Parana Journal of Science and Education, v.9, n.7, (1-11), December 1, 2023
PJSE, ISSN 2447-6153, © 2015-2023
https://sites.google.com/site/pjsciencea/
Received: October 26, 2023; Accepted: November 20, 2023; Published: December 1, 2023.
On Orthogonal Parts of a Solution to a Cauchy BVP
over Sobolev Spaces
Dejenie Alemayehu Lakew
1
* and Nar Rawal
1
**
Abstract
Let Ω be a smooth and bounded doamin in R
n
. Considering three BVPs.
(I ) First order: Let f ∈ L
2
(Ω) , g ∈ W
1
2
,2
(Ω). Then the first order Cauchy BVP :
Du = f in Ω
u = g on ∂ Ω
has a solution u given as W
1,2
(Ω) ∋ u =[u
g
] ⊎ [u
f
] where [u
g
] to be the part of the solution that is evolved from
the trace value g of u, and [u
f
] to be the part of the solution that is evolved from the value f of the differential
equation over the domain Ω.
(II ) Second order: Let f ∈ L
2
(Ω) , g
1
∈ W
3
2
,2
(∂ Ω) , g
2
∈ W
1
2
,2
(∂ Ω).
Then the BVP:
−D
2
u = f in Ω
τ u = g on ∂ Ω
where τ u =
(
u
|∂ Ω
, Du
|∂ Ω
)
=(g
1
, g
2
) has a solution u ∈ W
2,2
(Ω) with u =[u]
(g
1
,g
2
)
⊎ [u]
f
and
(III ) Higher order: Let f ∈ L
2
(Ω) , g
j
∈ W
k− j+
1
2
(∂ Ω) , j = 1, ..., k − 1. Then
D
k
u = f in Ω
τ u = g on ∂ Ω
where τ u =
(
u
|∂ Ω
, Du
|∂ Ω
, ..., D
k−1
u
|∂ Ω
)
=(g
1
, g
2
, ..., g
k−1
), for k ≥ 3, has a solution u ∈ W
k,2
(Ω) given as
u =[u]
(g
1
,g
2
,...,g
k−1
)
⊎ [u]
f
.
The symbol ⊎ represents an orthogonal sum of functions that are from orthogonal sum ⊕ of function subspaces
of a Sobolev space with inner product.
Keywords
Dirac Operator, Orthogonal Sum, Cauchy Problem, Sobolev Space.
1
Department of Mathematics, Hampton University, Hampton VA 23668, Hampton, Virginia, U.S.A.
*Corresponding author: dejenie.lakew@hamptonu.edu
**Corresponding author: nar.rawal@hamptonu.edu