CC BY
12
УДК 517.55
An Extremal Problem Related to Analytic Continuation
Olimjan Makhmudov*
Department of Mechanics and Mathematics, University of Samarkand, University Boulevard, 15, 703004, Samarkand,
Uzbekistan
Nikolai Tarkhanov^
Institute of Mathematics, University of Potsdam, Am Neuen Palais, 10, Potsdam, 14469
Germany
Received 06.07.2014, received in revised form 06.09.2014, accepted 06.10.2014 We show that the usual variational formulation of the problem of analytic continuation from an arc on the boundary of a plane domain does not lead to a relaxation of this overdetermined problem. To attain such a relaxation, we bound the domain of the functional, thus changing the Euler equations.
Keywords: extremal problems, Euler equations, p -Laplace operator, mixed problems.
Introduction
Let X be a bounded domain with smooth boundary in the complex plane C and S a nonempty open arc on the boundary of X. The problem of analytic continuation of functions given on S to X is of great importance in analysis, see [2]. It reads as follows: Given any function u0 on S, find an analytic function u in X whose limit values exist in a reasonable sense and coincide with u0 at S. This problem is not normally solvable unless S = dX, for no nonzero smooth function uo of compact support in S extends analytically to X.
Analytic functions in X are solutions of the Cauchy-Riemann system du = 0 in the domain. To get an approximate solution of the problem of analytic continuation, one can relax the system du = 0 and require that du be "small" in some sense in X. In other words, the problem is replaced by a variational problem which has the advantage of being constructively solvable. When looking for a solution u of Sobolev space W1,p(X) with p > 1, one considers the variational problem of minimizing the functional
I(u) := — |du|p dx ^ min (1)
Jx P
over the set A of all u e W1,p(X) satisfying u = u0 on S. By the Sobolev trace theorem, the condition p > 1 implies that u has boundary values belonging to W 1-1/p,p(dX), hence the equality u = u0 is well defined almost everywhere on S for all u0 e W 1-1/p,p(S).
In the language of partial differential equations the problem of analytic continuation from S is called the Cauchy problem with data at S for solutions of the elliptic system <9u = 0 in X. The variational approach to such problems was first elaborated in [10].
If u e W 1,p(X) is an analytic continuation of u0, then u e A and I(u) = 0. Hence, u is a solution of the variational problem I(u) ^ min over the set u e A. Conversely, if the functional I attains a minimum at a function u e A and this minimum just amounts to zero, then u is
*[email protected] [email protected] © Siberian Federal University. All rights reserved
actually an analytic continuation of u0. However, the minimum need not vanish. The infima of I over A belong to the set of all critical points of I in A. A function u e A is proved to be a critical point of the functional I in A if and only if it satisfies the so-called Euler equations for I. They look like
d* (|du|p-2du) = 0 in X,
u = u0 at S, (2)
n(|du|p-2 du) = 0 at d X\S,
where d* is the formal adjoint of d and n(|du|p-2du) = a(d*)( —iv)(|du|p-2du) the Cauchy data of |du|p-2du on dX with respect to d*. Here, a(d*)( —iv) stands for the principal symbol of d* evaluated at the cotangent vector —iv, v = (vi, v2) being the unit outward normal vector of dX.
The case p = 2 is of particular interest, since the Euler equations for functional (1) are linear in this case. If moreover S is the whole boundary, the extremal problem goes back at least as far as the theory of harmonic integrals on complex manifolds, see [5].
The nonlinear second order differential operator Lp(u) = d* (|du|p-2du) is called the complex p-Laplace operator. This is an analogue of the p-Laplace operator Ap(u) = div (|Vu|p-2Vu) in Rn which plays an important role in nonlinear potential theory and appears often in physics and engineering. A mixed boundary value problem for Ap similar to (2) was recently studied in [20]. For p = 2, this is precisely the well known mixed problem for the Laplace equation first studied by Zaremba [26].
The complex p -Laplace operator has been investigated also within complex analysis in the study of extremal problems in Bergman spaces of analytic function, see [4]. In [4], the regularity problem for solutions of the Dirichlet problem for the homogeneous equation Lpu = 0 is discussed, i.e. problem (2) in case S is all of dX.
The differential equation (2) in the domain X is the so-called degenerate elliptic equation. It can also be viewed as an elliptic system of two real such equations. Ellipticity fails exactly at the points where du = 0. In the case S = dX the problems similar to (2) or its variational source I(u) ^ min have been studied by Morrey in [14-16] and many others, cf., e.g., [7,8] and the references given there. The corollary from those investigations is that, if u0 is a polynomial, the Dirichlet problem for Lp possesses a unique solution of Holder class C 1,A(X) with exponent A > 0 depending on p (see [4], Theorem D).
In this paper we are interested in the variational problem I(u) ^ min over A in the case where S is nonempty and different from the whole boundary dX. For the study of mixed boundary value problem (2) we invoke the theory of weak boundary values of solutions to elliptic systems developed in [24]. We prove that a function u e W 1,p(X) satisfies (2) if and only if it is an analytic extension of u0 in X. Hence it follows that problem (1) has actually no solutions u different from the analytic continuation of u0 in X, if there is any. Thus, (1) is not suited to be a good relaxation of the Cauchy problem for the Cauchy-Riemann system in X with data on S. This result differs considerably from that of the paper [20] which asserts that the analogous mixed problem for the p -Laplace operator in Rn is uniquely solvable for all data u0 e W1,p (S). (In fact [20] allows also nonzero Neumann data |Vu|p-2n(Vu) = u1 at dX \ S, where u1 is a continuous linear functional on W 1-1/p.p(dX \ S).)
In order to achieve a true relaxation of the problem of analytic continuation one has to look for local infima of the functional I(u) on bounded closed subsets of A. To this end, given any R > 0, we denote by AR the set of all functions u e W 1,p(X), such that u = u0 at S and ||u||wi,p(x) < R. Obviously, AR is a convex bounded closed subset of A and the family AR increases and exhausts A, when R ^ ro. The main result of this paper is the following theorem which goes back at least as far as [18].
Theorem 0.1. Suppose S C dX is a nonempty arc different from the whole boundary and uo e W 1-1/p'p(S), where 1 < p < ro. For each R > 0, there is a unique function uR e W 1,p(X) in Ar minimising functional (1) over AR.
It is clear that the Euler equations for the critical points of I on the set AR are different from mixed problem (2). However, if the family {uR}R>0 is bounded, then it stabilises for R large enough. The limit function u is the minimum of I over all of A, and so u is an analytic extension of u0. As but one consequence we deduce that for a function u0 e W 1-1/p,p(S) to admit an analytic continuation in X it is necessary and sufficient that the family {uR}R>0 of Theorem 0.1 be bounded in W 1'p(X).
Note that the "approximate" solutions uR can be constructed, e.g., by the classical Ritz method, see [17].
The above results extend to the Cauchy problem for solutions of first order elliptic systems Au = 0 in a smoothly bounded domain X c Rn with data on an open part S of the boundary. More precisely, A is assumed to be a square matrix of first order scalar partial differential operators in a neighbourhood U of the closure of X and the principal symbol a(A)(x, £) of A to be invertible for all nonzero £ e Rn. Then the formal adjoint A* of A is also elliptic and we require A* to satisfy the so-called uniqueness condition for the local Cauchy problem in U, see [24, p. 185]. In particular, one can choose a square Dirac operator as A, e.g., the Cauchy-Riemann operator in Clifford analysis, etc. The corresponding p-Laplace operator is u ^ A*(|Au|p-2Au). The p-Laplace operator Ap in Rn does not belong to this class of operators, for the gradient operator fails to be elliptic unless n =1.
Let us dwell on the contents of the paper. Section 1. presents some preliminaries on the Cauchy-Riemann system from the viewpoint of partial differential equations. In Section 2. we give a variational formulation of the problem of analytic continuation from a part of boundary and derive the corresponding Euler equations which form a mixed boundary value problem in X. In Section 3. we show that the mixed problem actually reduces to the original problem of analytic continuation. Section 4. is devoted to further development of the variational approach to the problem of analytic continuation. Finally, in Section 5. we touch a few aspects of the theory of p -Laplace operators.
1. The Cauchy-Riemann system
The Cauchy-Riemann operator in the complex plane of variable z = xi + ix2 is defined by
du := 1 (di + 182) u,
d
where cj = —— for j = 1, 2. dxj
When identifying a complex-valued function u = u1 + iu2 with the two-column of real-valued functions u1 =: Ku and u2 =: 9u, one specifies the operator d within (2 x 2) -matrices of first order partial differential operators with real coefficients. More precisely,
1 ( di -82 \ ( ui
a" = 2( d2 ft A u2 j. (3)
Endowing the complex plane with the usual Hermitean structure we introduce the Hilbert space L2(C).
The formal adjoint d* of d is defined by requiring (du,g)L2(C) = (u,d*g)L2(c) for all smooth functions u and g of compact support. When identifying complex-valued functions with two-columns of real-valued ones and using the Hermitean structure in R2, we get precisely the same formal adjoint operator. That is
d*g = -dg =2(
1 / -di -d2 \ ( g 1
2 V 82 -di )\ g2
'<*>«> = 5( Iii -t) (4)
for g = g1 + lg2 •
The classical principal symbol of the Cauchy-Riemann operator is the family of (2 x 2)-matrices
1/ i£i -if 2 2 \ «fi
parametrised by f G R2. The operator d is elliptic in the sense that the family (4) is invertible for all f G R2 \ {0}.
By operation with symbols, we get <r(d*) = <r(d)*, the asterisk on the right-hand side indicating the adjoint matrix. Hence it follows that the formal adjoint is also elliptic. Moreover, it is easy to verify that
d*d = - - E2 A,
4 2 '
where E2 is the unit matrix of size 2 x 2 and A the Laplace operator in R2. This can be equivalently reformulated by saying that (2 times) d is a Dirac operator in the plane.
Solutions of the system du = 0 in a domain Xc R2 are known to be analytic (or holomorphic) functions in X. In this paper we restrict ourselves to functions u of Sobolev space W 1,p(X) with 1 < p < to. If X is bounded by a smooth curve, then each function u G W 1,p(X) possesses a trace on dX in the sense of Sobolev spaces which is an element of W1/p ,p(dX), where 1/p + 1/p' = 1, see for instance [1].
The following formula is known as the Green formula in complex analysis. It is a very particular case of Green formulas for general partial differential operators, see [24, p. 300].
Lemma 1.1. Suppose X is a bounded domain with piecewise smooth boundary in R2. Then
/ ((du, g)x - (u,d*g)x) dx = - (u, a(<9*)(-iv)g)xds, JX JdX
for all u G W 1,p(X) and g G W1,p (X), where v = (v1, v2) is the unit outward normal vector of the boundary.
The restriction of ct(<9*)(-iv)g to the boundary is called the Cauchy data of g at dX with respect to the operator <9*, see [24, p. 301]. It is usually denoted by n(g).
2. An extremal problem
Let X c R2 be a bounded domain with smooth boundary and S a nonempty open arc on dX. If u G W 1,p(X) is an analytic function in X vanishing at S, then u is identically zero in all of X, see, e.g., Theorem 10.3.5 of [24]. This raises the following problem of analytic continuation going beyond function theory: Given a function u0 G W1/p 'p(S), find u G W 1,p(X) which is analytic in X and satisfies u = u0 at S. Throughout the paper we tacitly assume that u0 = 0, since otherwise the problem is trivial.
If S' is a nonempty open arc on dX whose closure belongs to S, then the analytic continuation u is uniquely defined by the values of u0 at S'. Hence, the problem of analytic continuation from S is overdetermined. In particular, if u0 is a smooth function with compact support in S, then u0 extends to an analytic function u G W 1,p(X) if and only if u0 = 0.
To construct a variational relaxation of the problem of analytic continuation, we introduce the functional
I(u) := / - |<9u|p dx (5)
Xp
and give I the domain A consisting of all u G W 1,p(X), such that u = u0 at S. For the calculus of variations it is important that A is a convex closed subset of W 1,p(X).
Lemma 2.1. The functional I is strongly convex on A.
Proof. We have to show that if u, v e A then
I(tu + (1 - t)v) < 11(u) + (1 - t) I(v)
for all t e (0,1). Note that if u and v are two different elements of A, then du and <9v are different functions on X. Indeed, if du = <9v almost everywhere in X, then the difference u — v e W 1,p(X) is holomorphic in X and vanishes at S. By uniqueness, we get u — v = 0 in X, a contradiction. Now the strong convexity of the function |y|p, p > 1, implies
I(tu + (1 — t)v) = / 1 |d(tu +(1 — t)v) |p dx <
/X P
<t / 1 |du|p dx +(1 - t) / 1 |di|p dx = tl (u) + (1 - t) I (i) x P Jx P
for all u, v G A with u = v and all t G (0,1), as desired.
□
It follows from the lemma that there is at most one function u G A at which I attains its infimum over A.
Consider the extremal problem of finding the minimum of the functional I on the set A, if there is any. Clearly, if u G W 1,p(X) is an analytic extension of u0 into X, then u is a solution of the extremal problem, the minimal value of I on A being I(u) = 0.
The following lemma describes all critical points of the functional I on A. The corresponding equations are known as Euler equations for the extremal problem I(u) ^ min over u G A, see [16].
Lemma 2.2. Assume that the functional I : A ^ R attains a local minimum at a function u G A. Then u satisfies
d* (|<9u|p-2<9u) = 0 in X,
u = uo at S, n(|du|p-2du) = 0 at dX \ S,
Proof. Let v G Cœ(X) be an arbitrary complex-valued function vanishing on S. Write u = u1 + «u2 and v = v1 + iv2. For each e = (e1, e2) in R2, the variation (u1 + e1v1, u2 + e2v2) is left in A. Therefore, if I attains a local minimum at u, then the function F(e) = I(u1 + e1v1, u2 + e2v2) takes on a local minimum at e = 0. It follows that e = 0 is a critical point of F, i.e. both derivatives Fji and F'2 vanish at the origin. An easy computation shows that
f'i (0) = ix (|dur2du Ii;)) /x>
F2 (0) =
-2 â.. y d;v; 2
-2âu y d2v2 2
âur - I „ I I dx
d;i2
whence
F'ei (0) + f; (0) = i (|öu|p-2öu,9i)x dx = 0 J X
(6)
for all smooth functions v on the closure of X which vanish at S. In particular, equality (6) holds for all smooth functions with compact support in X, thus implying
d* (|<9w|p-2<9w) = 0
(7)
in the sense of distributions in X.
The function |<9w|p-2<9w is easily verified to belong to Lp (X), where p' is the real number with 1/p + 1/p' = 1. Since <9 is a Dirac operator, |du|p-2du is harmonic in the sense of distributions in X. By Weyl's lemma, this function is harmonic in X. We can now invoke the theory of [23], [24, 9.4] to conclude that |<9u|p <9u admits a distribution boundary value in W-1/pV (gX).
The distribution boundary value of |<9w|p-2<9w is referred to as weak limit value in [24]. In Section 9.4.3 ibid. the weak limit values are proved to be equivalent to the so-called strong limit values. These latter are precisely those values on the boundary which take part in the Green formula, see Section 9.4.1 in [24]. By Lemma 1.1, we get
(|S,,r2ft„dv},= -jax W^r2*,,«}.,
and so by (7) the right-hand side is equal to zero for all v e CTO(X) vanishing at S. Hence it follows that the distribution n(| du|p-2du) e W-1/p ,p (dX) has support in dX \ S. □
It is worth emphasizing that both d* (|du|p-2du) in X and n(|du|p-2du) at dX are understood in the sense of distributions.
L
3. Complex p-Laplace operator
The Euler equations for the variational problem I(u) ^ min over u e A form a mixed problem for solutions of the nonlinear differential equation Lpu = 0 in X, where
Lpu := d* (|du|p-2du}
for u e W 1,p(X). This operator is called the complex p-Laplace operator by analogy with the p-Laplace operator Ap(u) = div (|Vu|p-2Vu) in Rn.
An easy computation shows that Lp is a degenerate elliptic operator. Ellipticity is violated exactly at the points where du = 0. We endow this operator with the domain W 1,p(X) and interpret it in the weak sense, see (6). Moreover, given any u e W 1,p(X), the distribution Lp(u) in X extends (non-uniquely) to a continuous linear functional on W 1,p(X) by
(Lp(u), v) := (|du|p-2du, dv)^} (8)
for all v e W1,p(X).
If u e W 1,p(X) satisfies Lpu = 0 in X, then f = |<9u|p-2<9u is a weak solution of the equation <9*/ = 0 in X. Since dd* = -1/4 A, it follows by Weyl's lemma that f is a harmonic function in the domain X. Moreover, from f e Lp (X) we deduce that f is of finite order of growth near the boundary. Therefore, the function f admits boundary values at dX in the sense of distributions, see [23] and Section 9.4 of [24]. In the sequel by n(f) = ct(<9*)(-iv)f we mean the Cauchy data of f on dX with respect to d*.
Lemma 3.1. Assume that S = dX. If u e W1,p(X) is a solution of the mixed boundary value problem
9* (|du|p-2du} u
v n(|du|p-2du)
then <9u = 0 in X.
0 in X,
uo at S,
0 at dX \ S,
Proof. This lemma is actually a very particular case of Theorem 10.3.5 of [24] applied to the elliptic differential operator d* = — d. More precisely, set f = |<9u|p-2<9u. By the above, f is an antiholomorphic function in the domain X of finite order of growth at the boundary. Moreover, the Cauchy data of f with respect to d* vanish in the complement of <S. Since S is different from the whole boundary, it follows that the complement of S contains at least one interior point at the boundary. Applying Theorem 10.3.5 of [24] yields f = 0 in X. Since du e Lp(X), we get |f | = |du|p-1 whence du = 0 almost everywhere in X, as desired. □
If u e W 1,p(X) is an analytic extension of u0 into X, then u is obviously a solution of the mixed problem. Hence, the mixed problem of Lemma 3.1 is actually an equivalent reformulation of the problem of analytic extension from S into X, provided S = dX. The problem of analytic continuation from the whole boundary is stable. In the case S = dX the mixed problem becomes the Dirichlet problem for the complex p-Laplace operator. Its solutions need not be analytic functions in X. We conclude that the Euler equations of the extremal problem I ^ min over A can be thought of as relaxation of the problem of analytic continuation only in the case S = dX.
4. Relaxation of the problem of analytic continuation
We now restrict ourselves to the case where S is different from the whole boundary. Write m for the infimum of I(u) over u e A. We are aimed at finding those function u e A at which the functional I(u) takes on the value m. The integrand in (5) is
L(v) = — (K — v2)2 + (v1 + v2)2)p/2 , (9)
p
where v = (v1,v2) is a (2 x 2) -matrix whose columns are substitutions for d1u and d1u, respectively.
Lemma 4.1. For each fixed p ^ 1, the function L(v) is convex in the entries of v e R2x2.
Proof. We neglect the factor (p2p)-1 in (9), which is inessential for the proof. Write
•j=(j)
for j = 1, 2. A cumbersome but quite elementary computation shows that
E LVj(v) wj =
i,k=1,2 ¿,¡ = 1,2
p_4X22 2 \
= pL(v)^ (L(v)pL(w)p + (p — 2) ((vj —v2)(w1 —w2) + (v! +v2)(w1 + w?)) J (10)
for all wj = ^ w2 j = 1,2, in R2. Since p > 1, the right-hand side of (10) is greater than or equal to
pL(v)(L(v)pL(w)p — ((v1 —v22)(w! — w2^^1 +v2)(w1 +w2^2) > 0,
the last inequality being due to the Cauchy-Schwarz inequality for the scalar product in R2. The nonnegative definiteness of the quadratic form in (10) is equivalent to the convexity of L as a function of the entries of v, see Lemma 1.8.1 of [16] and elsewhere. □
Thus, the general hypothesis of [16, p. 91] concerning the integrand function L(v) are satisfied. It should be noted that L(v) fails obviously to be strongly convex.
If u G A furnishes a local minimum to (5), then necessarily
/2 2
j,l=1 2
fc,i=1 2
\
V j,i=i j,i=i J
> 0
(11)
is fulfilled for all £ G R2 and y G R2, the derivatives of L being evaluated at u '. This classical necessary condition is known as the Legendre-Hadamard condition, see [16, 1.5]. In this case, one says that the integrand function L is regular if the inequality holds in (11) for all £ G R2 and y G R2 which are different from zero.
Lemma 4.2. For p = 2, the integrand function L(v) is regular.
Proof. Using equality (10) we deduce that, for p = 2, the left-hand side in (11) is equal to
—* j i
i,fc=l,2 ¿,¡=1,2
where w is the (2 x 2) -matrix with entries wj = . Moreover,
L(w) = £ (y^)2,
i=1,2 ¿=1,2
showing that L(w) is positive for all £ G R2 and y G R2 different from zero, as desired. □
If p =2, the lemma seems to fail, for the complex p-Laplace operator degenerates at those x G X where <9w(x) vanishes.
Having disposed of this preliminary step, we proceed with searching for local minima of functional (5). By definition, there is a sequence } in A, such that I) \ m. It is called minimising. Any subsequence of a minimising sequence is also minimising. Were it possible to extract a subsequence } converging to an element u G A in the W 1,p(X) norm, then I) would converge to I(u) = m, and so u would be a desired solution of our extremal problem. It is possible to require the convergence of a minimising sequence in a weaker topology than that of W 1,p(X). However, the functional I should be lower semicontinuous with respect to correspondingly more general types of convergence. In order to find a convergent subsequence of a minimising sequence, one uses a compactness argument. The space W1,p(X) is reflexive, for we assume 1 < p < to. Hence, each bounded sequence in W 1,p(X) has a weakly convergent subsequence. Thus, any bounded minimising sequence {uv} has a subsequence {uVi} which converges weakly in W 1,p(X) to some function u. By a theorem of Mazur, see [25] and elsewhere, any convex closed subset of a reflexive Banach space is weakly closed. It follows that the limit function u satisfies u = u0 on S, i.e., it belongs to A. Moreover, Theorem 3.4.4 of [16] says that the subsequence {uVi} converges also strongly in L1(X) to u. Although L(v) is neither normal nor strictly convex in v, Theorem 4.1.1 of [16] applies to the functional I.
Lemma 4.3. If uv and u lie in W 1,p(X) and uv ^ u in L1(K) for each compact set K interior to X, then
I(u) ^ liminf I(uv).
Proof. See Theorem 4.1.1 of [16]. □
We have thus proved that if there is a bounded minimising sequence {uv} and u is a weak limit point of this sequence in W 1,p(X), then u G A and I(u) = m, i.e., u is a minimiser. It
1
1
*
y
y
2
2
y
y
is clear that any minimising sequence is bounded in W 1,p(X) if the functional 1 majorises the norm of u in A in the sense that
i |<9u|p dx > c ||u||W 1>p(x) - Q (12)
X
for all u G A, with c and Q constants independent of u. This is obviously a far reaching generalisation of A. Korn's (1908) inequality for the case of nonlinear problems.
Since the boundary value problem for d with boundary data Ku = u0 is elliptic, estimate (12) holds provided that S = dX. If however S = dX, no a priori estimate (12) is possible, see [22].
Hence, we are not able to prove the boundedness of any minimising sequence {uv}, so the above arguments do not apply. To get over this difficulty a general idea is to confine the set A of competing functions so that A be itself bounded. In linear analysis this idea goes back at least as far as [18]. While additional assumptions on A may depend on the concrete problem there is also an abstract prescription.
Given any R > 0, we denote by AR the subset of W1,p(X) consisting of all u G W 1,p(X), such that u = u0 at S and, moreover,
||Ku||Wi/p',p(dX\S) ^ R- (13)
Note that AR C A and each u G A lies in some AR, i.e. the family AR actually exhausts A. It is clear that the local minima of the functional 1 over AR need not satisfy Euler equations (2).
Theorem 4.4. Let R > 0. Then, for each data u0 G W1/p 'p(S), there is a unique function u G Ar minimising the functional 1 over AR.
Proof. One can assume without loss of generality that R is large enough, since otherwise the family Ar is empty and the assertion makes no sense. Since the boundary value problem
du = f in X, Ku = u0 at dX
is elliptic, there is a constant C depending only on X and p, such that
||u||W!.P(X) < C (||dHUp(X) + ||Ku||wi/p',p(dx)) (14)
for all u G W 1,p(X). Pick a minimising sequence {uv} in AR. Then the sequence {<9uv} is bounded in Lp(X). Combining (14) and (13) we readily deduce that the sequence {uv} is actually bounded in the W 1,p(X,Rl) -norm. We now argue as above. The bounded sets in W1,p(X) are relatively compact with respect to weak convergence in W 1,p(X). Hence, there is a subsequence {uVi} of {uv} which tends weakly in W 1,p(X) to some function u G W 1,p(X). Since Ar is a convex closed set in the reflexive Banach space W1,p(X), it is, by a theorem of Mazur cited above, weakly closed. Hence we conclude that the limit function u belongs to the set Ar. Moreover, Theorem 3.4.4 of [16] says that {uVi} converges also strongly in L1(X) to u. On applying the lower semicontinuity of I stated by Lemma 4.3 we obtain
I(u) ^ liminf I(uVi) = mR,
where mR is the infimum of I over AR. Since u G A, it follows that I(u) = mR, as desired. What is left is to show that the minimiser is unique. To this end, we assume that u and v are two different minimisers in Ar. Then (1/2)(u + v) G A and by Lemma 2.1
/u + v\ 11
^^^J < ^ 1 (u) + 2 1 (v) = m^
a contradiction. □
The same proof still goes when we replace condition (13) in the definition of AR by ||u||Wi,p(x) ^ R- This implies Theorem 0.1.
5. Generalisation to Dirac operators
Let A be an (l x k) -matrix of first order scalar partial differential operators with constant coefficients in Rn. It acts locally on k-columns of smooth functions or distributions in Rn by multiplying these with A from the left. As usual, the formal adjoint A* is defined for A by requiring (Au, v)L2(KnjCi) = (u, A*v)L2(Rn,Cfc) for all smooth functions u, v of compact support with values in Ck and C1, respectively. The operator A* is specified within (l x k) -matrices of first order scalar partial differential operators with constant coefficients in Rn. We say A is a Dirac operator if A* A = —A, where is the unit (k x k) -matrix. If A is a Dirac operator, then A* is a Dirac operator, too, if l = k, and is not, if l = k. Each Dirac operator A is overdetermined elliptic, i.e. its principal symbol <r(A)(£) has a left inverse for all £ e Rn \ {0}.
As developed above for the Cauchy-Riemann operator in one complex variable, the theory generalises to those Dirac operators A which are elliptic in the classical sense, i.e. with k = l.
Assume that X is a bounded domain with smooth boundary in Rn and S a nonempty open set in dX. If u e W 1,p(X, Ck) satisfies the system Au = 0 in X and the trace of u on S is zero, then u vanishes identically in X, see Theorem 10.3.5 of [24]. Hence, the following Cauchy problem has at most one solution: Given u0 e W1/p 'p(S, Ck), find a function u e W 1,p(X, Ck) satisfying Au = 0 in X and u = u0 at S.
For the study of the Cauchy problem for solutions of elliptic equations along more classical lines we refer the reader to [24] and more recent papers [12,13,21].
In order to construct a relaxation of the Cauchy problem we introduce the set A which consists of all functions u e W 1,p(X, Ck), such that u = u0 at S. One sees readily that A is a convex closed subset of W 1,p(X, Ck). Consider a functional I on A given by
I(u)= i |Au|p dx Jx
for u e A. Arguing as in the proof of Lemma 2.1 we deduce that the functional I is strongly convex on A.
The Euler equations for the extremal problem I(u) ^ min over u e A constitute the mixed boundary value problem
A* (|Au|p-2Au} =0 in X,
u = u0 at S, (15)
n(|Au|p-2Au) = 0 at dX\S,
where n(|Au|p-2Au) = a(A)*(—iv)(|Au|p-2Au) stands for the Cauchy data of |Au|p-2Au on dX with respect to A*. The second order partial differential operator u ^ A*(|Au|p-2Au) is called the p -Laplace operator associated with A. Since A* is elliptic, analysis similar to that in the proof of Lemma 3.1 shows that the function f := |Au|p-2Au has weak limit values on the boundary belonging to W-1/p 'p (dX, C1). If S is different from the entire boundary, then f = 0 in X, implying Au = 0 in X. Therefore, the extremal problem I(u) ^ min over u e A is equivalent to the Cauchy problem.
Let R > 0. Denote by AR the subset of W 1-p(X, Ck) consisting of all u e W 1-p(X, Ck), satisfying u = u0 at S and, moreover,
||u||W 1/p'.p(dX\S,Ck) ^ R,
cf. (13). By the Sobolev trace theorem, AR is a convex bounded closed subset of A and the family Ar exhausts A.
Theorem 5.1. Suppose R > 0. Then, for each data u0 e W1/p''p'ck (S, Ck) with 1 < p < the functional I takes on its minimum over AR precisely at one function uR e Ar.
Proof. This follows by the same method as in Theorem 4.4. The details are left to the reader. □
The behaviour of uR for large R proves thus to be crucial for the solvability of the Cauchy problem. If the family {uR}R>0 is bounded in W 1,p(X, Ck), then it stabilises for sufficiently large R. The limit function u is the minimum of I over all of A, and so u is a solution of mixed problem (15), provided that S is different from dX.
Corollary 5.2. Let u0 e W 1/p'>p(S, Ck), where S = dX. For the existence of a function u e W 1,p(X, Ck) satisfying Au = 0 in X and u = u0 at S it is necessary and sufficient that the family {uR}R>0 of Theorem 5.1 be bounded in W 1,p(X, Ck).
The first author gratefully acknowledges the financial support of the Deutscher Akademischer Austauschdienst.
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Экстремальная задача, относящаяся к аналитическому продолжению
Олимжан Махмудов Николай Тарханов
Мы показываем, что экстремальная задача, относящаяся к аналитическому продолжению с дуги на границе плоской области, не ведет к ослаблению этих переопределенных задач. Чтобы достигнуть такогоя ослабление, мы ограничиваем множество функционалов, следовательно, изменяем уравнения Эйлера.
Ключевые слова: экстремальные задачи, уравнения Эйлера, p -оператор Лапласа, смешанные проблемы.
