Variational Methods for Boundary Value Problems in ODE"s and PDE"s - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


V ariational metho ds for b oundary v alue problems Abstract. The v ariational form ulationofboundary v alue problems is v aluable in pro viding
remark ably easy computational algorithms as w ell as an alternativ e framew ork with whic h to
pro v e existence results. Boundary conditions imp ose constrain ts whic h can b e anno ying from a computational p oin tofview. The question is then p osed: what is the most general b oundary v alue problem whic h can b e p osed in v ariational form with the b oundary conditions app earing naturally? Sp ecial cases of t w o-p oin t problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjoin tness for the linear case. F urther cases under whic h a Lagrangian ma yor ma y not exist are explained. Keyw ords: natural b oundary conditions, self-adjoin tness, Hamiltonian, Lagrangian. 1. In tro duction 1.1. Bac kground V ariational metho ds pro vide a conceptual and computational framew ork for b ound-
ary v alue problems (BVPs). In the pap er w e deriv e conditions whic h guaran tee that a giv en BVP can b e reform ulated in v ariational form. This can b e a w a yofpro ving existence for BVPs (Graham-Eagle and W ak e [6] ) and it also pro vides a sc heme
for determination of a computational algorithm as in F radkin and W ak e[5]. In the
case of nonlinear eigen v alue problems, these can b e a w a y of pro ducing outcomes lik e parameter path-follo wing as has b een pro duced b y other means b y computer pac k ages lik e Auto (Do edel [2]). The question of what precise conditions need to b e imp osed on b oundary v alue problems so that they can b e written in v ariational form has b een discussed spas- mo dically in the literature, see for example Anderson and Thompson [1] , F els [3],
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V ariational metho ds for b oundary v alue problems Abstract. The v ariational form ulationofboundary v alue problems is v aluable in pro viding
remark ably easy computational algorithms as w ell as an alternativ e framew ork with whic h to
pro v e existence results. Boundary conditions imp ose constrain ts whic h can b e anno ying from a computational p oin tofview. The question is then p osed: what is the most general b oundary v alue problem whic h can b e p osed in v ariational form with the b oundary conditions app earing naturally? Sp ecial cases of t w o-p oin t problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjoin tness for the linear case. F urther cases under whic h a Lagrangian ma yor ma y not exist are explained. Keyw ords: natural b oundary conditions, self-adjoin tness, Hamiltonian, Lagrangian. 1. In tro duction 1.1. Bac kground V ariational metho ds pro vide a conceptual and computational framew ork for b ound-
ary v alue problems (BVPs). In the pap er w e deriv e conditions whic h guaran tee that a giv en BVP can b e reform ulated in v ariational form. This can b e a w a yofpro ving existence for BVPs (Graham-Eagle and W ak e [6] ) and it also pro vides a sc heme
for determination of a computational algorithm as in F radkin and W ak e[5]. In the
case of nonlinear eigen v alue problems, these can b e a w a y of pro ducing outcomes lik e parameter path-follo wing as has b een pro duced b y other means b y computer pac k ages lik e Auto (Do edel [2]). The question of what precise conditions need to b e imp osed on b oundary v alue problems so that they can b e written in v ariational form has b een discussed spas- mo dically in the literature, see for example Anderson and Thompson [1] , F els [3], and F orra y[4]. The question as to whic h b oundary v alue problems ha v eav aria- tional form ulation in whic h all of the b oundary conditions app ear naturally is largely still an op en one and is the sub ject of this pap er. Boundary conditions app e ar naturally b y the addition of suitable terms in the Hamiltonian (usually surface in tegrals on the b oundary of the region of in terest). It is usually dicult and sometimes imp ossible to determine these in a systematic w a y . Problems exist (lik ethe t w o-p oin t second order b oundary v alue problem in one dimension with Diric hlet b oundary conditions men tioned b elo w) for whic h this is imp ossible. In these cases the b oundary conditions need to b e imp osed on the space of functions on whic h the stationary v alues of the Hamiltonian is found,
and are called \essen tial b oundary conditions". This can be of n uisance v alue in the determination of the subspaces of functions on whic h the optimisation o ccurs. Th us it app ears useful to b e able to write the v ariational form ulation with all of the b oundary conditions app earing naturally . The more fundamen tal question as to whether there is ev en a Lagrangian is men tioned in Section 3. 1.2. Conceptual framew ork If a v ariational form ulation of our problem consists of the Hamiltonian functional T : S !< , on the space of functions S (suitably c hosen), then T , under suitable conditions, has a stationary v alue on S . This giv es rise to a b oundary v alue problem. Our question is: ho w to c ho ose T so that it has the righ t b oundary conditions whic h arises in the original problem. If a v ariational form ulation has the b oundary conditions imp osed on S , they are said to be essen tial . Ho w ev er, if T can be
constructed so as to mak e the b oundary conditions automatically satised on S , then they are said to b e natural . The problem b elo wsho ws, it is p ossible to ha v e part, all, or none of the b oundary conditions \natural" and the remaining therefore are \essen tial". W e initially though t that all self-adjoin t problems (see Stakgold [9] )had a v aria- tional form (VF) in whic h the b oundary conditions app ear naturally . The simple problem, y 0 0 + f ( x ) y + g ( x )= 0 ; a< x < b; (1 a ) y ( a )= y ( b )=0 ; (1 b ) is self-adjoin t. Ho w ev er, the b oundary conditions are essen tial and it can b e con-
sidered as an example of Case 1( vi ) b elo w. Hence not ev ery self-adjoin t bound- ary v alue problem has a VF in whic h the b oundary conditions app e ar naturally . Nonetheless it app ears that there is a substan tial o v erlap bet w een the class of b oundary v alue problems whic h can b e written in VF with natural b oundary con- ditions and those that are self-adjoin t. Ho w ev er, coun ter-examples lik e that in (1) pro vide evidence that there is not an exact corresp ondence. In Section 2, some of the conditions obtained are related to those of self-adjoin tness.
It app ears imp ossible to completely c haracterise the class of BVPs whic h ha v e VF with natural b oundary conditions. In this pap er w epro vide conditions on some
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V ariational metho ds for b oundary v alue problems Abstract. The v ariational form ulationofboundary v alue problems is v aluable in pro viding
remark ably easy computational algorithms as w ell as an alternativ e framew ork with whic h to
pro v e existence results. Boundary conditions imp ose constrain ts whic h can b e anno ying from a computational p oin tofview. The question is then p osed: what is the most general b oundary v alue problem whic h can b e p osed in v ariational form with the b oundary conditions app earing naturally? Sp ecial cases of t w o-p oin t problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjoin tness for the linear case. F urther cases under whic h a Lagrangian ma yor ma y not exist are explained. Keyw ords: natural b oundary conditions, self-adjoin tness, Hamiltonian, Lagrangian. 1. In tro duction 1.1. Bac kground V ariational metho ds pro vide a conceptual and computational framew ork for b ound-
ary v alue problems (BVPs). In the pap er w e deriv e conditions whic h guaran tee that a giv en BVP can b e reform ulated in v ariational form. This can b e a w a yofpro ving existence for BVPs (Graham-Eagle and W ak e [6] ) and it also pro vides a sc heme
for determination of a computational algorithm as in F radkin and W ak e[5]. In the
case of nonlinear eigen v alue problems, these can b e a w a y of pro ducing outcomes lik e parameter path-follo wing as has b een pro duced b y other means b y computer pac k ages lik e Auto (Do edel [2]). The question of what precise conditions need to b e imp osed on b oundary v alue problems so that they can b e written in v ariational form has b een discussed spas- mo dically in the literature, see for example Anderson and Thompson [1] , F els [3], and F orra y[4]. The question as to whic h b oundary v alue problems ha v eav aria- tional form ulation in whic h all of the b oundary conditions app ear naturally is largely still an op en one and is the sub ject of this pap er. Boundary conditions app e ar naturally b y the addition of suitable terms in the Hamiltonian (usually surface in tegrals on the b oundary of the region of in terest). It is usually dicult and sometimes imp ossible to determine these in a systematic w a y . Problems exist (lik ethe t w o-p oin t second order b oundary v alue problem in one dimension with Diric hlet b oundary conditions men tioned b elo w) for whic h this is imp ossible. In these cases the b oundary conditions need to b e imp osed on the space of functions on whic h the stationary v alues of the Hamiltonian is found,
and are called \essen tial b oundary conditions". This can be of n uisance v alue in the determination of the subspaces of functions on whic h the optimisation o ccurs. Th us it app ears useful to b e able to write the v ariational form ulation with all of the b oundary conditions app earing naturally . The more fundamen tal question as to whether there is ev en a Lagrangian is men tioned in Section 3. 1.2. Conceptual framew ork If a v ariational form ulation of our problem consists of the Hamiltonian functional T : S !< , on the space of functions S (suitably c hosen), then T , under suitable conditions, has a stationary v alue on S . This giv es rise to a b oundary v alue problem. Our question is: ho w to c ho ose T so that it has the righ t b oundary conditions whic h arises in the original problem. If a v ariational form ulation has the b oundary conditions imp osed on S , they are said to be essen tial . Ho w ev er, if T can be
constructed so as to mak e the b oundary conditions automatically satised on S , then they are said to b e natural . The problem b elo wsho ws, it is p ossible to ha v e part, all, or none of the b oundary conditions \natural" and the remaining therefore are \essen tial". W e initially though t that all self-adjoin t problems (see Stakgold [9] )had a v aria- tional form (VF) in whic h the b oundary conditions app ear naturally . The simple problem, y 0 0 + f ( x ) y + g ( x )= 0 ; a< x < b; (1 a ) y ( a )= y ( b )=0 ; (1 b ) is self-adjoin t. Ho w ev er, the b oundary conditions are essen tial and it can b e con-
sidered as an example of Case 1( vi ) b elo w. Hence not ev ery self-adjoin t bound- ary v alue problem has a VF in whic h the b oundary conditions app e ar naturally . Nonetheless it app ears that there is a substan tial o v erlap bet w een the class of b oundary v alue problems whic h can b e written in VF with natural b oundary con- ditions and those that are self-adjoin t. Ho w ev er, coun ter-examples lik e that in (1) pro vide evidence that there is not an exact corresp ondence. In Section 2, some of the conditions obtained are related to those of self-adjoin tness.
It app ears imp ossible to completely c haracterise the class of BVPs whic h ha v e VF with natural b oundary conditions. In this pap er w epro vide conditions on some classes of problems for whic h this is p ossible and giv e conditions under whic h this is true. The larger question is still op en. 2. Second order BVP in one-dimension Consider the second order ordinary dieren tial equation y 00 + g ( x; y ; y 0 )= 0 ; a< x < b; (2) where g ( x; y ; y 0 ) is a sucien tly smo oth function. W e are in terested in solutions of (2) that satisfy the non-homogeneous linear b oundary conditions m 11 y ( a )+ m 12 y ( b )+ n 11 y 0 ( a )+ n 12 y 0 ( b )= k 1 ; (3 a ) m 21 y ( a )+ m 22 y ( b )+ n 21 y 0 ( a )+ n 22 y 0 ( b )= k 2 : (3 b ) W e rewrite (3) in the v ector form, M y + N y 0 = k : (4) where M , N are 2  2 matrices and ( M : N ) has rank = 2 to ensure (3 a ), (3 b )are indep enden t. Supp ose that the BVP consisting of (2) and (3) has a corresp onding functional form, J [ y ]=
Z b a f ( x; y ; y 0 ) dx + h ( z 1 ;z
2 ) ; (5) where z 1 = y ( a )and z 2 = y ( b )and h ( z 1 ;z
2 ) 2 C 2 . The b oundary v alue problem is related to the corresp onding functional form b y the Euler-Langrange equation. By setting the rst v ariation J of J to zero, J = Z b a  @f
@y
y + @f
@y
0 y
0  dx + @h
@z
1 y ( a )+
@h
@z
2 y ( b ) ; =  @f
@y
0 y
 b a + Z b a  @f
@y
 d dx
 @f
@y
0  ydx + @h
@z
1 y ( a )+
@h
@z
2 y ( b ) ; = 0 ; w e nd that the follo wing conditions m ust b e satised b y the b oundary terms h in the functional J and the b oundary conditions: @h
@z
1 = @f
@y
0     x = a and @h
@z
2 =  @f
@y
0     x = b : (6) Let q 1 ( z 1 ;z
2 )=
@f
@y
0 j x = a and q 2 ( z 1 ;z
2 )=  @f
@y
0 j x = b . By partially in tegrating q 1 and q 2 with resp ect to z 1 and z 2 resp ectiv ely ,w e can nd the b oundary functional terms h ( z 1 ;z
2 ), pro vided that a certain exactness or in tegrabilit y condition is satised. That is @ 2 h @z
1 @z
2 = @ 2 h @z
2 @z
1 (7) as w e require that h ( z 1 ;z
2 ) 2 C 2 .
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V ariational metho ds for b oundary v alue problems Abstract. The v ariational form ulationofboundary v alue problems is v aluable in pro viding
remark ably easy computational algorithms as w ell as an alternativ e framew ork with whic h to
pro v e existence results. Boundary conditions imp ose constrain ts whic h can b e anno ying from a computational p oin tofview. The question is then p osed: what is the most general b oundary v alue problem whic h can b e p osed in v ariational form with the b oundary conditions app earing naturally? Sp ecial cases of t w o-p oin t problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjoin tness for the linear case. F urther cases under whic h a Lagrangian ma yor ma y not exist are explained. Keyw ords: natural b oundary conditions, self-adjoin tness, Hamiltonian, Lagrangian. 1. In tro duction 1.1. Bac kground V ariational metho ds pro vide a conceptual and computational framew ork for b ound-
ary v alue problems (BVPs). In the pap er w e deriv e conditions whic h guaran tee that a giv en BVP can b e reform ulated in v ariational form. This can b e a w a yofpro ving existence for BVPs (Graham-Eagle and W ak e [6] ) and it also pro vides a sc heme
for determination of a computational algorithm as in F radkin and W ak e[5]. In the
case of nonlinear eigen v alue problems, these can b e a w a y of pro ducing outcomes lik e parameter path-follo wing as has b een pro duced b y other means b y computer pac k ages lik e Auto (Do edel [2]). The question of what precise conditions need to b e imp osed on b oundary v alue problems so that they can b e written in v ariational form has b een discussed spas- mo dically in the literature, see for example Anderson and Thompson [1] , F els [3], and F orra y[4]. The question as to whic h b oundary v alue problems ha v eav aria- tional form ulation in whic h all of the b oundary conditions app ear naturally is largely still an op en one and is the sub ject of this pap er. Boundary conditions app e ar naturally b y the addition of suitable terms in the Hamiltonian (usually surface in tegrals on the b oundary of the region of in terest). It is usually dicult and sometimes imp ossible to determine these in a systematic w a y . Problems exist (lik ethe t w o-p oin t second order b oundary v alue problem in one dimension with Diric hlet b oundary conditions men tioned b elo w) for whic h this is imp ossible. In these cases the b oundary conditions need to b e imp osed on the space of functions on whic h the stationary v alues of the Hamiltonian is found,
and are called \essen tial b oundary conditions". This can be of n uisance v alue in the determination of the subspaces of functions on whic h the optimisation o ccurs. Th us it app ears useful to b e able to write the v ariational form ulation with all of the b oundary conditions app earing naturally . The more fundamen tal question as to whether there is ev en a Lagrangian is men tioned in Section 3. 1.2. Conceptual framew ork If a v ariational form ulation of our problem consists of the Hamiltonian functional T : S !< , on the space of functions S (suitably c hosen), then T , under suitable conditions, has a stationary v alue on S . This giv es rise to a b oundary v alue problem. Our question is: ho w to c ho ose T so that it has the righ t b oundary conditions whic h arises in the original problem. If a v ariational form ulation has the b oundary conditions imp osed on S , they are said to be essen tial . Ho w ev er, if T can be
constructed so as to mak e the b oundary conditions automatically satised on S , then they are said to b e natural . The problem b elo wsho ws, it is p ossible to ha v e part, all, or none of the b oundary conditions \natural" and the remaining therefore are \essen tial". W e initially though t that all self-adjoin t problems (see Stakgold [9] )had a v aria- tional form (VF) in whic h the b oundary conditions app ear naturally . The simple problem, y 0 0 + f ( x ) y + g ( x )= 0 ; a< x < b; (1 a ) y ( a )= y ( b )=0 ; (1 b ) is self-adjoin t. Ho w ev er, the b oundary conditions are essen tial and it can b e con-
sidered as an example of Case 1( vi ) b elo w. Hence not ev ery self-adjoin t bound- ary v alue problem has a VF in whic h the b oundary conditions app e ar naturally . Nonetheless it app ears that there is a substan tial o v erlap bet w een the class of b oundary v alue problems whic h can b e written in VF with natural b oundary con- ditions and those that are self-adjoin t. Ho w ev er, coun ter-examples lik e that in (1) pro vide evidence that there is not an exact corresp ondence. In Section 2, some of the conditions obtained are related to those of self-adjoin tness.
It app ears imp ossible to completely c haracterise the class of BVPs whic h ha v e VF with natural b oundary conditions. In this pap er w epro vide conditions on some classes of problems for whic h this is p ossible and giv e conditions under whic h this is true. The larger question is still op en. 2. Second order BVP in one-dimension Consider the second order ordinary dieren tial equation y 00 + g ( x; y ; y 0 )= 0 ; a< x < b; (2) where g ( x; y ; y 0 ) is a sucien tly smo oth function. W e are in terested in solutions of (2) that satisfy the non-homogeneous linear b oundary conditions m 11 y ( a )+ m 12 y ( b )+ n 11 y 0 ( a )+ n 12 y 0 ( b )= k 1 ; (3 a ) m 21 y ( a )+ m 22 y ( b )+ n 21 y 0 ( a )+ n 22 y 0 ( b )= k 2 : (3 b ) W e rewrite (3) in the v ector form, M y + N y 0 = k : (4) where M , N are 2  2 matrices and ( M : N ) has rank = 2 to ensure (3 a ), (3 b )are indep enden t. Supp ose that the BVP consisting of (2) and (3) has a corresp onding functional form, J [ y ]=
Z b a f ( x; y ; y 0 ) dx + h ( z 1 ;z
2 ) ; (5) where z 1 = y ( a )and z 2 = y ( b )and h ( z 1 ;z
2 ) 2 C 2 . The b oundary v alue problem is related to the corresp onding functional form b y the Euler-Langrange equation. By setting the rst v ariation J of J to zero, J = Z b a  @f
@y
y + @f
@y
0 y
0  dx + @h
@z
1 y ( a )+
@h
@z
2 y ( b ) ; =  @f
@y
0 y
 b a + Z b a  @f
@y
 d dx
 @f
@y
0  ydx + @h
@z
1 y ( a )+
@h
@z
2 y ( b ) ; = 0 ; w e nd that the follo wing conditions m ust b e satised b y the b oundary terms h in the functional J and the b oundary conditions: @h
@z
1 = @f
@y
0     x = a and @h
@z
2 =  @f
@y
0     x = b : (6) Let q 1 ( z 1 ;z
2 )=
@f
@y
0 j x = a and q 2 ( z 1 ;z
2 )=  @f
@y
0 j x = b . By partially in tegrating q 1 and q 2 with resp ect to z 1 and z 2 resp ectiv ely ,w e can nd the b oundary functional terms h ( z 1 ;z
2 ), pro vided that a certain exactness or in tegrabilit y condition is satised. That is @ 2 h @z
1 @z
2 = @ 2 h @z
2 @z
1 (7) as w e require that h ( z 1 ;z
2 ) 2 C 2 . 2.1. Case 1: g indep enden tof y 0 F or simplicit y and without loss of generalit y ,w e let [ a; b ]= [0 ; 1],
y 00 + g ( x; y )=0 ; 0 <x < 1 ; (8) M y + N y 0 = k : The corresp onding v ariational form ulation is J [ y ]=
Z 1 0  ( y 0 ) 2 2  G ( x; y )  dx + h ( z 1 ;z
2 ) ; (9) where G is a partial primitiv eof g with resp ect to y , i.e. @G
@y
= g . The rst v ariation of J is J =[ y 0 y ] 1 0  Z 1 0  @G
@y
+ y 00  ydx + @h
@z
1 y (0) + @h
@z
2 y (1) : (10)
There are sev eral t yp es b oundary functional terms h ( z 1 ;z
2 ) formed in the v ari- ational form ulation whic h are sho wn as the follo wing. Let [ ~ N : ~ M ] be the ro w reduced ec helon form of the augmen ted matrix [ N : M ]. There are only six p o s- sible cases and these are analysed b elo w. Cases ( i )  ( iv ) are BVPs with at least one natural b oundary condition in their v ariational forms. Cases ( v )and( vi )are imp ossible to write in this form with an y natural b oundary condition, that is, they ha v e to b e imp osed as essen tial b oundary conditions. ( i ) Mixed b oundary conditions I [ ~ N : ~ M ]  =  1 0  1  2 0 1  3  4  The b oundary conditions are  1 y (0) +  2 y (1) + y 0 (0) = k 1 ;  3 y (0) +  4 y (1) + y 0 (1) = k 2 ; whic h are Neumann b oundary conditions if all the  ’s are zero. Then putting J = 0 in equation (10), w e see that the b oundary terms yield the conditions, @h
@z
1 = k 1   1 z 1   2 z 2 ; (11)
@h
@z
2 =  ( k 2   3 z 1   4 z 2 ) : (12)
Then in tegrating equations (11) and (12) partially to get h ,w end, h ( z 1 ;z
2 )=  1 2  1 z 2 1   2 z 1 z 2 + 1 2  4 z 2 2 + k 1 z 1  k 2 z 2 :
Page 5


V ariational metho ds for b oundary v alue problems Abstract. The v ariational form ulationofboundary v alue problems is v aluable in pro viding
remark ably easy computational algorithms as w ell as an alternativ e framew ork with whic h to
pro v e existence results. Boundary conditions imp ose constrain ts whic h can b e anno ying from a computational p oin tofview. The question is then p osed: what is the most general b oundary v alue problem whic h can b e p osed in v ariational form with the b oundary conditions app earing naturally? Sp ecial cases of t w o-p oin t problems in one-dimension and some higher dimensional problems are addressed. There is a deep connection with self-adjoin tness for the linear case. F urther cases under whic h a Lagrangian ma yor ma y not exist are explained. Keyw ords: natural b oundary conditions, self-adjoin tness, Hamiltonian, Lagrangian. 1. In tro duction 1.1. Bac kground V ariational metho ds pro vide a conceptual and computational framew ork for b ound-
ary v alue problems (BVPs). In the pap er w e deriv e conditions whic h guaran tee that a giv en BVP can b e reform ulated in v ariational form. This can b e a w a yofpro ving existence for BVPs (Graham-Eagle and W ak e [6] ) and it also pro vides a sc heme
for determination of a computational algorithm as in F radkin and W ak e[5]. In the
case of nonlinear eigen v alue problems, these can b e a w a y of pro ducing outcomes lik e parameter path-follo wing as has b een pro duced b y other means b y computer pac k ages lik e Auto (Do edel [2]). The question of what precise conditions need to b e imp osed on b oundary v alue problems so that they can b e written in v ariational form has b een discussed spas- mo dically in the literature, see for example Anderson and Thompson [1] , F els [3], and F orra y[4]. The question as to whic h b oundary v alue problems ha v eav aria- tional form ulation in whic h all of the b oundary conditions app ear naturally is largely still an op en one and is the sub ject of this pap er. Boundary conditions app e ar naturally b y the addition of suitable terms in the Hamiltonian (usually surface in tegrals on the b oundary of the region of in terest). It is usually dicult and sometimes imp ossible to determine these in a systematic w a y . Problems exist (lik ethe t w o-p oin t second order b oundary v alue problem in one dimension with Diric hlet b oundary conditions men tioned b elo w) for whic h this is imp ossible. In these cases the b oundary conditions need to b e imp osed on the space of functions on whic h the stationary v alues of the Hamiltonian is found,
and are called \essen tial b oundary conditions". This can be of n uisance v alue in the determination of the subspaces of functions on whic h the optimisation o ccurs. Th us it app ears useful to b e able to write the v ariational form ulation with all of the b oundary conditions app earing naturally . The more fundamen tal question as to whether there is ev en a Lagrangian is men tioned in Section 3. 1.2. Conceptual framew ork If a v ariational form ulation of our problem consists of the Hamiltonian functional T : S !< , on the space of functions S (suitably c hosen), then T , under suitable conditions, has a stationary v alue on S . This giv es rise to a b oundary v alue problem. Our question is: ho w to c ho ose T so that it has the righ t b oundary conditions whic h arises in the original problem. If a v ariational form ulation has the b oundary conditions imp osed on S , they are said to be essen tial . Ho w ev er, if T can be
constructed so as to mak e the b oundary conditions automatically satised on S , then they are said to b e natural . The problem b elo wsho ws, it is p ossible to ha v e part, all, or none of the b oundary conditions \natural" and the remaining therefore are \essen tial". W e initially though t that all self-adjoin t problems (see Stakgold [9] )had a v aria- tional form (VF) in whic h the b oundary conditions app ear naturally . The simple problem, y 0 0 + f ( x ) y + g ( x )= 0 ; a< x < b; (1 a ) y ( a )= y ( b )=0 ; (1 b ) is self-adjoin t. Ho w ev er, the b oundary conditions are essen tial and it can b e con-
sidered as an example of Case 1( vi ) b elo w. Hence not ev ery self-adjoin t bound- ary v alue problem has a VF in whic h the b oundary conditions app e ar naturally . Nonetheless it app ears that there is a substan tial o v erlap bet w een the class of b oundary v alue problems whic h can b e written in VF with natural b oundary con- ditions and those that are self-adjoin t. Ho w ev er, coun ter-examples lik e that in (1) pro vide evidence that there is not an exact corresp ondence. In Section 2, some of the conditions obtained are related to those of self-adjoin tness.
It app ears imp ossible to completely c haracterise the class of BVPs whic h ha v e VF with natural b oundary conditions. In this pap er w epro vide conditions on some classes of problems for whic h this is p ossible and giv e conditions under whic h this is true. The larger question is still op en. 2. Second order BVP in one-dimension Consider the second order ordinary dieren tial equation y 00 + g ( x; y ; y 0 )= 0 ; a< x < b; (2) where g ( x; y ; y 0 ) is a sucien tly smo oth function. W e are in terested in solutions of (2) that satisfy the non-homogeneous linear b oundary conditions m 11 y ( a )+ m 12 y ( b )+ n 11 y 0 ( a )+ n 12 y 0 ( b )= k 1 ; (3 a ) m 21 y ( a )+ m 22 y ( b )+ n 21 y 0 ( a )+ n 22 y 0 ( b )= k 2 : (3 b ) W e rewrite (3) in the v ector form, M y + N y 0 = k : (4) where M , N are 2  2 matrices and ( M : N ) has rank = 2 to ensure (3 a ), (3 b )are indep enden t. Supp ose that the BVP consisting of (2) and (3) has a corresp onding functional form, J [ y ]=
Z b a f ( x; y ; y 0 ) dx + h ( z 1 ;z
2 ) ; (5) where z 1 = y ( a )and z 2 = y ( b )and h ( z 1 ;z
2 ) 2 C 2 . The b oundary v alue problem is related to the corresp onding functional form b y the Euler-Langrange equation. By setting the rst v ariation J of J to zero, J = Z b a  @f
@y
y + @f
@y
0 y
0  dx + @h
@z
1 y ( a )+
@h
@z
2 y ( b ) ; =  @f
@y
0 y
 b a + Z b a  @f
@y
 d dx
 @f
@y
0  ydx + @h
@z
1 y ( a )+
@h
@z
2 y ( b ) ; = 0 ; w e nd that the follo wing conditions m ust b e satised b y the b oundary terms h in the functional J and the b oundary conditions: @h
@z
1 = @f
@y
0     x = a and @h
@z
2 =  @f
@y
0     x = b : (6) Let q 1 ( z 1 ;z
2 )=
@f
@y
0 j x = a and q 2 ( z 1 ;z
2 )=  @f
@y
0 j x = b . By partially in tegrating q 1 and q 2 with resp ect to z 1 and z 2 resp ectiv ely ,w e can nd the b oundary functional terms h ( z 1 ;z
2 ), pro vided that a certain exactness or in tegrabilit y condition is satised. That is @ 2 h @z
1 @z
2 = @ 2 h @z
2 @z
1 (7) as w e require that h ( z 1 ;z
2 ) 2 C 2 . 2.1. Case 1: g indep enden tof y 0 F or simplicit y and without loss of generalit y ,w e let [ a; b ]= [0 ; 1],
y 00 + g ( x; y )=0 ; 0 <x < 1 ; (8) M y + N y 0 = k : The corresp onding v ariational form ulation is J [ y ]=
Z 1 0  ( y 0 ) 2 2  G ( x; y )  dx + h ( z 1 ;z
2 ) ; (9) where G is a partial primitiv eof g with resp ect to y , i.e. @G
@y
= g . The rst v ariation of J is J =[ y 0 y ] 1 0  Z 1 0  @G
@y
+ y 00  ydx + @h
@z
1 y (0) + @h
@z
2 y (1) : (10)
There are sev eral t yp es b oundary functional terms h ( z 1 ;z
2 ) formed in the v ari- ational form ulation whic h are sho wn as the follo wing. Let [ ~ N : ~ M ] be the ro w reduced ec helon form of the augmen ted matrix [ N : M ]. There are only six p o s- sible cases and these are analysed b elo w. Cases ( i )  ( iv ) are BVPs with at least one natural b oundary condition in their v ariational forms. Cases ( v )and( vi )are imp ossible to write in this form with an y natural b oundary condition, that is, they ha v e to b e imp osed as essen tial b oundary conditions. ( i ) Mixed b oundary conditions I [ ~ N : ~ M ]  =  1 0  1  2 0 1  3  4  The b oundary conditions are  1 y (0) +  2 y (1) + y 0 (0) = k 1 ;  3 y (0) +  4 y (1) + y 0 (1) = k 2 ; whic h are Neumann b oundary conditions if all the  ’s are zero. Then putting J = 0 in equation (10), w e see that the b oundary terms yield the conditions, @h
@z
1 = k 1   1 z 1   2 z 2 ; (11)
@h
@z
2 =  ( k 2   3 z 1   4 z 2 ) : (12)
Then in tegrating equations (11) and (12) partially to get h ,w end, h ( z 1 ;z
2 )=  1 2  1 z 2 1   2 z 1 z 2 + 1 2  4 z 2 2 + k 1 z 1  k 2 z 2 : pro vided that  3 =   2 ,whic hfollo ws from the exactness condition (7). ( ii ) Mixed b oundary conditions I I [ ~ N : ~ M ]  =  1  1 0  2 0 0 1  3  The b oundary conditions are  2 y (1) + y 0 (0) +  1 y 0 (1) = k 1 ; y (0) +  3 y (1) = k 2 : In this case, w e nd that h = 1 2  2  3 z 2 2   3 k 1 z 2 ; pro vided that  1  3 = 1 whic h again follo ws from the exactness condition (7). Note
that in this case only the second b oundary condition is essen tial in that it m ust b e imp osed on the space of functions considered. ( iii ) Mixed b oundary conditions I I I [ ~ N : ~ M ]  =  1  1  2 0 0 0 0 1  The b oundary conditions are  2 y (0) + y 0 (0) +  1 y 0 (1) = k 1 ; y (1) = k 2 ; then h is h =   2 2 z 2 1 + k 1 z 1 ; pro vided  1 = 0 from the exactness condition (7). In this case only the second b oundary condition is essen tial. ( iv ) Mixed b oundary conditions IV [ ~ N : ~ M ]  =  0 1 0  1 0 0 1  2  The b oundary conditions are  1 y (1) + y 0 (1) = k 1 ; y (0) +  2 y (1) = k 2 ; then h is h =  1 2 z 2 2  k 1 z 2 ;