Variational p-harmonious functions: existence and convergence to p-harmonic functions
In a recent paper, the last three authors showed that a game-theoretic p -harmonic function v is characterized by an asymptotic mean value property with respect to a kind of mean value ν p r [ v ] ( x ) defined variationally on balls B r ( x ) . In this paper, in a domain Ω ⊂ R N , N ≥ 2 , we consid...
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| Published in: | Nonlinear differential equations and applications Vol. 28; no. 5 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01-10-2021
Springer Nature B.V |
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | In a recent paper, the last three authors showed that a game-theoretic
p
-harmonic function
v
is characterized by an asymptotic mean value property with respect to a kind of mean value
ν
p
r
[
v
]
(
x
)
defined variationally on balls
B
r
(
x
)
. In this paper, in a domain
Ω
⊂
R
N
,
N
≥
2
, we consider the operator
μ
p
ε
, acting on continuous functions on
Ω
¯
, defined by the formula
μ
p
ε
[
v
]
(
x
)
=
ν
p
r
ε
(
x
)
[
v
]
(
x
)
, where
r
ε
(
x
)
=
min
[
ε
,
dist
(
x
,
Γ
)
]
and
Γ
denotes the boundary of
Ω
. We first derive various properties of
μ
p
ε
such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function
u
ε
∈
C
(
Ω
¯
)
satisfying the Dirichlet-type problem:
u
(
x
)
=
μ
p
ε
[
u
]
(
x
)
for
every
x
∈
Ω
,
u
=
g
on
Γ
,
for any given function
g
∈
C
(
Γ
)
. This result holds, if we assume the existence of a suitable notion of barrier for all points in
Γ
. That
u
ε
is what we call the
variational
p
-harmonious function with Dirichlet boundary data
g
, and is obtained by means of a Perron-type method based on a comparison principle. We then show that the family
{
u
ε
}
ε
>
0
gives an approximation for the viscosity solution
u
∈
C
(
Ω
¯
)
of
Δ
p
G
u
=
0
in
Ω
,
u
=
g
on
Γ
,
where
Δ
p
G
is the so-called game-theoretic (or homogeneous)
p
-Laplace operator. In fact, we prove that
u
ε
converges to
u
, uniformly on
Ω
¯
as
ε
→
0
. |
|---|---|
| ISSN: | 1021-9722 1420-9004 |
| DOI: | 10.1007/s00030-021-00714-7 |