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5Динамические системы, том 3(31), No. 1-2, 69-84
УДК 517.972
Necessary conditions for ^-extrema of variational functionals in Sobolev spaces on multi-dimensional domains
I. V. Orlov, E. V. Bozhonok, E. M. Kuzmenko
Taurida V. Vernadsky University,
Simferopol 95007. E-mail: [email protected], [email protected], [email protected]
Abstracts. This paper deals with generalized Euler-Ostrogradsky equations and necessary conditions of Legendre-type in the case of compact extrema of variational functionals in Sobolev spaces on multi-dimensional domains. The inverse problem of smoothness refinement for solutions of generalized Euler-Ostrogradsky equations is considered. It is shown that a certain refinement of smoothness of solutions of generalized Euler-Ostrogradsky equations is achieved. Some related problems are considered.
Keywords: variational functional, Sobolev space on domains, compact extremum, generalized Euler-Ostro-gradsky equation, generalized Legendre condition.
1. Introduction
Starting with the work by L.Tonelli [22] variational problems in Sobolev spaces attracted much attention by many mathematicians.
In most cases (see, e.g., [7], [9], [11], [12], [13]) the corresponding contributions are connected with so-called direct methods of the calculus of variation which do not use the second variation.
At the same time, the various generalizations of the classical approach were considered. Such way enables to eliminate direct methods (see, e.g., the works by R. Klotzler [14], [15]).
Recently, in our papers [5], [18], [19], [20] a new method of semi-classical type has been developed in the one-dimensional case. It is based on the concept of so-called compact-analytic (or, K-analytic) properties of variational functionals as well as on the determination of compact extrema of variational functionals in Sobolev spaces.
Subsequently, this method has been extended to the multi-variate case ([16], [17]). Here, following the above mentioned approach fundamental K-extreme necessary conditions for variational functionals acting in Sobolev spaces on multi-dimensional domains are studied.
In Section 2 necessary definitions and theorems of K-analytical properties of variational functionals in Sobolev spaces W1p, p € N, on a multi-dimensional (Lipschitz) domain D are given. The main results of the paper are contained in Sections 3-5. In Section 3 a generalized Euler-Ostrogradsky equation for compact extrema of variational functionals in Sobolev spaces W1,p, p € N, is established. Next, in Section 4 the inverse problem of smoothness refinement for solutions of generalized Euler-Ostrogradsky equations is considered. It is shown that under some natural conditions a certain refinement of smoothness of K-extremals is achieved.
Finally, in the fifth section a generalized necessary condition of Legendre-type condition for compact extrema of variational functionals in W1p is derived. At first, the concept of a scalarly non-negative quadratic form is introduced. Then it is proved that the Hessian of the
© I. V. ORLOV, E. V. BOZHONOK, E. M. KUZMENKO
integrand in the leading variable is scalarly non-negative almost everywhere at the point of K-minimum. Moreover, an example is considered as model case.
2. The X-analytical properties of variational functionals in W1p(D) (review)
Let us introduce now general K-analytic properties for a functional acting on an arbitrary real locally convex space (LCS). In what follows, E is an arbitrary real LCS, $ : E ^ R is a real functional, C(E) is the system of all absolutely convex compacts from E. For any C € C(E) denote by EC the linear hull of C equipped with the Banach norm || • ||C generated by the set C. Recall (see [20]) that an arbitrary real Frechet space E is topologically isomorphic to inductive limit of a spectrum {EC}Cec(E) and the expansion
E t=> lim EC (2.1)
cgc(e)
holds true.
Definition 1. A functional $ : E ^ R is called K-continuously (K-differentiable, twice K-differentiable, etc.) at a point y € E if all restrictions of $ to (y + EC) are continuously (Frechet differentiable, twice Frechet differentiable, etc.) at y with respect to the norm || • ||C. Analogously we say that $ attains a compact extremum (K-extremum) at y if all restrictions $\y+Ec attain a local extremum at y with respect to the corresponding compact norms in Ec .
Remark 1. 1) All K-properties mentioned above are in general weaker than the usual local ones.
2) Due to expansion (2.1) in case of a Frechet space E the K-derivatives of any order are multilinear forms of corresponding order which are continuous in the usual sense.
It is well-known (cf. [7], [9]) that well-posedness of the basic variational functional
Ф(у)= I f (x,y, Vy)dx (2.2)
Jd
in Sobolev spaces W 1,p(D), p € N, where D is a compact domain in Rra with Lipschitz boundary, is usually closely connected with an estimate of the integrand f of type
f (x,y,z) > a + f3HzHP, (13 > 0).
Such severe constraints substantially restrict the class of admissible integrands. In our paper [16] we have introduced an essentially larger class of the admissible integrands, so called K-pseudopolynomials, using the concept of dominating mixed smoothness (see, for example [21], Chapter 2 and the references given there).
Definition 2. A mapping f : R™ x Ry x R™ ^ R is called K-pseudopolynomial of the order p € N if it can be represented in the form
f (x,y,z) = J2 Rk(x,y,z)(z)k , (2.3)
k=0
where the coefficients Rk, taking values in the space of fc-linear forms on Rn, are Borel mappings satisfying dominating mixed boundedness in x, y. More precisely, for any compacts Cx C R™ and Cy C Ry the coefficients Rk (k = 0,p) are bounded on Cx x Cy x R™. For the sake of shortness we shall write f € Kp(z). Here (z)k = (z,... ,z) is diagonal polyvector in
k
(Rn)k.
The following theorem shows that the functional (2.2) is well-defined if f € Kp(z).
Theorem 1. If the integrand f of the variational functional (2.2) belongs to the K-pseudopolynomial class Kp(z) then the functional (2.2) is well-defined on the space Wl'p(D). Moreover, for any compact set Ca C W1,p(D) the estimate |$(y)| < acA + ficA ■ (I|y||^i>p)p holds. Here the coefficients acA > 0, (3cA > 0 depend only on the choice of the compact set Ca.
In order to pass to K-continuity conditions for variational functionals in Sobolev spaces a suitable subclass of integrands from Kp(z) will be selected.
Definition 3. Let f € Kp(z) be continuous. The mapping f is called a Weierstrass K-pseudo-polynomial of p-th order (f € WKp(z)) if there exists a representation (2.3) such that all coefficients Rk possess dominating mixed continuity with respect to x, y. More precisely, for any compact sets Cx C R™, Cy C Ry the coefficients Rk (k = 0,p) are uniformly continuous and bounded on Cx x Cy x R™.
The condition f € WKp(z) provides K-continuity of the functional (2.2).
Theorem 2. If the integrand f of the variational functional (2.2) beJongs to the Weierstrass class WKp(z) then the functional &(y) is K-continuous everywhere on the space W1,p(D).
Furthermore, m-th order K-differentiability is provided by introducing the following K-pseudopolynomial Weierstrass classes.
Definition 4. Let f € Cm n Kp(z). The mapping f is called a Weierstrass K-pseudo-polynomial of the class WmKp(z) if there exists a representation (2.3) such that all m-th order jets (Rk , VyzRk,..., V^,Rk) of the coefficients Rk (k = 0,p) possess dominating mixed smoothness with respect to x, y.
The condition f € WmKp(z) provides m-times K-differentiability of the functional (2.2).
Theorem 3. If the integrand f of the variational functional (2.2) belongs to the class WmKp(z), m € N, then the functional &(y) is m-times K-differentiable on the space Wl'p(D). In addition, the classical formula of m-th order variation remains true for the K-variation of m-th order, i. e. it holds
Ф{K](y)(h)m = j
D
E
.1=0
C
д mf
(x,y, Vy)hm-i • (Vh)
Qym-i dzl
dx
(2.4)
I
Let us emphasize that in cases m = 1 and m = 2, which are of practical relevance for extremal problems, equality (2.4) reads as
Ф'к (y)h =
D
df df
dy (x, y, Vy)h + dz (x, y, Vy) ■ Vh
(2.5)
Ф'к(y)(h)2 = I (x,y, Vy)h2 + 2^-(x,y, Vy)h ■Vh+
f.. „„л,2 , nd2f
D
-dy2 dydz
д 2 f ]
+ (x,y, Vy) ■ (Vh)2\ dx. (2.6)
In the end we want to give a reformulation of Fermat's lemma for K-differentiable functionals (see [6]).
Theorem 4. Let E be an arbitrary real LCS. Assume that the functional $ : E ^ R attains a K-extremum at a point y € E. If $ is K-differentiable at the point y then $K(y) = 0.
3. The generalized Euler-Ostrogradsky equation for K-extremals in W1p (D)
Here we consider the variational functional
$(y) = i f (x,y, Vy)dx, yt) € W 1,p(D), p € N, (3.1) Jd
with additional boundary condition
y\dD = yo , (3.2)
where y0 € Wl'p(dD), D is a compact domain in Rn with a Lipschitz boundary dD.
Note that the boundary condition (3.2) means, in particular, an additional smoothness of y near dD. The definition of Sobolev space Wl'p(dD) can be found in [8].
To determine the K-extremals of the functional (3.1)-(3.2), we need an almost everywhere-analog of the corresponding classical (C1) necessary condition, in other words an analog of the Euler-Ostrogradsky equation (see [10]).
Theorem 5. Let f € W 1Kp(z). Suppose that
(i) the functional (3.1) attains a K-extremum at the point y(^) € Wl'p(D),
(ii) the mapping (df/dz)(x,y, Vy) belongs to the Sobolev space W1,1 (D). Then the generalized Euler-Ostrogradsky equation
~ d
% <xy - £ £ (% <xy Vy))=0 (33)
holds true almost everywhere (a.e.) on D. In particular, condition (ii) is fulfilled if
df, „. ,
dz(x,y,z) € C*(R£ x Ry x R") and y(-) € W2'p(D)
Proof. 1) If f € W 1Kp(z) then the functional (3.1) is K-differentiable everywhere in Wl'p(D) by virtue of Theorem 3. Therefore, in view of Theorem 4 equality (y)h = 0 holds for any h € Wl'p(D). In detail, this means that
D
df df
ду (x,y, Vy)h + dz(x,y, Vy) •Vh
dx = О (У h e Wl'p(D)).
(3.4)
2) Now, note that the condition (df /dz)(x,y, Vy) € W1,1 (D) implies representability of this function by means of indefinite Lebesgue integrals of its partial derivatives with respet to Xi, i = l,n. Applying now the Green formula [1] to the second summand in (3.4) and taking into account hldD = 0 we obtain
(y)h = J (x, y, Vy)hdx + J dfe. X y, Vy) ■ dXdx
D i=1 ID
/O /» n n o /»
— (x,y, Vy)hdx + j> h ■ — (x,y, Vy)cos(lf, e$ )dl
T^ i=1 r-,J~i
dD
4 f x y• hdx =О • (35)
D
where it = cos(!rt, et)et stands for the external normal vector on D. Since the line integral k=1
in (3.5) vanishes we get therefrom
/
Ф'к (y)h =
D
\
df, „ ч д ( df , _ .
- (x,y,vy - g -dxt{ {x,y,Vy)
V
L(f)(y)
hdx = О .
(3.6)
/
3) Next we show that identity (3.6) implies the equality L(f )(y) = 0 almost everywhere on D. Assume, on the contrary, L(f )(y)(x0) = 0 at some point x0 € D of approximate continuity of L(f )(y). Without loss of generality we may suppose L(f )(y)(x0) > 0. Then it follows
y L(f )(y) dx > 0
Os (xo)
for ô > 0 small enough, where Os (xo) is a ¿-neighborhood of the point x0. We choose ô' <6 as much as close to ô such that
L(f )(y)dx< j L(f )(y)dx,
(3.7)
A Os, (xo)
where A = Os(x0)\Os' (x0). Now, we put
1 , if x € OS' (xo)
h(x) = { 0 , if x€ Os (xo) , h € Wi'p (D)
"radially linear" , if x € Os(x0)\Os'(x0)
We have f L(f )(y)hdx =
D
= j L(f )(y)hdx = j L(f )(y)hdx + j L(f )(y)hdx =: I1 + I2. (3.8)
Os (xo) A Os, (xo)
By virtue of (3.7) we get \h\ < f \L(f )(y)\dx < f L(f )(y)dx = I2. Together with (3.8)
A Os, (xo)
this implies /L(f)(y)hdx > 0. The last inequality contradicts condition (3.6).
D
4) If, in particular, (df/dz)(x,y,z) € C 1(Rn x Ry x RJ) then the mapping f locally satisfies a Lipschitz condition. For y(^) € W2'p(D) the function x — f (x,y, Vy) belongs to the space Wl'p(D). Hence, the composition (df/dz)(x,y, Vy) belongs to W 1'1(D). Thus, condition (ii) of the theorem is fulfilled. □
In what follows solutions of the generalized Euler-Ostrogradsky equation (3.3) with condition (ii) of Theorem 5 are called K-extremals of the variational functional (3.1). Note, in addition, that equation (3.3) is satisfied a priori at any point of approximate continuity of Vy.
4. Smoothness of K-extremals in Sobolev spaces W1,p
It is well known that under sufficiently general conditions in the classical C1 -case a solution of the Euler-Ostrogradsky equation even belongs to the class C2. We look at a similar problem in the Sobolev case (see [4]). Here the question is whether a solution of the generalized Euler-Ostrogradsky equation belongs to the space W2'p under natural conditions. A related problem is whether such a solution possess at least some additional smoothness properties.
Our first result in this direction is not immediately connected with the Euler-Ostrogradsky equation and Sobolev spaces, respectively .
Theorem 6. Let f : x Ry x ^ R, f € C2, and let the function y(^) : D R be continuous and a.e. differentiable on D. Suppose that
(i) the gradient (df/dz)(x,y, Vy) is differentiable a.e. on D;
(ii) the Hessian (d2f/dz2)(x,y, Vy) is non-degenerate a.e. on D, i.e.
(d2f \ detyddz2 (x,y, Vy)) = 0 a.e.
Then the function y( ) is twice approximately differentiable a.e. on D. In addition, it holds
V2ap(y)(x) • Ax = Vap(Vy)(Ax) = (dz2 (x, y, Vy)) •
d2 f 1
(4.1)
f 2f 2f V ( dz (x, y, Vy)) ■ Ax - idx, (x, y, Vy) ■ Ax - (Vy, Ax)^(x, y, Vy)
z x z y
Proof. 1) We fix i = 1,n and apply the mean value theorem [23] to the function
f
= Fi(x,y,z)
dzi
(4.2)
on the vector interval [(x, y, z);(x + Ax, y + Ay, z + Az)] = [h; h + Ah]. It follows
+ Ay,z + Az) (x,y,z) = ( V\ f
^ (x + Ax,y + Ay, z + Az) - f (x, y, z) = (v ( f (С)) , Ah^J , (4.3)
f
))zí
for some £ € [h; h + Ah]. Because any measurable function is approximately continuous almost everywhere ([3]) we can choose a point x € D in which Vy exists and is approximately continuous. Let the conditions (i)-(ii) of the theorem be satisfied. We choose a measurable subset Ai c D having x as a density point such that Vy(x + Ax) — Vy(x) if x + Ax — x in Ai. Now we substitute Ay = y(x + Ax) — y(x) and Az = = Vy(x + Ax) — Vy(x) in (4.3). Moreover, Ay — 0 as Ax — 0 by continuity of y(^) and Az — 0 as x + Ax — x in Ai in view of approximate continuity of Vy at the point x. Thus, we obtain
f (x + Ax, y + Ay, z + Az) - f (x, У, z) =
+
)2f )zí )y
(С) ■ Ay +
' ))2f )zí dz
(С), A(Vy)\ . (4.4)
Using notation (4.2) we find
Fí(x + Ax) - Fi(x) =
{rzk(С ),Ax) +
2f
_ df
)zídxy^n ""J ' )zíy ' \)zídz
(OAy +
(С), A(Vy)
.
(4.5)
Now we determine the principal linear part of (4.5). It holds
Fí(x + Ax) - Fi(x) = (VFí, Ax) + o(||Ax||) = ((С\ A^ +
, )zí dx
d2f ( d2f \ +dz)y. (С) ■ ((Vy, Ax) + o(|| Ax||)) + ( ^(С), (Vap(Vy) ■ Ax + o (||Ax||))J =
)zí dy
)f :(x) + ((С) - -êL(x)
)zí dx
)zí dx
)zí dx
o(1)
\
, Ax
+
+
2f 2f 2f d f -(x)+ ¿fL(С) - éí-(x)
))zí dy
))zí dy ))zí dy
o(1)
+
2f 2f 2f df :(x)+ ¿fh(С) - éi-(x)
))zí dz
))zí dz )zí dz
o(1)
■ ((Vy, Ax) + o(||Ax||)) +
x
(Vap(Vy)Ax + o(|| Ax||))
)
д 2f
dzjdx
(x) + x)
,Ad +
d2f
(x)+ ß (Z,x)
+
d2f dzjdz
(x)+ j(C,x)
dzidy
(Vap(Vy)Ax + o(||Ax||)))
((Vy, Ax) + o(|| Ax||))
where a({, x) t 0, /(£, x) t 0, x) t 0 as Ax t 0 in Ai — x.
Neglecting the small terms we are led to the existence of the approximate gradient of Fi at x and to the equality
СVFt,Ax) =
dFt(;Ax)
{ß-x (x)'Ax) + ЩЦ{x) ' (V'y- Ax)+(f
(x), Vap(Vy) • Ax), (4.6
,
respectively.
n
2) Next, we observe that the set A = Ai also has x as its density point. Thus, via the
i=1 _
limiting argument Ax t 0 in (A — x) we see that all equalities (4.6), i = 1, n, are fulfilled. As a result we have the system
{(VF- Ax)" (Wx{x)' Ax) - Шй,x(Vу•Ax =
/ д 2f
dzdz (VaP(V')- Ax), U=1
о к
(4.7)
Now, we introduce the matrices
e
d2f д 2f \n / A = ( VFi (x) - т^д,, (x) -Vy) , B =[
bzSx
c)ziöy
д 2f dzídz
(x)
/ i=i
Then the system (4.7) can be rewritten as A ■ Ax = B ■ (Vap(Vy) ■ Ax). Therefore it follows V2ap(y) ■ Ax = Vap(Vy) ■ Ax = B-1 ■ (A ■ Ax) = (B-1 ■ A) ■ Ax, that is
V2ap(y)(x) ■ Ax = Vap(Vy)(x) ■ Ax = (djz (x, y, Vy)) ■
(df ) d 2f d 2f V ( dz (x, y, Vy)) ■ Ax — dzdx(x, y, Vy) ■ Ax — (Vy, Ax)(x, y, Vy)
The last expression can be rewritten in matrix form as '02f
3zc)y
(4.8)
(д2^ \-1
V2ap(y)(x) = д^2 (x,', Vy))
v(kVy)) --k
д 2f
(x, y, Vy) - (Vy, •) д^гу (x, y, Vy)
(4.9) □
As an application we can establish a result on some strengthening of smoothness of K-extremals in Sobolev spaces.
Corollary 1. Under the assumptions of Theorem 6 let the function y(-) € Wl'p(D), y\dD = y°> be a K-extremal of the functional (3.1). Then, at all points x € D of both approximate continuity of the gradient Vy(x) and non-degeneracy of the Hessian (d2 f/dz2)(x,y, Vy) the trace function
(x,y, Vy) •V2ap(y)(x)^ is approximately continuous as well. Moreover, equality
Тг(Ш(x,y, Vy) ■V2aP(y)(x)) =
)
= f x У' Vy) — Tr{jzbx у, Vy) + ■) ■ Щу x у, Vy)) (41°)
holds true.
Proof. Multiplying both parts of equality (4.9) by (d2f/dz2)(x,y, Vy) from the left side we get
2f 2
dz2(x,y> Vy)' Vap(y)(x) =
(d f ) d2f d2 f = V(dZVy)) - gzkVy) - (Vy, ) ■ dZdy(xy Vy) ■ (411)
Applying the trace operator to both sides of (4.11), we obtain the equality
Tr (V(dz(x,y, Vv))) = ± f (x,y, Vv>).
This yields equality (4.10) using the generalized Euler-Ostrogradsky equation (3.3). In view of the approximate continuity of right-hand side of (4.10) the trace function
Tr ((d2f/dz2)(x,y, Vy) ■V2aP(y)(x))
is approximately continuous at those points where the gradient Vy(x) is approximately continuous as well. □
The previous results can be essentially improved under the assumption of usual almost everywhere continuity of the gradient of K-extremal. In particular, we can show the usual repeated almost everywhere differentiability of K-extremal (i.e., almost everywhere differentiability of usual gradient of K-extremal).
Theorem 7. Under the assumptions of Theorem 6 let the gradient Vy(x) be continuous almost everywhere on D. Then the following statements hold true.
(i) There exists V2(y)(x) a.e. on D. Moreover, we have
V\y)(x) = (df (x,y, Vy)) •
• (v(dz(x,y,vy>)-Oxvy -)my,(x,y,Vy)) ■ <412)
(ii) It holds the formula
Tr(j%(x,y, Vy) -V2(y)(x)} = Tr(v(%(x,y, Vy^ -
( d2f d2 f \ -Tr\ddi(x-y-v') + (vy,- > - idy(x','vy))) <413)
for the trace function Tr ((d2 f/dz2)(x,y, Vy) - V2(y)(x)).
(iii) If, in particular, y(-) satisfies the generalized Euler-Ostrogradsky equation <3.3), then the function Tr ((d2 f/dz2)(x,y, Vy) - V2(y)(x)) is also continuous at all points of continuity of Vy(x) and the trace formula <4.10) can be rewritten as
Tr(d%(xy Vy) - V2(y)(x)) =
df ( d2 f d2 f \ = (x,y, Vy) - T^QZdxxy> Vy) + (Vy> -) - ddyxy> Vy))) ■ <4-14)
Proof. <i) Extracting the principal linear part in <4.5) and passing to the limit as Ax ^ 0 arbitrarily we find the system of equations
r d2f d2 f
{(VF-Ax) - (d*(x) Ax) - dzdy <x)(Vy- Ax) =
= ( izk:(x) v(v»)ax) l ■ <«5)
Equality <4.12) follows immediately from <4.15).
<ii) Multiplying both sides of <4.12) by (d2 f/dz2)(x,y, Vy) - V2(y)(x) from the right side and passing to traces we easily obtain equality <4.13).
<iii) Suppose, in particular, that y(-) satisfies the generalized Euler-Ostrogradsky equation <3.3). Then equality <4.13) can be rewritten as <4.14). Moreover, the function
Tr ((d2f/0z2)(x,y, Vy) -V2(y)(x))
is continuous simultaneously with Vy(x). □
Now, we consider an important special case of the integrand which leads to explicit representations of weighted and usual Laplacians of K-extremals y(-).
Corollary 2. Under the assumptions of Theorem 7 let the integrand f be given as
f (x, y, z) = P(x, y) + Q(x, y) ■ (z) + R(x, y) ■ (z)2 .
Suppose that both the coefficients
P : D x Ry — R, Q : D x Ry — Li(R™) ^ Rn, R : D x Ry — L2(R™) = Mn(R),
(where Mn(R) is set of n x n matrices on R) and the gradient Vy are continuous almost everywhere in D. Then the following statements hold true. (i) The trace function (4.13) can be represented as
Tr (R(x, y) ■ V2(y)(x)) = Tri V(f (x,y, Vy))j -
-Tr{izl;{x-y-Vy> + (Vy- > iziy{x-y-V«»). (416)
In addition, in the special case of a diagonal matrix R(x,y) = diag (pu(x,y))n=i representation (4.16) can be rewritten as
Tr (R(x,y) ■ V2(y)(x)) = ^ pu(x,y) ■ ^xy =: Apy(x) =
i=l i
=tk V(f x y> Vy))) - x y> Vy) + &y> ■) ■ dZf x y, Vy))) ■ (4ir)
Here Apy denotes the weighted Laplacian of y with the weights {pii(x,y)}'n=l. In particular, in case of the unit matrix R(x, y) = E we obtain the representation of the usual Laplacian Ay
Ay(x)= Tr ( V(f (x,V, Vy))) —
)
-Tr{B-xV»> +■) ■ syVy>)). (418)
(ii) Let y(■) be a K-extremal of the functional (3.1) and let the gradient Vy be continuous a.e. on D. Then both the weighted Laplacian Apy and the usual Laplacian Ay are continuous a.e. on D as well. Moreover, the right-hand sides of representations (4.17)-(4.18) read as
f (x, y, Vy) - x y' Vy) + (Vy' ■) ■ izjfy x y' Vy))) ■ (419)
Proof. The corollary is an immediate consequence of Theorem 7. □
We have shown that the solutions of the generalized Euler-Ostrogradsky equation possess some additional smoothness properties. However, in spite of this fact the question whether or not K-extremals belong to the Sobolev space W2'p must be answered negatively in general. Let us discuss an appropriate counter example.
Example 1. Let us consider the most simple variational functional
H(v) = j\Vy(x)\2dx, (y(■) & W 1,2(D), D — [0; 1] x [0; 1])
r\,2/
\VV(x)\ dx, yy(■) t W
D
Here f (x,y,z) = (yXl)2 + (yX2)2, dy = 0; dZ~ = 2Vxi; dZ = 2Vx2; дХт (т^) = 2Vx1x1; d_ дХ2
f -0- f — 2y f — 2y .±
O — 0 o — 2yxi) n — 2yX21 O
dy dzl dz2 ox\
—— ( df) — 2yX2X2. Hence , the generalized Euler-Ostrogradsky equation reads as \Oz2j
Vxx + VX2X2 a—e' 0. (4.20)
Let x(t) be the "Cantor ladder"on [0; 1] (see , e.g. , [2]). We put
Xl X2
yo(x) — Jx(t)dt + Jx(t)dt, 0 < xi < 1, 0 < X2 < 1. 0 0
Then (yo)xi — x(xi); (yo)x2 — x(x2); (yo)xixi — x'(xi) — 0 a.e. on [0; 1] c Rxi; (y0)X2X2 — x'(x2) — 0 a.e. on [0; 1] c RX2. Hence, y0( ■ ) satisfies the generalized Euler-Ostrogradsky equation (4.20). Nevertheless, y0(■ ) & W2'2(D) because Vy0(■ ) & W1,2 (D). Thus, in contrast to the classical Cl-case an essential strengthening of smoothness of K-extremals in Sobolev case does not occur.
5. Generalized Legendre necessary conditions for X-extrema
Definition 5. Let y be a quadratic form acting on a real vector space E. We call y the
seal
scalarly non-negative form (y > 0) if the condition y < 0 is not fulfilled, that is, if there exists h & E (h — 0) such that y(h) > 0.
Theorem 8. Assume that the variational functional (3.1) attains a K-minimum at y(^) & Wl,p(D). Moreover, suppose that
(i) the integrand f belongs to the Weierstrass class W2Kp(z);
(ii) the mapping (d2f/dydz) (x,y, Vy) belongs to the Sobolev space W 1'1(D). Then the generalized Legendre necessary condition
2 f seal
(x,y(x), Vy(x)) > 0 (5.1)
is fulfilled for the K-extremal y(^) almost everywhere on D.
Proof. We transform the canonical expression of the second K-variation of H
y(h)'— /[0x V■ Vy)h2 +21 vV) h ■ £+
D %-1
2
д2 f ] + д^2 (x, V, Vy) ■ (Vh)2 dx (h t W 1'P(D)) . (5.2)
To this end we use the repeated K-differentiability of $ ([17], Theorem 2) and condition (ii) of the theorem. Application of Green's formula ([1]) to the second summand on the right-hand side in (5.2) and taking into account the boundary condition
(v\od = yo) ^ (h\dD = 0)
we obtain
Ф'К(v)(h)2 = J df (x, v, Vv)h2 + ¿ fh2^ df (x,v, Vv)cos(Jl, 4)dl-
n
■ ( x v \ / v I I h 2
d2f
D íl1 dD
/3 í &2 f \
äü( avk(x'v' VvV hdx.
/Q2 f
(x,v, Vv)^(Vh)2dx =
DD
n
D
( &2f n д í &2f \ KW2 (x,v Vv) - s exuy evk (x>v Vv))
h2+
d2 f,
+ 1 оф(x,v, Vv)^(Vh)2^ dx. (5.3)
D
n
Here n = cos(n , ek)ek stands for the external normal vector on dD. We give the proof by
k=l ^ ^
contradiction. Suppose that (5.1) is not fulfilled. Then there exists a subset AO C D, ¡Al > 0 such that inequality
d2f
<fi(x) '■= dz? (x,y(x), Vy(x)) < 0
holds for any x € Al. The last inequality can be replaced by
^(x, h) = p(x) ■ (h)2 < 0
for any x € AO, h € R™, ||h|| = 1. Next, we choose a compact subset A1 c Al with positive measure ¡A^ > 0. Applying the Weierstrass theorem to the function ^(x, h) on the compact set Al x (||h|| = 1) we obtain the inequality
^(x, h) <-ko < 0 (V x € AO , h = 1).
Hence, using the second order homogeneity of ^ in h it follows immediately
^(x, h) <-ko ■ h2 (Vx € AO , h € R™).
Here k0 does not depend on the choice of x € AO and h € R™. In particular, this implies 2f
dZz (x,y(x), Vy(x)) ■ (Vh(x), Vh(x)) < —k2 ■ HVh(x)f (x € AO). Now, we choose a set AO C D with ¡A^ > ¡D — ¡AO such that the inequality
d2 f n я ( d2 f \
4 (x.v.W) - £ ^ bv.™) < C2 <
oo
holds for all x € A2. Then the set A0 := A1 n A2 has a positive measure as well.
Next, let x0 be an arbitrary density point of A0 ([3]). We choose a neighborhood O$0(xo) (¿o > 0) such that the inequality
ß(Ao П Os(xo))
ß(O& (xo))
holds for ö < ö0. Now we define a function h0(x) by
> 1 - £o (0 < eo < 1)
Vs , as x = x0;
ho(x) = ^ 0 , as \\x — x0\\ > ö;
is "radially linear" , as \\x — x0\\ < ö.
Then, in a ö-neighborhood 0 < \\x — x0\\ < ö we get
h20 < S, Vho = (±V=±V ).
1
'VS'
1
'VSJ
(5.4)
Combining (5.4) and (5.3) we find
if
фк ШЫ? = /( 0 (x,y. vy)- £ (x,y,
D L г=1
h0+
+
df dz2
д
(x,y, Vy)(Vho, Vho)^jdx
<
Os (xo)
df y2
(x,y, Vy)-
<
n d ( d2f \i d2f )
- E -^.[d^jtz(x,y' Vy))\h2+dà x y vy)(vh*' vh°))dx
h
< C2 • S • [(1 - ea) • 2S + [(1 - ec) • 25 • - • (-k2) =
= 2Ca2(1 - ea) • S2 + 2 • (1 - ea) • h • (-k20) < 0
for S > 0 small enough. Finally, using Taylor's formula of second order in direction ha we immediately obtain the inequality
$(y + tha) - $(y) < 0
for t > 0 small enough. Hence, $ does not realize a minimum on any absolutely convex compact Ce C Wl'p(D) for which Cef| R • ha = {0} holds. Therefore, $ does not possess a K-minimum at the point y(). The last result contradicts the assumption of the theorem. □
At the end, let us consider an example of a two-dimensional variational functional having a non-smooth K-extremal but satisfying the generalized Legendre necessary condition.
Example 2. We put
ф(у) =
1-11-1
'■^/vl-y +vl
cos2t2dt\dx (y e W 1'2(D),D = [-1; 1] x [-1; 1]).
r1,2
2
o
1. In this case we have
Thus we obtain 1)
f(z1, z2) =
f
dzi
cos2t2dt.
Vz2 + z2
2)
д 2 f
дф =Cos2(Z2 + Z22)
V(z'2 + z2)3
cos2(2(z2 + z2));
— sin2(z2 + z2)-
2z2
Z + z2 '
d2 f 2/ 2 2N zi • z
dzidzj
= — cos (z2 + z2)
V(z 2 + z2 )3
J — sin2(z2 + z2) 2Zi ^ Zj
z 2 + 4
We introduce the mapping
Since
(i,j = 1, 2; i= j). , ч f (z 1,z2)
V(zi, Zj) = 72 + 72 z1 + z2
= ¡p'z. M = M = 0 (i,j = 1, 2)
the jet (p,dp/dz,d2p/dz2) possesses dominating mixed continuity. Hence f € W2K2(z). Moreover, (df/dz)(x,y, Vy) € W 1'1(D).
2. It is evident that &(y) attains a minimum at any point y(-) € Wl'2 satisfying the condition
п
\Vy\2 = y2xi + y22 = 2 + nk almost everywhere (k € Z).
We consider the special point of minimum yO(xl,x2) = ^Jп1(\xl\ + \x2\). In this case the generalized Euler-Ostrogradsky equation reads as
z1
— (
dxi ^z2 + z2
d
cos2(2(z2 + z2))) + (
z2
cos2(2(z2 + z2))) a= О.
dx2 ^z2 + z2
(5.5)
Because of
9vo f* г I o\ ivy 12 (9vo\ , (9vo\ п
dx¡ = \j-4sgnxi (i = 1,2) \VvO\ пdx;) 4dx-J =2
a.e.
the function yO() satisfies equation (5.5). 3. Finally, in the case under consideration we obtain
d 2 f ae
dz2(x, vo(x), Vvo(x)) a='
yX1 cos1 \Vy\1 2yi1sin2\Vy\1
\Vy\3
\Vy\
Ух1 yx1 cos2 \Vy\1 '2yyx1yx2sin2\Vy\1
Ух1 yx1 cos1 \Vy\1 2yx1 yx1 sin 2\Vy\1
\Vy\3 . ..
y1 cos1 \Vy\1 2yX1 sin 2\Vy\1
\Vy\
\Vy\3
\Vy\
\Vy\3
\Vy\
This implies
(x,vO(x), vvo(x)) a='2п ^ о О )
seal
О a.e. on D
for the K-extremal yO(). Thus, yO satisfies the generalized Legendre necessary condition but it does not satisfy the usual one because of nonsmoothness.
O
z
2
z
References
1. Azizov T. Ya, Kopachevsky N. D. Abstract Green formula and it's application. — Simferopol: FLP «Bondarenko O.A.», 2011. — 145 p. (in Russian)
2. Berezansky Yu.M., Sheftel Z.G., Us G.F. Functional Analysis, Vol.1. — Basel-Boston-Berlin: Birkhauser, 1996. — 600 p.
3. Bogachev V.I. Basic Measure Theory, Vol.1. — Moscow: R&C Dynamics, 2006. — 544 p. (in Russian)
4. Bozhonok E. V. On solutions to "almost everywhere"Euler-Lagrange equation in Sobolev space H1 // Methods of Functional Analysis and Topology. — 2007. — Vol. 123, no. 3. — P. 262-266.
5. Bozhonok E. V. Some existence conditions of compact extrema for variational functionals of several variables in Sobolev space H1 // Operator Theory: Advances and Applications, Basel: Birkhauser. — 2009. — Vol. 190. — P. 141-155.
6. Cartan H. Calcul différentiel. Formes différentielles. — Paris: Hermann, 1967. — 392 p.
7. Dacorogna B. Introduction to the calculus of variations. — London: Imperial College Press, 2004. — 228 p.
8. Functional Analysis. / Eds.: S.G.Krein. — Moscow: Nauka, 1972. — 544 p. (in Russian)
9. Galeev E.M, Zelikin M.I, et al. Optimal Control Theory. / Eds.: N. P. Osmolovsky, V. M. Tihomirov. — Moscow: MCNMO, 2008. — 320 p. (in Russian)
10. Gelfand I.M., Fomin S. V. Calculus of variations. — Moscow: Fizmatgiz, 1961. — 230 p. (in Russian)
11. Giaquinta M., Hildebrandt S. Calculus of Variations I. — New York: Springer Verlag, 1996. — 474 p.
12. Giusti E. Direct Methods in the Calculus of Variations. — Singapore: World Scientific Publishing Co., 2003. — 403 p.
13. Jost J., Li-Jost X. Calculus of variations. — Cambridge: Cambridge University Press, 1998. — 323 p.
14. Klotzler R. Mehrdimensionale Variationsrechnung. — Boston: Birkhauser, 1980. — 299 p. (in German)
15. Klotzler R. Minimal surfaces as webs of optimal transportation flows // Optimization. — 2001. — Vol. 49, no. 1-2 — P. 151-159.
16. Kuzmenko E. M. Conditions of well-posedness and compact continuity of variational functionals in W1,p // Uchenye Zapiski Tavricheskogo Natsyonal'nogo Universiteta. — 2011. — Vol. 24(63), no. 1. — P. 76-89. (in Russian)
17. Kuzmenko E. M. Conditions of compact differentiability and repeated compact differentiability of variational functionals in Sobolev spaces W1p of functions of several variables // Uchenye Zapiski Tavricheskogo Natsyonal'nogo Universiteta. — 2011. — Vol. 24(63), no. 3. — P. 39-60. (in Russian)
18. Orlov I. V. ^-differentiability and ^-extrema // Ukrainian Math. Bulletin. — 2006. — Vol. 3, no. 1. — P. 97-115. (in Russian)
19. Orlov I. V. Compact extrema: general theory and its applications to the variational functionals // Operator Theory: Advances and Applications, Basel: Birkhauser. — 2009. — Vol. 190. — P. 397417.
20. Orlov I. V., Bozhonok E. V. Additional chapters of modern natural science. Calculus of variations in Sobolev space H1: tutorial. — Simferopol: DIAYaPI, 2010. — 156 p. (in Russian)
21. Schmeisser H.-J., Triebel H. Topics in Fourier Analysis and Function Spaces. — Chichester: Wiley, 1987. — 300 p.
22. Tonelli L. Fondamenti di Calcolo delle Variazioni. — Bologna: Zanichelli, 1921-23. — 466 p.
23. Zorich V. A. Mathematical analysis, Vol.2. — Moscow: Nauka, 1984. — 650 p. (in Russian)
Получена 10.04.2013
