The application of the intersect index to quasilinear eigenfunction problems
First time the intersect index was applied to non-linear problems in L. Lusternick's research. This direction is investigating at voronezh school now [1,2]. Small eigenfunctions and its global branches was considered by the intersect index in [3-5].
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| Published in: | Нелинейные граничные задачи |
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| Date: | 1999 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
1999
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| Cite this: | The application of the intersect index to quasilinear eigenfunction problems / Ya.M. Dymarsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 17-119. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860057920962560000 |
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| author | Dymarsky, Ya.M. |
| author_facet | Dymarsky, Ya.M. |
| citation_txt | The application of the intersect index to quasilinear eigenfunction problems / Ya.M. Dymarsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 17-119. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Нелинейные граничные задачи |
| description | First time the intersect index was applied to non-linear problems in L. Lusternick's research. This direction is investigating at voronezh school now [1,2]. Small eigenfunctions and its global branches was considered by the intersect index in [3-5].
|
| first_indexed | 2025-12-07T17:02:20Z |
| format | Article |
| fulltext |
THE APPLICATION OF THE INTERSECT INDEX
TO QUASILINEAR EIGENFUNCTION PROBLEMS
c© Ya.M.Dymarsky
First time the intersect index was applied to non-linear problems in L. Lusternick’s
research. This direction is investigating at voronezh school now [1,2]. Small eigenfunc-
tions and its global branches was considered by the intersect index in [3-5].
1. Definitions. We are interested in eigenvalues (e.v.) λ ∈ R and eigenfunctions
(e.f.) u ∈ W 1
2 (Ω) of the quasilinear problem
∆u + p(u, grad(u), x)u + λu = 0, u|∂Ω= 0 (1)
u ∈ S∞R = {u :
∫
Ω
u2 = R2} (R > 0), (2)
where W k
2 (Ω) is Sobolev’s space with norm ‖ · ‖k, Ω ⊂ Rn is a bounded domain with
smooth boundary ∂Ω, x ∈ Ω, ∆ is Laplas operator, p is a continue function. For
simplicity of a priori estimates we have to suppose m < p(u, y, x) < M ((u, y, x) ∈
Rn+1 × Ω).
The pair (λ, u) which satisfy (1),(2) is called normalised solution (n.s.). If (λ∗, u∗)
is a n.s. then λ∗ is an e.v. of the linear problem
∆u + q(x)u + λu = 0, u|∂Ω= 0, (3)
where
q(x) = p(u∗(x), grad(u∗(x)), x). (4)
The e.f. u∗ is among eigenfunctions of the problem (3),(4) certainly. The linear prob-
lem (3),(4) is symmetric that is why λ ∈ R. Eigenvalues of (3) form the nondecreasing
sequence λ0 < λ1 ≤ λ2 ≤ ...; λn →∞.
D e f. 1. The n.s. (λ∗, u∗) of the problem (1),(2) is named the simple (n-multiple)
, if λ∗ is simple (n-multiple) for the linear problem (3),(4). (The multiplicity of e.v. is
finite always.)
D e f. 2. The n.s. (λ∗, u∗) of the problem (1),(2) and its elements have such number
which the e.v. λ∗ has as a eigenvalue of the linear problem (3),(4).
We need a priori estimates of normalised solutions which have bounded numbers.
L e m m a 1. Eigenvalues λ with number n of the problem (1),(2) satisfy estimates
λn − M < λ < λn + m where λn is the e.v. with number n of the problem (3) with
q(x) ≡ 0.
L e m m a 2. Normalised solutions (λ, u) with number n of the problem (1),(2) satisfy
the estimate |λ | +‖u‖2 < C where the constant C depends on R, n, m, M only.
The problem (1) is equal to the operator equation
u + (λ + M)A(u)u = 0, (5)
due to lemma 1 where A is a continue mapping from W 1
2 (Ω) to Banach space L of linear
symmetric compact operators.
We consider the family of linear equations
u + (λ + M)Bu = 0. (6)
An operator B ∈ L is the parameter of the family. Let T∞R = {(B, u) ∈ L × S∞R :
u is an e.f. of the problem (6)}. The set T∞R is a smooth Banach manifold with
model space L [6]. The manifold T∞R is stratificated by numbers and multiplicity of its
eigenfunctions: T∞R (n, l) = {(B, u) ∈ T∞R : u is an e.f. of (6) with e.v. λ, moreover
λn−1(B) < λ = λn(B) = ... = λn+l−1(B) < λn+l(B)}. Thus T∞R =
⋃
n,l∈N T∞R (n, l).
According to [7] it’s possible to prove that T∞R (n, l) is the smooth submanifold of T∞R
end codimT∞R (n, l) = (l − 1)l/2. Notice codimT∞R (n, 1) = 0, codimT∞R (n, 2) = 1. We
give those number end multiplicity to a point (B, u) ∈ T∞R which the e.f. u has.
We examine the mapping
GrA : S∞R −→ L× S∞R , GrA(u) = (A(u), u), (7)
which is important for us.
T h e o r e m 1. A function u is an e.f. of the equation (6) only in the case
GrA(u) ∈ T∞R . The number of solution (λ, u) and its multiplicity are defined by the
index (n, l) of stratum T∞R (n, l): GrA(u) = (A(u), u) ∈ T∞R (n, l) ⊂ T∞R .
D e f. 3. A mapping A is called n-typical if the image of the mapping (7) doesn’t
intersect stratums T∞R (n, l) where the multiplicity l ≥ 2. Other words solutions with
number n are simple.
We will show that simple solutions can be obtained by the intersect index.
2. Intersect index. At first we consider the finite dimensional problem
v + γK(v)v = 0, v ∈ Sk−1, (8)
which is analogous to the problem (5); K is a continue mapping from Sk−1 to the
space Lk of real symmetric k-dimensional matrixes. Definitions 1-3 have the sense in
the problem (8). Manifolds T k, T k(n, l), the mapping GrK are determined by analogy
with T∞R , T∞R (n, l), GrA accordingly. Theorem 1 is true in case of the problem (8).
L e m m a 3.The set of n-typical mappings K is opened and dense in the space of
continue mappings from Sk−1 to Lk.
Since dimT k = dimLk for any n ≤ k and an n-typical mapping K is determined
the orientated intersect index χ(T
k
(n, 1), GrK) = χ(n,K) (T
k
(n, 1) is the closure of
the stratum T k(n, 1)). If the index isn’t equal to zero then the equation (8) has a n.s.
with number n. The calculation of the index is a difficult problem due to the manifold
T
k
(n, 1) has the boundary.
Let {u0, u1, ...} be the set of eigenfunctions of some operator B ∈ L. Let Rk ⊂
W 1
2 (Ω) (k = 1, 2...) be the finite dimensional subspace which is generated by the basis
{u0, u1, ..., uk−1}. Let P k be the orthogonal projection on Rk. We replace the problem
(5),(2) by the approximate equation
v + (λ + M)P kA(v)v = 0, v ∈ Sk−1, (9)
which has type of (8). If a mapping A is n-typical than the mapping P kA is n-typical
for any big k too. Therefore the index χ(n, P kA) is determined for any big k.
T h e o r e m 2. Index χ(n, P kA) has not change for any big k.
D e f. 4. Let L be a n-typical mapping. We determine that the orientated intersect
index χ(T
∞
R (n, 1), GrA) = χ(n, P kA), where k is big enough.
If the index isn’t equal to zero then the problem (5),(2) has a n.s. with number n.
Moreover, the solution is the limit (k →∞) of solutions of equations (9) due to a priory
estimates (lemma 2).
The intersect index is an invariant of a homotopy in the class of n-typical mappings.
In our opinion a control of n-typeness isn’t easy. For small eigenfunctions n-typeness
are checked in a finite dimensional kernel of the linear problem
∆u + p(0, 0, x)u + λ∗u = 0, u|∂Ω= 0, (10)
where λ∗ is the e.v. of the problem (10) [3,4].
References
1. Borisovich Yu.G., Zvyagin V.G., Sapronov Yu.I., Non-linear Fredholm mappings, Uspehi Matem.
Nauk 32 (1977), no. 4, 3-52.
2. Borisovich Yu.G., Kunakovskaya O.V., Intersection theory methods, Stochastic and global analysis.
Voronezh. (1997).
3. Dymarsky Ya.M., On typical bifurcations in a class of operator equations, Russian Acad. Sci. Dokl.
Math. 50 (1995), no. 2, 446-449.
4. Dymarsky Ya.M., On branches of small solutions of some operator equations, Ukr. Math. Jour. 48
(1996), no. 7, 901-909.
5. Dymarsky Ya.M., Unbounded branches of solutions of some boundary-value problems, Ukr. Math.
Jour. 48 (1996), no. 9, 1194-1199.
6. Uhlenbeck K., Generic properties of eigenfunctions, Amer. Jour. Math. 98 (1976), no. 4, 1059-1078.
7. Fujiwara D., Tanikawa M., Yukita Sh., The spectrum of the Laplacian, Proc. Japan Acad. 54, Ser.
A (1978), no. 4, 87-91.
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| id | nasplib_isofts_kiev_ua-123456789-169281 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0236-0497 |
| language | English |
| last_indexed | 2025-12-07T17:02:20Z |
| publishDate | 1999 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Dymarsky, Ya.M. 2020-06-09T16:37:30Z 2020-06-09T16:37:30Z 1999 The application of the intersect index to quasilinear eigenfunction problems / Ya.M. Dymarsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 17-119. — Бібліогр.: 7 назв. — англ. 0236-0497 https://nasplib.isofts.kiev.ua/handle/123456789/169281 First time the intersect index was applied to non-linear problems in L. Lusternick's research. This direction is investigating at voronezh school now [1,2]. Small eigenfunctions and its global branches was considered by the intersect index in [3-5]. en Інститут прикладної математики і механіки НАН України Нелинейные граничные задачи The application of the intersect index to quasilinear eigenfunction problems Article published earlier |
| spellingShingle | The application of the intersect index to quasilinear eigenfunction problems Dymarsky, Ya.M. |
| title | The application of the intersect index to quasilinear eigenfunction problems |
| title_full | The application of the intersect index to quasilinear eigenfunction problems |
| title_fullStr | The application of the intersect index to quasilinear eigenfunction problems |
| title_full_unstemmed | The application of the intersect index to quasilinear eigenfunction problems |
| title_short | The application of the intersect index to quasilinear eigenfunction problems |
| title_sort | application of the intersect index to quasilinear eigenfunction problems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169281 |
| work_keys_str_mv | AT dymarskyyam theapplicationoftheintersectindextoquasilineareigenfunctionproblems AT dymarskyyam applicationoftheintersectindextoquasilineareigenfunctionproblems |