is known that the standard initial boundary value problem for the heat operator consists of the recovering of the function u over the cylinder QT satisfying the heat equation
Lu = f in QT, (2)
the boundary and initial conditions
a(x,t)u(x,t) + b(x,t)dvu(x,t)=0 on dQ.T, u(x, 0) = uo(x) on Q. (3)
for fixed real-valued functions a and b and given data f and u0. Problem (2), (3) is well-posed over the scales of anisotropic Holder and Sobolev spaces, see, for instance, [12-18]. Instead, we consider the Cauchy problem in the cylinder QT in the sense of the Cauchy-Kowalevski Theorem with respect to the space variables, cf. [19].
Problem 1. Let functions ui e C1 '0(rT), u2 e C(rT) and f e C(QT) be given. Find a function u e C2 '1(QT) fl C1'0((Q U r)T) satisfying the heat equation (2) and boundary conditions
u(x,t) = ui(x, t) on rT, dvu(x,t) = u2(x, t) on rT. (4)
The motivation of Problem 1 is rather transparent. It describes the situation where for some reasons at each time t e [0, T] only part r of the boundary of the "body" Q is available for measurements.
Example 1. If n > 1 then the famous Hadamard example for the Cauchy problem for the Laplace operator is fit to demonstrate the absence of the continuity of the solution with respect to the data in all reasonable standard spaces (Holder spaces, Sobolev spaces, etc., see [19]). For instance, denote by Qn,r the n-dimensional cube {0 < xm < R, 1 < m < n} and take the cube Q", 1 as the base Q of the cylinder QT. Let r be the face {xn = 0} of the cube Qn, 1 and rT = Qn-1 ,i x (0, T). Then, for each aj e 1, j e N, the function uj(x,t) = aj cos(jx1) cosh(jx") is a solution to problem (2), (4), with the data
fj = 0, ui,j = aj cos(jxi), u2j = 0.
If lim j2s+kaj = 0 then for each k e Z+ and each A e [0,1] we have
fj 0 in C^(QT), uij 0 in C2s+k's'x'x/2(rT), u2jj 0 in C2s+k-1's'X'X/2(TT).
On the other hand, the sequence \uj(x, t)\ is unbounded for each (x,t) G QT.
If n = 1 then we could not directly provide a corresponding example but we reduce the problem to an ill-posed one in Theorem 2 below. However, Problem 1 become ill-posed if we additionally impose rather mild restrictions on the growth of its solutions in QT, see [10]. Indeed, consider the sequence of smooth solutions uj (x, t) = aj e (t-T)+jxn to problem (2), (4), with the data
fj = 0, uij = aj e »j2(t~T), u2j = jaj e ^j2(t-T).
If Hm j2(s+1^aj = 0 then for each k G Z+ and each A G [0,1] we have
j^+tt
fj 0 in Ctt(QT), u1,j 0 in C2s+k's'x'x/2(TT), u2,j 0 in C2s+k-1's'x'x/2(TT).
On the other hand, if we choose aj = j-2(s+1)-1 then for any q ^ 1 we have
j-^+tt ^uj ^qQn.i^o, t]) = +
2. Solvability conditions and Carleman formula
If the surface r and the data of the problem are real analytic then the Cauchy-Kowalevski Theorem implies that problem (2), (4) has at most one solution in the class of (even formal) power series. However the theorem does not imply the existence of solutions to Problem 1 because it grants the solution in a small neighbourhood of the (analytic) surface rT only (but not in a given domain QT!). We begin this section proving that Problem 1 can not have more than one solution in the spaces of differentiable (non-analytic) functions.
To investigate Problem 1, we use an integral representation constructed with the use the fundamental solution to heat operator L, see, for instance, [13,17]:
{H2
e
(2^t)n if t> 0 0 if t < 0.
Consider the cylinder type domain QTl,T2 = QT2 \ QTl with 0 < T1 < T2 and a closed measurable set S c dQ. For functions f G C(QTl T2), v G C(ST), w G C(ST), h G C(QTl T2) we introduce the following potentials:
In,Ti (h)(x,t)= i $(x-y,t)h(y,T1)dy, On ,Ti (f )(x,t)= i f $(x-y, t-r )f (y,T )dydr, Jn J Jn
Vs,Ti (v)(x,t)= / §(x - y,t - r)v(y,r)ds(y)dr, JTiJ S
Ws,Ti (w)(x,t) = - / dUy $(x - y,t - r )w(y, r )ds(y)dr, JTi JS
(see, for instance, [12, Ch. 4, Sec. 1], [13, Ch. 1, Sec. 3 and Ch. 5, Sec. 2], [16, Ch. 3, Sec. 10]). The potential In,Ti (h) is sometimes called Poisson type integral and the functions On,Ti (f), VS,Ti (v), WStTi (w) are often referred to as heat potentials or, more precisely, volume heat potential, single layer heat potential and double layer heat potential, respectively. By the construction, all these potentials are (improper) integrals depending on the parameters (x,t). Next, we need the so-called Green formula for the heat operator.
Lemma 1. For all 0 < T1 < T2 and all u G C2'1(QTiT2) n C 1'o(QTiT2) with Lu G C(QTiT2) the following formula holds:
u(x,t), (x,t) G . lT . . ^ . Tr . TTr . . w .
■ ' ' ' (5)
0, (x,t)G Qti
—| = (jn,Ti (u) + Gn,Ti (Lu)+ V9n,Ti (dvu) + wq^Ti (u)j(x,t).
Proof. See, for instance, [10], [20, Ch. 6, Sec. 12] (cf. also [21, Theorem 2.4.8] for more general linear operators). □
As is known, the heat equation is hypoelliptic. Moreover, any C2,1(QTi,T2)-solution P to the heat equation LP = 0 in the cylinder domain QTi,T2 belongs to C^(QTi,T2) and, actually P(x,t) is real analytic with respect to the space variable x G Q for each t G (T1,T2), see, for instance, [17, Ch. VI, Sec. 1, Theorem 1]. Then Green formula (5) and the information on the kernel $ provide us with a uniqueness theorem for Problem 1.
Theorem 1 (A uniqueness theorem). If r has at least one interior point in the relative topology of dQ then Problem 1 has no more than one solution.
Proof. See, for instance, [10, Theorem 1, Corollary 1]. □
Now we are ready to formulate a solvability criterion for Problem 1. As before, we assume that r is a relatively open connected set of dQ. Then we may find a set Q+ C 1" in such a way that the set D = Q U r U Q+ would be a bounded domain with piece-wise smooth boundary. It is convenient to set Q- = Q. For a function v on DT we denote by v+ its restriction to Q+ and, similarly, we denote by v- its restriction to QT. It is natural to denote limit values of v± on rT, when they are defined, by v±,T. Actually, a solvability criterion for Problem 1 was obtained in [10]. In this section we would like to improve these results in order to invoke Hilbert space methods. More precisely, the following theorem is a modification of [10, Theorem 2], related to the use of anisotropic Holder spaces. Though the proofs of the theorems are very similar, we obtain additional essential information about the solution to Problem 1 allowing us to use anisotropic Sobolev spaces and so called bases with double orthogonality property.
Theorem 2 (Solvability criterion). Let X G (0,1), dQ belong to C1+A and let V be a relatively open connected subset of dQ. If f G C0'0'x'x/2(Qt), u1 G C 1'0'X'X/2(TT), u2 G C0'0'x'x/2(Tt) then Problem (2), (4) is solvable in the space C2'1'X'X/2(QT) n C 1'0'X'X/2(QT U rT) if and only if there is a function F G C(DT) satisfying the following two conditions: 1) LF = 0 in DT, 2) F = Gn,o(f) + Vr,o(u2) + Wr,o(u1) in Q+.
Proof. Necessity. Let a function u(x,t) G C2'1'x'x/2(Qt ) nC}'0'X'X/2 (Qt U rt ) satisfy (2), (4). Clearly, the function u(x,t) belongs to the space C2,1,X,X/2(Q'T) for each cylindrical domain Q'T with such a base Q' that Q' C Q and Q7 n dQ C r. Besides, Lu = f G Co_^X'X/2(^). Without loss of the generality we may assume that the interior part r' of the set Q' n dQ is non-empty. Consider in the domain DT the functions
F = Gn,o(f) + Vr,o(u2) + WF o(u1) and F = F - \Qtu, (6)
where xm is the characteristic function of a set M C 1"+1. By the very construction condition 2) is fulfilled for it. Note that xqtu = Xn'Tu in D'T, where D' = Q'Ur'UQ+. Then Lemma 1 yields
F = Gq\q7 o(f) + Vr\r',o(u2) + WF\r , o(u1) - ,o(u) in DT. (7)
Arguing as in the proof of Theorem 1 we conclude that each of the integrals on the right-hand side of (7) is smooth outside the corresponding integration set and each satisfies homogeneous heat equation there. In particular, we see that F G CX(D'T) and LF = 0 in D'T because
of [17, Ch. VI, Sec. 1, Theorem 1]. Obviously, for any point (x,t) G DT there is a domain D'T containing (x, t). That is why LF = 0 in DT, and hence F belongs to the space Ctt(DT). Thus, this function satisfies condition 1), too.
Sufficiency. Let there be a function F G Ctt(DT), satisfying conditions 1) and 2) of the theorem. Consider on the set DT the function
U = F - F. (8)
As f G Co'0-x-x/2(QT) then the results of [12, Ch. 4, Secs. 11-14], [13, Ch. 1, Sec. 3] imply
On, o(f) G C2-1-x-x/2(Q±) n C1 -0 -x -x/2(Dt) (9)
and, moreover,
LO--o(f) = f in Qt, LO+- o(f) =0 in Q+ . (10)
Since u2 G Co-o-x-x/2(rT) then the results of [12, Ch. 4, Secs. 11-14], [13, Ch. 5, Sec. 2] yield
Vr oo(u2) G Ctt(Q±) n C1 -0-x-x/2((Q± U r)T), LVYo0(u2) =0 in QT U Q+ . (11)
On the other hand, the behaviour of the double layer potential Wp ^u^ is similar to the behaviour of the normal derivative of single layer potential Vp ^u^. Hence
Wr oo(u1) G Ctt(Q±) n Co-0-x-x/2((Q± U r)T), LWr o¿ux) = 0 in QT U Q+. (12)
Lemma 2. Let S c T G C1+x. If u1 G C1+x -x(rT), then the potential W-o(u1) belongs to the space C1 -0-x-x/2(Qt U St) if and only if W+o(u2) G C1 -0-x-x/2(Q+ U ST).
Proof. It is similar to the proof of the analogous lemma for Newton double layer potential (see, for instance, [6, Lemma 1.1]). Actually, one needs to use Lemma 1 instead of the standard Green formula for the Laplace operator, see [10, Lemma 2] for a different function class. □ Since F G Ctt(DT) C C1 -0-x-x/2((Q+ U T)t) then it follows from the discussion above that W+o(u2) G C1 -0-x-x/2((Q+ U T)t). Thus, formulas (8)-(12) and Lemma 2 imply that U belongs
C2 -1 -x - x/2(Q±) n C1 -0-x-x/2((Q± U T)t) and
LU = xdt f in Qt U Q+.
In particular, (2) is fulfilled for ULet us show that the function U- satisfies (4). Since F G Ctt(DT) we see that daF- = daF+ on rT for a G Z+ with H < 1 and
daF+T = (d aO+- o(f) + daV+o(u2) + daW+o(u1)\ .
V - - / | LT
It follows from formulas (9) and (11) that the volume potential O+ o(f) and the single layer potential V+o(u2) are continuous if the point (x,t) passes over the surface rT. Then
U-T = W-o(u1)lTT - W+fl(u1)lTT = u1.
because of the theorem on jump behaviour of the double layer potential Wr o(u1) (see, for instance, [13, Ch. 5, Sec. 2, theorem 1] for the corresponding heat potential), i.e. equality the first equation in (4) is valid for U-.
Formula (9) means that that the normal derivative of the volume potential On o(f) is continuous if the point (x,t) passes over the surface rT. Therefore
dv Ui TT = dv Vfo(u2) l TT ~ dv V+^2) | ^ + W-^) ^ - dv W+^1) | ^ . (13)
By theorem on jump behaviour of the normal derivative of the single layer potential (see, for instance, [16, Ch. 3, Sec. 10, Theorem 10.1] for the corresponding heat potential),
dvV-0(U2) I rT - dvV+0(«2)I rT = U2• (14)
Finally, we need the following lemma.
Lemma 3. Let r G C1+A and u2 G C0'0'A'A/2(F^). If W-0(ui) G C1'0'A'A/2((^ U r)T) or W^0(ui) G C 10,A,A/2((Q+ U r)T) then '
dvW-0(ui)ir t - dvW+0(ui) | r T = 0. (15)
Proof. It is similar to the proof of the analogous lemma for the the heat double layer potential (see, for instance, [10, Lemma 3] for a different function class). □
Using Lemma 3 and formulas (13), (14), we conclude that dvU—t = u2, i.e. the second equation in (4) is fulfilled for UThus, function u(x,t) = U-(x,t) satisfies conditions (2), (4). The proof is complete. □
We note that both [10, Theorem 2] and Theorem 2 are analogues of Theorem by Aizenberg and Kytmanov [5] describing solvability conditions of the Cauchy problem for the Cauchy-Riemann system (cf. also [6] in the Cauchy Problem for Laplace Equation or [8] in the Cauchy problem for general elliptic systems). Formula (8), obtained in the proof of Theorem 2, gives the unique solution to Problem 1. Clearly, if we will be able to write the extension F of the sum of potentials Gn,0(f) + Vr0(u2) + Wp 0(u1) from Q+ onto DT as a series with respect to special functions or a limit of parameter depending integrals then we will get Carleman-type formula for solutions to Problem 1 (cf. [5]). The simplest formulas of this type for the Cauchy-Riemann system was constructed by Goluzin and Krylov, see [22] or [4], for the special domains (the so-called "lunes", i.e. parts of a disc Q on the complex planes, separated from the origin by a smooth curve r c Q). Aizenberg and Kytmanov [5] supplement the Goluzin-Krylov formula with a simple solvability criterion for the Cauchy problem for holomorphic functions in the lunes based on the Cauchy-Hadamard formula for power series. Let us extend this approach for the case n = 1.
With this purpose we introduce the following Carleman kernels:
N k
Cn(x,y,t - t) = $(x - y,t - t) - $(y,t - T) ^
/
^ (t - T)k
fc=o v '
E
m+j = k
(2y)m-j(-l)j(t - t)j (4M)m(m - j)!j!
Corollary 1. Let n = 1, a > 0, Q = (a, 1) and D = (-1,1). Under the hypotheses of Theorem 2, Problem 1 is solvable in the space C2-1 -A -A/2(QT) n C1'0 -A -A/2(QT U rT) if and only if
limsup (\ck (t)| j < 1 for each t G (0,T)
fc^oo ^ '
where ck (t) are the Taylor coefficients of the function F(x,t) with respect to the variable x at the origin for fixed value t G (0, T) given by (19) below. Besides, the solution, if exists, equals to
i(x,t)= lim / / Cn(x,y,t - t)f (y,T)dydT + / / Cn(x,y,t - t)u2(y,T)ds(y)dT+ NJo JQ Jo Jr
+ J J dv(y)CN(x,y,t - T)ui(y,T)ds(y)d^j . (16)
Proof. The first part of the corollary follows immediately from Theorem 2, formula (8) for the solution to Problem 1 and the Cauchy-Hadamard formula for the radius of the convergence of a power series. In particular,
N
U(x,t)= lim (f(x,t) -V ck (t)xk), (x,t) G QT■ (17)
N
... )x
k=0
On the other hand, we may write easily the Taylor series with respect to the variable x near the origin for the kernel §(x — y,t — t) if t > t:
2 2
2xy— x m ,
§(x — y,t — t ) = ®(y,t — t )e 4mt—T) (y,t — t )xk (18)
k=0
where the series converges uniformly on the compacts in the set {|x| < a, y £ Q,t > t > 0} and
<£k (y,t - T) = --T1
kyy' ' (t - T)k
(
(2y)m-j(-l)j(t - t)j
j (4^)m(m - j)! j!
As y £ Q then y = 0 and the kernels $k (y,t — t) and dv$k (y,t — t) are integrable over subsets of QT. Thus, using (6), after termwise integration of the Taylor series (18), we identify the Taylor coefficients ck (t) as follows:
ck(t) = f f (y,t — T)f(y,T)dydT+
Jo Jn
+ J J (&k(y,t — T)u2(y,T) + dv(y,t — T)ui(y,T))ds(y)dT. (19)
Finally, combining formulas (6), (17), (19), we arrive at Carleman-type formula (16). □
For n > 1 the situation differs drastically because of the nature of the multi-dimensional Cauchy-Hadamard formula for multiple power series. Instead, for the multidimensional Cauchy-Riemann system Aizenberg and Kytmanov [5] suggested to use Hilbert space theory and the so-called bases with the double orthogonality property (cf. [6-8] for elliptic systems).
Following this idea, we assume that the surface dQ and that the data uj are smoother than in Theorem 2. Namely, we need the following lemma.
Lemma 4. Let dQ £ C3+x and let r be a relatively open subset of dQ with boundary dr £ C2+x. If u\ £ C2-1 -A' X/2(TT), U2 £ C2^<x<X/2(TT) then there exist functions Uj £ C2^•x'x/2(3Qt) such that Uj = Uj on rT, j = 1, 2 and a function U £ C2-1 -x-X/2(QT) such that U = U1 on (dQ)T and dvu = U2 on (0Q)t .
Proof. We may adopt the standard arguments from [23, Lemma 6.37] related to isotropic spaces. Indeed, according to it, under our assumptions, for every v0 £ C2 -x(r) there is v £ C2- x(dQ) such that v = v0 on r. The construction of the extension involves the rectifying diffeomorphism of dr and a suitable partition of unity of a neighbourhood of dr, only. Thus, we conclude there are functions Uj £ C2 -1 - x - x/2(dQT) such that Uj = Uj on rT, j = 1, 2.
Next, we use the existence of the Poisson kernel P&2, n(x,y) for the Dirichlet problem related to the operator A^, see [24]. It is known that the problem is well-posed over the scale of Holder spaces in Q. Namely, as dQ £ C3-x, for each 1 < s < 3 and v1 £ Cs-x(dQ), v2 £ Cs-1 -x(dQ) the integral
PA2-n(v1,v2)(x) = ((dv(y)PA2-n)(x,y)v1(y) + PA2-Q(x,y)v2(y)] ds(y) ■Jan v 7
belongs to Cs'a(Q) and satisfies A^v =0 in Q and v = v1, dvv = v2 on dQ. Now, we may set
u(x, t) = VA2, n(u1(-, t), 0)(x) + VA2, n(0, u2(-, t))(x).
Then the first integral belongs to C2 'A(Q) and the second belongs to C3'A(Q) with respect to the variable x for each t G [0, T]. By the construction, u(^,t) = u1(^,t) on dQ and dvu(,t) = u2(,t) on dQ for each t G [0, T]. It remains to check that u belongs to C2,1,A,A/2(QT) as the sum of integrals depending on the parameter (x,t) G QT. □
The statements of such type actually reflect the fact that the system of boundary operator, related to the investigated problem, is compatible. In the theory of elliptic operators this means that the system {I,dv} is a Dirichlet system on r, see, for instance, [7,8]. For example, under the assumptions of Lemma 4, for each pair v1 G C2A(r) and v2 G C 1A(r) there is v G C2A(Q) such that v = v1 on r and dvv = v2 on r. In Lemma 4 we use smoother data u2 on rT because the heat operator under the consideration is not elliptic and the behaviour with respect to the variable t can not be improved by actions with respect to the variable x. Under the assumptions of Lemma 4, we set
F = Gn,0(f) + Van,0(u2) + WSn,0(u1) + Infi(u). (20)
Corollary 2. Let X G (0,1), dQ belong to C3+A and let r be a relatively open connected subset of dQ with boundary dr G C2+A. If f G C0'0'A'A/2(QT), u1 G C2'1'A'A/2(TT), u2 G C2'1'A'A/2(TT) then Problem (2), (4) is solvable in the space C2-1-a-a/2(Qt) n C1-0-a-a/2(Qt U Ft) n H2'1(Qt) if and only if there is a function F G C^(DT) n H2,1(DT) satisfying the following two conditions:
1') LF = 0 in Dt, 2') F = F in Q+.
Proof. First of all, we note that, by Green formula (5), we have F = Gn,0(f — Lu) and then F G C2'1'A'A/2(Q±) because of (9). On the other hand,
F — F = Vafi\r ,0(u2) + Wafi\r ,0^1) + In,0(u). (21)
This means that the function F — F satisfies the L(F — F) = 0 in DT and hence the function F extends to DT as a solution of the heat equation if and only if function F extends to DT as a solution of the heat equation, too.
Let Problem (2), (4) be solvable in the space C2-1-a-a/2(Qt) nC 1-0-a-a/2(QtUFt) nH2-1(Qt). Then formulas (6) and (21) imply
^ = F — xq,tu G H2'1(Q±) and LF = 0 in DT.
Now, as F G H2'1(Q±) n C^(Dt) (see [17, Ch. VI, Sec. 1, Theorem 1]) we conclude that F G H2,1(DT), i.e. conditions 1'), 2') of the corollary are fulfilled.
If conditions 1'), 2') of the corollary hold true then conditions 1), 2) of Theorem 2 are fulfilled, too. Moreover, formulas (8) and (21) imply that in DT we have
U = F — F = F — F G H2-1(Q±) (22)
and the U- is the solution to Problem 1 in the space C2'1'A'A/2(QT)nC1'0'A'A/2(QTUTT)H2-1(Q±) by Theorem 2. □
Let us describe the way for the construction of the bases with the double orthogonality property related to Problem 1. For s G Z+, denote by H2iS's(QT) the space of weak (i.e. in the sense of distributions) solutions to the heat equation (2) belonging to H2s's(QT). Obviously, it is a closed subspace of H2s's(QT).
Lemma 5. Let w be a subset of D with piece-wise smooth boundary. There exists an orthonormal basis {bv} in the space H2>1(DT) such that the system {bv| UT} is an orthogonal basis in L2C(wT).
range of the operator R is dense in L\ (wT).
Moreover, by Fubini Theorem the anisotropic Sobolev space H21(DT) can be continuously
Proof. Indeed, by the definition the space H2'1(DT) is continuously embedded to L2(wT).
As any weak solution to the heat equation in DT is a weak solution the heat equation in wT,
too, we conclude that H21(DT) is continuously embedded to L2C(wT). Let us denote by RD
the corresponding embedding operator, RD: H21(DT) ^ L2C(wT). By the construction, the
2LL
Jc
anisotropic Sobolev space Hc embedded to the Bochner type space B((0,T, H2(D), L2(D)) consisting of maps v : [0,T] ^ H2(D) such that dtv G L2(D), see [25, Ch 1, Sec. 5]. According to the Kondrashov Theorem, the embedding H2(D) ^ L2(D) is compact. Applying famous compactness theorem for Bochner type spaces (see, for instance, [25, Ch 1, Sec. 5, Theorem 5.1] we see that the space B((0,T,H2(D),L2(D)) is embedded compactly to L2((0,T),D) = L2(DT). Thus, the space H21(Dt) is embedded compactly to L2C(DT) and then to L2C(wT), i.e. the operator RDis compact.
Let R*d u be the Hilbert space adjoint operator for RD,u, i.e. R*D u : L2c(wT) ^ H21(DT). Then the Hilbert-Schmidt spectral theorem grants the existence of an orthonormal basis {bv} in the space H21(DT), consisting of the eigen-vectors of the compact self-adjoint operator R*d uRd,u : H21(DT) ^ H21(DT). Finally, applying we see that the system {bv} is the basis with the double orthogonality property, we are looking for, see [7, Example 1.9]. □
Let w be a relatively compact subset of Q+ c D with piece-wise smooth boundary and let {bv} be the basis with the double orthogonality property granted by Lemma 5. We introduce now the following Carleman kernels:
C
CN
N
(u)
(x,y,t,r) = $(x - y,t,T)-Y^ (bv (x,t) bv (z,f)$(z - y,f - r)dzdfj/\\bv\\]
v=0 Ut
Also, let cv (F) be the Fourier coefficients of the function F with respect to the orthogonal basis {bv\UT} in the space L2c(wt):
Cv (F) =( f bv (z,f)Jf(z,f)dzdf)/\\bv\\2Ll (ut )■
\ j Ut /
(23)
Corollary 3. Let w be a relatively compact subset of Q+ with piece-wise smooth boundary. Under assumptions of Corollary 2, Problem (2), (4) is solvable in the space C2^'x'x/2(QT) n C1'0'x'x/2(Qt U rT) n H2¿(Qt) if and only if
TO
5>(F)? <
v=l
Besides, the solution, if exists, is given by
et
u(x,t) = Jim ( / Cfp(x, y, t, r)f (y,r)dydr + / / (x,y,t,r)u2(y,r)ds(y)dr+
N \ Jo Jo Jo J da
+ f f dv(y)C{N;)(x,y,t,r)U1(y,r)ds(y)d^i . (24) JO JdO J
Proof. Again, we note that the heat operator is hypoelliptic, i.e. all weak L2(QT)-solutions to the homogeneous heat equations are in fact the smooth ones in QT. This means that Hz£1(Qt) = = h21(Qt) n C^(Qt). Then [17, Ch. VI, Sec. 1, Theorem 1] implies that conditions 1'), 2') of Corollary 2 are equivalent to the following one: F = F in wT, or, the same, RD,uF = F. Thus, the first statement of the corollary follows immediately from Lemma 5 and [7, Example 1.9]. On
c
the other hand, according [7, Example 1.9], if F e H21(DT) is the extension of the function FF from wT to DT then
cv(F )bv(
v=0
N
F = (F)bv(x,t), (x,t) e Dt.
Therefore, (22) yields
u(x,t) = lim [F(x,t) cv (F)bv (x,t)), (x,t) e QT. (25)
N —V z—' J
v=0
We note that if y e 0 and x e w then x = y and the kernel $(x — y,t — t) is integrable over ilT x ilT. Thus, we may use integral formula (20) for F and Fubini Theorem in order to change the order of the integrations in (23). Hence it follows from (25) that (24) holds true. □
Example 2. What is still lacking is an example of a basis with the double orthogonality property granted by Lemma 5 for a pair D and w. However we may easily give an example of a system with the double orthogonality. Indeed, for each multi-index k e Zn we set
n n
b(1\x,t) = e-4n2^1 k 12t cos (2^kj x,), bk2\x,t) = 1 k 12t sin (2^kj x^
j=i j=i
where \k\2 = £ Then Cbf =0 in 1" x (0, T) for each k e Zn and the system {b<j}), b^jkezn 3 = 1
is orthogonal in L2(Qnj x (0, T)) for every cube Qn,j, j e N, and T > 0 because the trigonometrical system is orthogonal on the cube Qnj1 and it is periodic. Thus, {b^,b(2'>}keZn is a system with the double orthogonality property for any pair D = Qnj and w = Qn,i, if j > l. Unfortunately, this system can not be complete in H2'1([Qnj)T) because each function bk has equal values on the opposite faces of the cube Qnj and the elements of the space H2'1([Qnj )T) admit traces on (dQnj )T.
We note also that the scheme, realized in the paper, is valid for a more general parabolic operator L, too, if it admits a fundamental solution with the following properties: the analyticity with respect to x for each fixed t and the proper behavior of the corresponding integrals Iq, Gq, Vdo, and Wq on the scale of the (anisotropic) Holder spaces (see, for example, [13,26] for the conditions providing the existence of such kernels).
The work was supported by the Ministry of Education and Science of the Russian Federation 1.2604.2017/PCh.
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О формулах карлемановского типа для решений уравнения теплопроводности
Илья А. Куриленко Александр А. Шлапунов
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Мы применяем метод интегральных представлений к исследованию некорректной задачи Коши для уравнения теплопроводности. Более точно, используя подходящую формулу Грина, мы восстанавливаем комплекснозначную функцию, удовлетворяющую уравнению теплопроводности в цилиндре, по заданным ее значениям и значениям ее нормальной производной на части боковой поверхности цилиндра. Мы показываем, что задача является некорректной в естественных для нее (анизотропных) пространствах (Соболева, Гельдера и т.д.). В итоге нами получены теорема единственности для задачи Коши, а также необходимые и достаточные условия ее разрешимости и формула карлемановского типа для ее решения.
Ключевые слова: уравнение теплопроводности, некорректные задачи, метод интегральных представлений, формулы Карлемана.
