CC BY
11
YflK 517.955
Hyperbolic Formulas in Elliptic Cauchy Problems
Dmitry P. Fedchenko*
*
Institute of Mathematics, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041,
Russia
Nikolai Tarkhanov^
Institute of Mathematics, University of Potsdam, Am Neuen Palais, 10, Potsdam, 14469
Germany
Received 10.06.2010, received in revised form 10.07.2010, accepted 20.08.2010
We study the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of it's boundary which is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of Hans Lewy (1927).
Keywords: Laplace equation, Cauchy problem, wave equation, Carleman formulas.
Introduction
The question of the well-posedness of the Cauchy problem was first raised by Hadamard who proved in [1] that it is ill-posed in the case of linear second order elliptic equations. Hadamard's proof is based on the analytic regularity of linear boundary value problems. This regularity has been extended to nonlinear elliptic equations in [2] so that Hadamard's argument also applies to general nonlinear elliptic equations.
Hadamard also pointed out in [1] that the problem occurring in wave propagation is not at all analytic problem, but a problem with real, not necessarily analytic data. For general linear equations it is well known that the hyperbolicity is a necessary condition for the well-posedness of the noncharacteristic Cauchy problem in CX, that is for the existence of solutions for general CX data, cf. [3], [4]. Moreover, for several classes of nonhyperbolic equations, explicit conditions on the initial data necessary for the existence of solutions were given in [5]. For nonlinear equations, [6] proves that the existence of a smooth stable solution implies hyperbolicity, stability meaning that one can perturb the initial data and the source terms in the equations.
The nonlinear theory yields difficult new problems, see [7], [8], etc. There are many interesting examples, for instance in multiphase fluid dynamics, where the equations are nor everywhere hyperbolic. As but one occurrence of this phenomenon, we consider Euler's equations of gas dynamics in Lagrangian coordinates
mentioned in [8]. The system is hyperbolic, when p'(u) > 0, and elliptic, when p'(u) < 0. For van der Waals state laws, it happens that p is decreasing on an interval [u*, u*]. A mathematical
*[email protected] [email protected] © Siberian Federal University. All rights reserved
(0.1)
example is p(u) = u(u2 — 1). Hadamard argument shows that the Cauchy problem with data taking values in the elliptic region is ill-posed. If u(0,x) = u0(x) is real analytic near x and u0(x) belongs to the elliptic interval, then any local C1 solution is analytic, see e.g. [2]. Thus, the initial data u0(x) must be actually analytic for the initial value problem to have a solution.
It was Hans Lewy who first used hyperbolic techniques to study problems for elliptic equations, cf. [9]. The solutions of elliptic equations with real analytic coefficients prove to be real analytic, and so they extend to holomorphic functions in a complex neighbourhood of their domain. For a holomorphic function obtained in this way the derivative d/dxk just amounts to the derivative d/d(iyk) where zk = xk + iyk are complex variables with k = 1,..., n. One can go to a complex space in only one variable, say xn, and the change d/dxn ^ —id/dyn leads to a drastical modification of the characteristic variety. The Laplace equation written in the coordinates (x', xn) with x' = (x1,..., xn-1) transforms to the wave equation in the coordinates (x',yn).
This idea is especially useful in the study of the Cauchy problem for elliptic equations. This problem is overdetermined even in the case of data given on an open part of the boundary, hence it does not admit any simple formulas for solutions, see however [10] and [11].Since the problem is unstable, the left inverse operator fails to be continuous. On the other hand, the Cauchy problem for hyperbolic equations is of textbook character and it admits many explicit formulas for solutions like d'Alembert, Kirchhoff, Poisson, etc. formulas, cf. [1]. Outstanding contribution to the Cauchy problem for hyperbolic equations is due to Leray who developed multidimensional residue theory in complex analysis to handle the problem, see [12], [13], etc. Having granted a solution u(x', iyn) of the Cauchy problem for a hyperbolic equation, how can one restore the solution u(x',xn) of the Cauchy problem for the original elliptic equation? The simple substitution iyn ^ xn does not make sense in general. For this purpose we invoke a formula of [14] which restores the values of holomorphic functions in a corner on the diagonal through their values on an arc connecting to faces of the corner. The resulting formula for the solution of an elliptic Cauchy problem includes a limit passage and agrees perfectly with the general observation that the character of instability in an elliptic Cauchy problem is similar to that in the problem of analytic continuation, cf. [15].
As mentioned, the idea to use hyperbolic formulas for elliptic Cauchy problems goes back at least as far as [9]. In the 1960s it was directly applied in a number of papers by Krylov, see for instance [16]. In [16], an integral representation for holomorphic solutions of a partial differential equation in a complex domain is constructed through the Cauchy data of solutions on an analytic surface. However, the formula does not manifest any instability of the Cauchy problem, which shows its local character.
The approach we develop in this paper has the advantage of providing a large parameter to perturb the solution of the problem. This might give rise to a calculus of Cauchy problems for elliptic equations. Since these problems are unstable, no operator calculus similar to that including elliptic boundary values problems and their parametrices on compact manifolds with boundary is possible. On introducing a large parameter into operators we are able to describe their perturbations which lead to solutions.
Let us dwell on the contents of the paper. In Section 1 we formulate the Cauchy problem for a second order elliptic equation in a domain X in Rn. The principal part of the equation is given by the Laplace operator while the lower order part may include nonlinear terms. The Cauchy data are given on a nonempty open set S of the boundary. Our standing assumption is that X is a cylinder over a bounded domain B with smooth boundary in the space Rn-1 of variables x' and S a smooth cross-section of X.
In Section 2 we reformulate the same Cauchy problem for a hyperbolic equation. Namely, we assume that the solution u(x',xn) is a real analytic function of xn e (b(x'),t(x')) for each fixed x' e B. Then it extends to a function u(x', zn) holomorphic in a narrow strip —e < yn < e around the interval (b(x'), t(x')) in the plane of complex variable zn = xn + iyn. The Cauchy-Riemann
equations force u(x',zn) to fulfill (d/dxn)u = — i(d/dyn)u in the strip (b(x'), t(x')) x (— e,e). Hence, we rewrite the original elliptic equation as a hyperbolic equation for a new unknown function of variables (x', yn). Since S is the graph of some smooth function xn = t(x') on B, the Cauchy data transform easily for the new unknown function.
In Section 3 we test our approach in the case of two variables. It is precisely the case treated in [9], and the approach of [9] does not work for n > 2. For n = 2, the geometric picture is especially descriptive because the complexification of x2 does not lead beyond R3.
On solving the Cauchy problem for a hyperbolic equation in a conical domain in the space of variables (x',yn), we are left with the task of continuing the solution given on the base of an isosceles triangle analytically along the bisectrix of the angle at the vertex, for each fixed x' e B. To this end we invoke the classical formula of Carleman established precisely for this configuration, see [14]. Of course, the use of Carleman's formula is justified only for real analytic solutions of the original elliptic Cauchy problem. In Section 4 we give a simple proof of this formula. Numerical simulations with Carleman's formula failed to manifest its striking efficiency. However, nowadays more efficient formulas of analytic continuation are available, cf. [17].
In Section 5 we investigate the Cauchy problem for the inhomogeneous Laplace equation in the space Rn of variables (x', xn) with odd n. As is shown in Section 2, it reduces to the Cauchy problem for the inhomogeneous wave equation in the space of variables (x',yn). The case n = 1 deserves a special study, for it concerns the initial problem for ordinary differential equations. If n = 3, the Cauchy problem for the wave equation possesses a very explicit solution constructed by Poisson. For odd n > 5 an explicit solution formula was derived by Hadamard in [1] by his method of descent. On substituting it into Carleman's formula and changing integrations over yn and x', we get a formula for solutions of the Cauchy problem for harmonic functions.
In Section 6 we restrict our attention to the Cauchy problem for the inhomogeneous Laplace equation in the space Rn of variables (x', xn) with even n. By the above it reduces to the Cauchy problem for the inhomogeneous wave equation in the space of variables (x',yn). The latter Cauchy problem admits a very explicit solution formula due to d'Alembert in the case n = 2 and Kirchhoff in the case n = 4. For general even n the formula seems to be first published in [1]. We combine it with Carleman's formula and change the integration over yn and over x '. This yields an explicit formula for solutions of the Cauchy problem for the inhomogeneous Laplace equation. To our best knowledge, this formula has never been published.
In Section 7 we analyse if our approach applies to Cauchy problems for elliptic equations of order different from two. Yet another question under study is whether the method of quenching functions in the Cauchy problem for the Laplace equation presented in [10] is actually a very particular case of formulas elaborated in this paper.
1. The Cauchy Problem
Let X be a bounded domain with piecewise smooth boundary in Rn. We require X to be of cylindrical form, i.e., X is a part of the cylinder B x R intercepted by two surfaces yn = b(x') and yn = t(x') over B, where B is a bounded domain with smooth boundary in the space Rn-1 of variables x ' = (x1;..., xn-1). For simplicity we assume that t(x') > b(x') for all x' e B, the case t(x') = b(x') for some or all x ' e dB is not excluded. The Cauchy data will be posed on the top surface S := {(x',t(x')) : x ' e B} which is tacitly assumed to be real analytic, cf. Fig. 1.
For an elliptic second order differential operator on the closure of X the Cauchy data on S look like
{u = uo on S, du
- = ui on S,
where v is the outward unit normal vector at S. Obviously, v = Vq/|Vq| where q = xn — t(x').
- 4
x'
xn t(x )
x„ = b(x')
B
Fig. 1. A typical domain under consideration
Lemma 1.1. If u is a smooth function near S satisfying u = u0 on S, then
du 1 f ,„ . „ du
dv
VIVx' t|2 + i
(-(Vx't, Vx'uü) +
5.
Proof. This is an easy exercise. □
Consider a nonlinear second order partial differential equation ^u = f (x, u, Vu) in X, where f (x,u,p) is a real analytic function on X x R x Rn. By Lemma 1.1, the Cauchy problem for solutions of this equation with data on S can be formulated in the following way. Given functions u0 and ui on S, find a function u in X smooth up to S which satisfies
Au
u
uX
f (x, u, Vu) uü Ul
in X,
on S, on S.
(1.1)
Lemma 1.2. There is at most one real analytic function u in XUS which is a solution of (1.1).
Proof. Let ui and u2 be two real analytic functions in XUS satisfying (1.1). Set u = ui — u2, then u is real analytic in XUS and vanishes up to the order 2 on S. Hence it follows that Au = f (x, u1, Vu1) — f (x, u2, Vu2) vanishes on S. Since A is a second order elliptic operator, we readily deduce that uX x =0 on S, and so u vanishes up to order 3 on S. Hence it follows that Au vanishes up to order 2 on S, and so (d/dxn)3u = 0 on S. Arguing in this way, we conclude that u vanishes up to the infinite order on S. Since u is real analytic in X U S, we get u = 0 in X, as desired. □
2. Hyperbolic Reduction
Assume that u is a real analytic function in X U S which satisfies (1.1). Then, for each fixed x' £ B, the function u(x', xn) can be extended to a holomorphic function u(x', xn + iyn) in some complex neighbourhood of the interval (6(x'),t(x')j. Without loss of generality we can assume that this neighbourhood is a triangle T(x') in the complex plane zn = xn + iyn with vertexes at b(x') and t(x') ^«£, where e > 0 depends on x'. We write U(x', xn, yn) for the extended function, so that u(x) just amounts to U(x', xn, 0).
Since u(x', zn) is holomorphic in a complex neighbourhood of (b(x'),t(x')], it follows from the Cauchy-Riemann equations that
/8 \j ' ( d \j ' \dx~J U (x,x„,y„)=(^ - «d—) U (x,x„,y„)
d
x
n
for all j = 1,2,.... Therefore, the Cauchy problem (1.1) for u transforms to the problem
¿x'U — = f (x', Zn,u, Vx'u, —»uyn), if x ' e b, Zn e t(x'),
U(x ', xn, 0) = uo(x ', zn), if x ' e B, zn = t(x'), (2.1)
Uyn(x',xn,0) = iui(x',zn), if x' e B, zn = t(x'),
relative to the new unknown function U(x ',xn,yn).
Hardly can (2.1) be specified within Cauchy problems for second order differential equations, for the number of independent variables is n +1 while the Cauchy data are given on a surface of dimension n — 1. Since the differential equation in (2.1) does not contain the derivative UXn, it is easy to deduce that the smooth solution to this problem is by no means unique. This no longer holds true for the holomorphic solution because of uniqueness theorems for holomorphic functions. Moreover, if U(x ',xn,yn) is holomorphic in zn = xn + «yn, then the differential equation in (2.1) is satisfied for all x' e B and zn e T(x') provided it is fulfilled for all x' e B and Zn = t(x ') + iyn with |yn | < e.
Thus, when one looks for a holomorphic solution to (2.1), this problem actually reduces to the Cauchy problem for a quasilinear hyperbolic equation in the space of variables (x', yn), whose principal part is given by the wave operator. More precisely,
uynyn = ¿x'U — f(x',xn + iyn, U, Vx'U, —iUyn), if x' e B, |yn| < e(x '), U(x ',xn, 0) = uo(x ', xn), if x ' e B, (2.2)
Uyn(x',xn,0) = «ui(x',xn), if x' e B,
where the variable xn is thought of as a parameter which runs over the interval (b(x'),t(x')). We are actually interested in the solution of this problem corresponding to the special choice xn = t(x') of the parameter. In other words, we study problem (2.2) on the hypersurface xn = t(x') in the space of variables (x,yn), the Cauchy data being given on the intersection of the hypersurface with the hyperplane {yn = 0}.
When passing to the Cauchy problem on the hypersurface xn = t(x') in Rn+1, one should interpret equations (2.2) adequately in accordance with the presence of parameter xn. Namely, each equations has to be fulfilled together with all derivatives in xn on xn = t(x').
Lemma 2.1. There is at most one function U(x',xn,yn) in a neighbourhood of S, which is real analytic in yn at yn = 0 and satisfies (2.2) with xn = t(x').
Proof. Let U1 and U2 be two functions in a neighbourhood of S, which are real analytic in yn at yn = 0 and satisfy (2.2) with xn = t(x'). In the coordinates (x ',xn,yn) the surface S is given as intersection of two hypersurfaces xn = t(x'), where x' e B, and yn = 0. Set U = U1 — U2, then U is real analytic in yn at yn = 0. We shall have established the lemma if we prove that each derivative (d/dyn)jU with j = 0,1,... vanishes for xn = t(x') and yn = 0. For j =0,1 this follows immediately from the conditions which U1 and U2 fulfil on S. For j < 2 this follows from the differential equation in (2.2) by induction. We check it only for the initial value j = 2, for the induction step is verified in much the same way. From (2.2) we get
Uy'nyn = ¿X' U1 — ¿x' U2
(f (x ', xn + iyn, U1, Vx' U1, — iU1,yn) — f (x ', xn + iyn, U2, Vx' U2, —«U2yn))
provided that xn = t(x ').
Since (d/dxn)j(U1 — U2) = 0 for xn = t(x'), yn = 0, and all j = 0,1,..., it follows that
U1,xfc (x',t(x'), 0) = (U1(x',t(x'), 0))xfc — U1,x„ (x ',t(x '), 0) *xfc (x' ) = = (U2(x',t(x'), 0))xfc — U2,x„ (x',t(x'), 0) txfc (x') = = U2,xfc (x ',t(x '), 0)
for each k = 1,..., n — 1. Moreover, we get
da' Ui = da' U2 (2.3)
on the surface xn = t(x'), yn =0 for all multi-indices a' = (a1,..., an-1). This yields readily Ax' U1 = Ax' U2 for xn = t(x') and yn = 0. Substituting these equalities into the formula for U.','y we obtain U.',' y (x ',t(x '),0) = 0 for all x ' £ B, as desired. □
ynyn ynyn v ' K ' ' ' '
Note that equalities (2.3) generalise to dad^n^1 U1 = dad^^1 U2 for xn = t(x'), yn = 0, and all multi-indices a = (a1,..., an) and an+1 =0,1,..., as is easy to check.
We have thus reduced the Cauchy problem for the Laplace equation perturbed by nonlinear terms of order < 1 to the Cauchy problem for the wave equation perturbed in the same way. The reduction is justified as long as the solution under study is real analytic in xn.
Perhaps the reduction does not make sense in the case n =1, for it leads to no simplification.
3. The Planar Case
To test the hyperbolic reduction of Section 2., we consider the case n = 2 in detail, assuming f to depend on x £XUS only.
Let X be a strip domain in R2 consisting of all x = (x1,x2), such that x1 £ (a, b) and b(x1) < x2 < t(x1), where (a, b) is a bounded interval in R and b, t are smooth functions of x1 £ (a, b). Write B := (a, b) and denote by S the curve {(x1, t(x1)) : x1 £ (a, b)} which is a part of dX. We focus on the Cauchy problem for the inhomogeneous Laplace equation given by (1.1). When looking for a solution u of this problem which extends to a holomorphic function u(x1, z2) of z2 = x2 + iy2 in a neighbourhood of {(x2,0) : x2 £ (b(x1), t(x1)]}, for each fixed x1 £ (a, b), we arrive at
U''2y2 = UX'ixi — f (x1, x2 + if x1 £ (a,b) |y2| <e(x1),
U(x1,x2,0) = uo(x1,x2), if x1 £ (a, b), (3.1)
U'2 (x1,x2,0) = iu1(x1,x2), if x1 £ (a, b),
which is a Cauchy problem for the inhomogeneous wave equation with parameter x2 relative to the unknown function U(x1,x2,y2) = u(x1,x2 + iy2), cf. (2.2). We are actually interested in finding a function U which satisfies (3.1) only on the surface x2 = t(x1), see Fig. 2.
x2 = i(xi)
Fig. 2. The case n = 2
It is an easy exercise to verify that the function
'2 x1+y2
(Gf)(xi,x2,y2) = - 2/dy2 J f(x'i,x2 + i(y2 - y2)) dx'i
X1 -y'2
satisfies the inhomogeneous wave equation and homogeneous (i.e., corresponding to u0 = «i = 0) initial conditions in (3.1). On the hand, d'Alembert's formula gives a function satisfying the homogeneous (i.e., corresponding to f = 0) wave equation and the inhomogeneous initial conditions in (3.1), see [18, Ch. I, § 7.1]. In fact, this is
D/ W \ u0 (xi + ^2,x2) + u0(xi -y2,x2K 1 f ' '
P (uo,ui)(xi,X2,y2) = -2--+2 ui(xi,x2)dxi, (3.2)
Xl—y2
where the right-hand side is well defined for all (xi, x2, y2) satisfying xi + y2 G (a, b) and xi — y2 G (a, b). The pairs (xi,y2) with this property form two cones C± in the plane, C± being the set of all (xi,y2), such that xi G (a, b) and ±y2 G [0, e(xi)), where
b — a
£(xi) = ——
a + b
x1--
1 2
Thus, given any twice differentiable function u0(xi,x2), differentiable function ui(xi,x2) of xi G (a, b) and any differentiable function f (xi,z2) of both variables, the formula U = Gf + P(u0, ui) yields a solution to the Cauchy problem (3.1) for all values of parameter x2 that do not lead beyond the domains of u0, ui and f. Had we known u0(xi, x2) and ui(xi, x2) for all values x2 G (b(xi), t(xi)], then the first initial condition of (3.1) would give U(xi,x2, 0) = w0(xi,x2) and so the solution to the Cauchy problem (1.1) by u(x) = u0(xi,x2). This just recovers the reduction but is not of use to solve the original Cauchy problem. However, on substituting x2 = t(xi) into U(xi,x2,y2) we obtain
y2 xl+y2
u(xi,t(xi)+ iy2) = — 2y dy2 J f(x'i,i(xi) + i(y2 — y2)) dxi+
2
0
xi+y2
0 xi y2 (3.3)
+ u0(xi + y2,t(xi)) + u0(xi —y2,t(xi)) +2 J ui(x'i,i(xi))dx'i
Xl— y2
for all xi G (a, b) and |y2| < e(xi). Note that (xi,t(xi)) fails to lie on the curve S for all xi G [xi — y2, xi + y2] unless t(xi) is constant. Therefore, u(xi,t(xi) + «y2) is determined by the Cauchy data of u in some neighbourhood of S. This forces us once again to confine ourselves with solutions which are real analytic in the variable x2.
For fixed xi G (a, b), formula (3.3) gives the restriction of the function u(xi, z2), holomorphic in z2 in the triangle with vertexes at b(xi) and t(xi) ^ «e(xi), to the side t(xi) +«[—e(xi), e(xi)] of the triangle. This limits application of hyperbolic theory. Our next objective is to continue the function from the side of the triangle analytically along the bisectrix of the angle at b(xi). This is a problem of analytic continuation.
4. Carleman Formula
Let D be a domain in the complex plane C of variable z bounded by lines BO and OA and by a smooth curve c = AB lying inside the angle BOA. Write ZBOA = an with 0 < a < 2.
Choose the univalent branch of the analytic function in the complex plane with a slit along the ray arg w = n, which takes the value 1 at w =1.
Lemma 4.1. If u is a holomorphic function in D continuous up to the boundary, then
u<z) = I u(Z )exp N ((r — 1) ;
holds for any point z £ D on the bisectrix of the angle BOA, where Co is a complex number corresponding to the vertex O of the angle.
This formula is due to Carleman [14]. To our best knowledge it was the first formula of analytic continuation using the idea of quenching function. Since that time such formulas in complex analysis and elliptic theory are called Carleman formulas, see [17], [15].
Proof. Fix any z £ D lying on the bisectrix of the angle BOA. For N = 1, 2,..., we apply the Cauchy integral formula to the function
»<< )exP N (( — 1
which is holomorphic in D and continuous in the closure of D. Since its value at C = z is u(z), we get
1 r ^ „((C — Co)1/c
«")= ¿1 /«(Z)expN(fcf) - 0 C^+ + à / «(Z)expN((Z-f )1/a - 1)
dC C - z'
(4.1)
If c e dD \ c, then
C - Co)1/c z - Co J
C - Co
z - Co
1/a (-L n , ,
eXP I ± 2 = ±
C - Co
Z - Co
1/a
(( C — Co )1/a )
and so the modulus of expN^-c~) — 1J equals e-N. Letting N ^ to in (4.1) establishes
the lemma. □
Having disposed of this preliminary step, we now turn to the problem of analytic continuation we have encountered in Section 3. We apply Lemma 4.1 in the plane of complex variable z2 = x2 + iy2. Given any fixed x1 £ (a, b), we take the triangle T(x1) with vertexes O := b(x1) and A := t(x1) — ie(x1), B := t(x1) + ie(x1) as D, cf. Fig. 3.
i
In this case
Fig. 3. Recovering a holomorphic function
2 ( e(x1)
— arctan '
n Vt(x1) - b(x1)
depends on xi and the bisectrix of the angle BOA coincides with the real axis. The solution w(xi,z2) is given on the edge AB and we are aimed at reconstructing it in the interval (6(xi),t(xi)).
a
Theorem 4.2. Let n = 2. For each solution u of the Cauchy problem (1.1) in X which is real analytic up to S, the formula
eOi) 1
u(x)= lim 2L fv(x1,t(x1),y2)exp N((t(x) — b(xi) +a —1
N^to 2n J W x2 — b(x1) / /t(x1)— x2 + iy2
-e(xi)
holds for all x £ X.
Proof. This follows immediately from Lemma 4.1 and formula (3.3) giving an explicit continuation of the solution u(x1,x2) along S to the plane of complex variable z2 = x2 + iy2. □ This formula is especially simple if S is a segment x2 = to, i.e. the graph of a constant function t(x1) = to of x1 £ (a, b). If moreover f = 0 then formula (3.3) transforms to
TT, . \ uo(x1 + y2,to)+ uo(x1 — y2,to) i f . ' w ' v (x1,to,y2) =-2--+2 u1(x1 ,to)dx1
X1 — y2
for all x1 £ (a, b) and |y2| < e(x1). Substituting this into the formula of Theorem 4.2 we get
xi+e(xi)
u(x) = lim / u(x1, to) KKn(x1,x2,x1 — x1) dx1 —
N ^TO J
Xi—£(Xi) (4.2)
xi+e(xi) e(xi)
Jini^ J (x!,to)( J 9KN(x1,x2,y2) dy^dx1,
x
1—e(xi)
where
exp N fft(x ') - 6(X;} + ) 4 - 1 ' 1 W x„ - b(x') y
Kn (x ,x„,y„) =
2n t(x ') - x„ +
Formula (4.2) can be regarded as an elliptic analogue of the d'Alembert formula for the wave equation.
Note that nowadays there are many explicit formulas of analytic continuation which are simpler than the original formula of [14]. We refer the reader to [17].
Xi —xi
5. Poisson Formula
In this section we discuss the case n = 3 in detail, assuming the function f to depend on x £ X U S only. The Cauchy problems for the inhomogeneous Laplace equation reduces to the Cauchy problem for the inhomogeneous wave equation. This latter reads
{U^ya = Ax' — f (x ',x3 + iy3), if x ' £ B, |y31 < e(x '), U(x ',x3, 0) = uo(x ', x3), if x ' £ B, (5.1)
Uy3(x',x3,0) = iu1(x',x3), if x' £ B,
x3 being thought of as parameter. We are aimed at finding a function U which fulfills (5.1) on the surface x3 = t(x ).
The advantage of the reduction lies in the fact that the Cauchy problem for hyperbolic equations is well posed in the class of smooth functions. For n = 3, there is an explicit formula for its solution due to Poisson, see [18, Ch. III, § 6.5]. More precisely,
y3
U (x',x3,yS ) = - 2. I « I f ^f-f
0 |x''-x'|<|y3|
d sgn y3 f uo'x ",X3 ) „ sgn y3 f rn^x ",X3 ) ,,
+ ^--—, dx +--—. dx
dy3 2n J ^y2 — |x'' - x '|2 2n J ^y2 — |x'' - x '|2
| x'' —x' | < | y31 | x'' —x' | < | y3 |
(5.2)
for all x ' e B and |y3| < e(x ').
For formula (5.2) to make sense it is certainly required that, for any y3, the ball |x ''— x'| < |y3| would belong to the domain B in Re'1, where the Cauchy data u0(x ',xn) and ui(x',xn) are given. Since y3 varies in the interval (—e(x '), e(x')), we get readily the formula e(x') = d(x ', dB), the distance from x ' to the boundary of B, cf. Fig. 4.
yn
d'x dB)
B
x
Fig. 4. Reduction to imaginary cones
Theorem 5.1. Let n = 3. For each solution u of the Cauchy problem (1.1) in X which is real analytic up to S, the formula
E(x')
u'x) = lim — N2n
-e(x')
TTt ' \ AT((t(x ')—b(x ') + ly3V
U (x ,t(x ),y3)exp nl x3 — b'x ' ) j
-1
dy3
t'x ' )— x3 + «y3
2 / e(x' ) \
holds for all x G ^ where a = — arctan ( ——--——- )
n vt(x ')- b(x')/
U'x ' ) — b'x ' ) /
Proof. This is a direct consequence of Lemma 4.1 and formula (5.2) which gives an explicit continuation of the solution u(x ', x3) along S to the plane of complex variable z3 = x3 + iy3. □ On substituting (5.2) into the Carleman formula of Theorem 5.1 we arrive at an explicit formula for solutions of the Cauchy problem for the inhomogeneous Laplace equation. The computations are cumbersome, and so we confine ourselves with the case f = 0, as in (4.2). By the very construction of the Carleman kernel, (x ',x3,e(x ')) tends to zero as N ^ to, for any
i
x ' £ B and x3 £ (b(x '),t(x ')). Hence
' d e(x ) — K Kn (x ',x3,y3)
t(x) = — Km f u(x'',-(x'))( f 1 dy3 2 , ,, dy$)dx''
N ^TO J V J n — |x '' — x ' |2 '
|x'' —x '| <e(x') |x '' —x '| (5 3)
lim / (x'',t(x '))( / 1 9Kn(x ',x3,y3) dy^dx''
N-TO J ^ ' V J n ^yl — |x '' — x ' |2 ^
|x '' —x ' |<e(x') |x '' —x ' |
for all x £ X.
Formula (5.3) can be thought of as an elliptic analogue of the Poisson formula for the wave equation.
6. Kirchhoff Formula
The solution of the Cauchy problem for the wave equation bears certain structure which changes in odd and even dimensions. For this reason we consider also the case n = 4 in detail. The corresponding formula for solutions of the Cauchy problem for the wave equations is known as the Kirchhoff formula, see [18, Ch. III, § 6.4] and elsewhere.
By the above, the Cauchy problem for the Laplace equation in a cylindrical domain X c R4 reduced to
vy4y4 = ^x' — f (x',x4 + iy4), if x' £ B, |y41 < e(x'),
U(x ',x4, 0) = uo(x ', x4), if x ' £ B, (6.1)
Uy4(x',x4,0) = iu1(x',x4), if x' £ B,
where x ' = (x1,x2,x3) varies in a domain B c R3, e(x ') stands for the distance from x ' £ B to the boundary of B, and x3 is thought of as parameter in (b(x '),t(x ')]. The Cauchy data uo and u1 are in C3(B) and C2(B), respectively. The Kirchhoff formula gives
1 f f (x'', x4 + l(y4 — |x '' — x ' |))
|x — x |
|x '' —x '| <|y4 |
4 (6.2)
1 f f(x'',x4 + l(y4 — |x '' — x '„
U (x ,x4,y4) = -7—r.-Tj- dx +
4n J |x ' ' — x ' |
|x '' —x ' | < | y4 |
+ —--i uo(x'', x4)da(x'') +--i iu1(x'', x4)da(x'')
dy4 4ny4 J 4ny4 J
| x '' x ' | = | y41 | x '' x ' | = | y41
for all x ' £ B and |y4| < e(x ').
The substitution x4 = t(x ) into U gives the restriction of the function U, holomorphic in z4 = x4 + iy4, to the edge t(x ') + i[—e(x'),e(x ')] of the triangle T(x ') c C, where U is holomorphic. Using Carleman's formula of Lemma 4.1, we arrive at a formula for u(x) similar to that of Theorem 5.1. It reads in much the same way, with x3 and y3 replaced by x4 and y4, respectively. For short we restrict our attention to a formula like (5.3).
Corollary 6.1. Let n = 4. For each solution u of the Cauchy problem (1.1) with f = 0 in X, which is real analytic up to S, we get
1 (d— KKn) (x ',x4, |x'' — x '|)
u(x) = — lim u(x'',t(x '))--—-:-:-dx ''-
V 7 N-to J K ' V 2n |x '' — x '|
|x'' —x'| <e(x ') (6.3)
lim i (x'',t(x '))
N^TO y dx4
1 9Kn(x ',x4, |x ' ' — x '|) h — dx
n —to J dx4 ' 2n |x ' ' — x ' |
| x '' —x ' | <e(x ')
for all x EX.
Proof. The proof is quite elementary although cumbersome. We first substitute the integral of mo on the left-hand side of (6.2) into Carleman's formula. Integration by parts yields
d f 1
j j Mü(x//,t(x/))da(x")j Kn (x',^,^) dy4 =
|x''-x'| = |y4|
(- I uo(x '',t(x ' Kn (x ',X4 ,^4)
V4ny4 J )
|x''-x'| = |y4|
e(x')
1
—e(x') |x'' —x'| = |y4
The first integral on the right-hand side is equal to f1
j uo(xt(x '))d<r(x d— Kn(xX4, ^4) dy4.
V2ne(x ')
|x'' — x'|=e(x')
which vanishes as N ^ to by the construction of the kernel KN(x ',x4,e(x ')). Indeed, the point t(x') + ie(x') belongs to the top leg of the angle BOA, and x4 to its bisectrix.
Furthermore, we write the second integral on the right-hand side as the sum of two integrals. The first integral is over y4 e (—e(x'), 0) and the second one over y4 e (0,e(x')). In the second integral we change the variable by y4 ^ —y4, and then evaluate the sum, obtaining
e(x')
i ( 1 i ,,, ^d
\ d
uo(xt(x'))d<r(x '') ) —— K»(x x4, ^4) dy4 =
V4ny4 J ' / dy4
—e(x') |x'' — x' | = | y41
e("') 1 d
(---Mo(x",t(x'))da(x"n -— K Kn (x',x4,y4) dy4.
„ V2ny4 J J dy4
0 |x'' —x'| = |y4|
Since dx '' = da(x")dy4, we deduce from Fubini's theorem that the latter integral just amounts to
(—KKN) (x ', x4, |x'' — x '|)
L /
u(x ' ',t(x ' -T-^-J.-dx ' '
2n |x ' ' — x ' |
| x'' —x' | <e(x')
as desired.
The same (even easier) reasoning applies when one substitutes the integral of w on the left-hand side of (6.2) into Carleman's formula. The details are left to the reader. □
Formula (6.3) is an exposition of Kirchhoff's formula for the wave equation in the context of elliptic theory. We have already mentioned another interpretation of Kirchhoff's formula in [16]. Unfortunately, we could not understand this latter paper.
7. Concluding Remarks
The developed method of analytic continuation in the plane of complex variable zn = xn + iyn still works if the Cauchy problem under study is nonlinear. Having granted a holomorphic
solution U(x',xn, yn) to the Cauchy problem (2.2) on the surface xn = t(x'), we use Carleman's formula to extend U to all of X. The extension looks like
£(>' )
u(x) = lim
N—œ
U(x ', t(x '), y„) Kn(x ', x„, y„) dy„
(7.1)
-e(x ')
for all x eX.
Formula (7.1) allows one to construct explicit formulas similar to (4.2), (5.3) and (6.3) for arbitrary n. To this end one uses classical formulas for the solution of the Cauchy problem for a second order hyperbolic equation by the descent method of Hadamard, cf. [1], [18, Ch. VI, §. 5.2]. We were rather interested in equations of mathematical physics.
The simplest formula is obtained for even n ^ 4, thus generalising Kirchhoff's formula (6.3). If u0 e C(n+2)/2(S) and ui e Cn/2(S), then every solution u of (1.1) with f = 0 represents by
u(x) = lim / dx ' 'x
N—œ J
X u(x '',t(x' ))
| x'' —x ' | <e(x ' )
(-1) n2
d 1 n —2
((---) y„ K Kn (x',x„, |x ' ' x' |)
ct„_i1 • 3 •.. .• (n-3)
(-1)n-1 2
+ (7.2)
+ (x'',t(x' ))
dx4 an-11 ^3 •...• (n-3)
n —4 2
d 1 ^ ---" 9 Kn )(x',x„, |x ''-x'|)
|x - x |
for all x e X, where <rn-i stands for the area of the (n — 2) -dimensional unit sphere in Rn 1. We used here an exotic designation for the integral by purely technical reasons.
Remark 7.1. Formula (7.2) has much in common with the familiar formula of [10].
The method of proof carries over to right-hand sides f (x, u, Vw) which are affine functions of u and Vw. This is the case, e.g., for the Helmholtz equation, cf. [18, Ch. VI, §. 5.7].
Another class of equations which may be handled in much the same way consists of those of the form
Au + = f (x^
where A is a linear differential operator containing at most the derivative wXn but no higher order derivatives in xn, see [18, Ch. III, § 6.4].
The research of the first author was done in the framework of the Mikhail Lomonosov Fellowship which is supported by the Russian Ministry of Education and the Deutsche Forschungsgemeinschaft.
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Гиперболические формулы в эллиптической задаче Коши
Дмитрий П.Федченко Николай Тарханов
Мы изучаем задачу Коши для уравнения Лапласа в цилиндрических областях с начальными данными, заданными на части границы. Сводя данную задачу к задаче Коши для волнового уравнения в комплексной области и используя гиперболическую теорию, получаем точные формулы для решения, развивая тем самым классический подход Леви (1927).
Ключевые слова: уравнение Лапласа, задача Коши, волновое уравнение, формулы Карлемана.
