Diffusion with non-local boundary conditions
Wolfgang Arendt
Ulm University
Będlewo, April 2017
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
1 / total
Non-local boundary conditions
Part 1
local Dirichlet b.c.: u|∂Ω = 0.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
2 / total
Non-local boundary conditions
Part 1
local Dirichlet b.c.: u|∂Ω = 0.
non-local Dirichlet b.c.: u = hµ( · ), ui.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
2 / total
Non-local boundary conditions
Part 1
local Dirichlet b.c.: u|∂Ω = 0.
non-local Dirichlet b.c.: u = hµ( · ), ui.
Part 2:
local Robin b.c.: ∂ν u + βu|∂Ω = 0.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
2 / total
Non-local boundary conditions
Part 1
local Dirichlet b.c.: u|∂Ω = 0.
non-local Dirichlet b.c.: u = hµ( · ), ui.
Part 2:
local Robin b.c.: ∂ν u + βu|∂Ω = 0.
non-local Robin b.c.: ∂ν u + βu|∂Ω = hµ( · ), ui.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
2 / total
Non-local boundary conditions
Part 1
local Dirichlet b.c.: u|∂Ω = 0.
non-local Dirichlet b.c.: u = hµ( · ), ui.
Part 2:
local Robin b.c.: ∂ν u + βu|∂Ω = 0.
non-local Robin b.c.: ∂ν u + βu|∂Ω = hµ( · ), ui.
Here: µ(z) measure on Ω for all z ∈ ∂Ω.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
2 / total
Non-local boundary conditions
Part 1
local Dirichlet b.c.: u|∂Ω = 0.
non-local Dirichlet b.c.: u = hµ( · ), ui.
Part 2:
local Robin b.c.: ∂ν u + βu|∂Ω = 0.
non-local Robin b.c.: ∂ν u + βu|∂Ω = hµ( · ), ui.
Here: µ(z) measure on Ω for all z ∈ ∂Ω.
References:
Part 1: S. Kunkel, M. Kunze, W. A.: Diffusion with nonlocal boundary
conditions (J.F.A., 2016)
Part 2: S. Kunkel, M. Kunze, W. A.: Diffusion with nonlocal Robin
boundary conditions (submitted)
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
2 / total
Holomorphic semigroups
Definition
An operator A generates a holomorphic semigroup if there exist ω ∈ R,
M ≥ 0 such that {λ ∈ C : Re λ > ω} ⊆ ρ(A) and
kλR(λ, A)k ≤ M
Wolfgang Arendt (Ulm University)
(Re λ > ω).
Non-local boundary conditions
April 2017
3 / total
Holomorphic semigroups
Definition
An operator A generates a holomorphic semigroup if there exist ω ∈ R,
M ≥ 0 such that {λ ∈ C : Re λ > ω} ⊆ ρ(A) and
kλR(λ, A)k ≤ M
Then T (t) =
1
2πi
R
Γe
λt R(λ, A) dλ
Wolfgang Arendt (Ulm University)
(Re λ > ω).
defines a hol. semigroup on a sector Σθ .
Non-local boundary conditions
April 2017
3 / total
Holomorphic semigroups
Definition
An operator A generates a holomorphic semigroup if there exist ω ∈ R,
M ≥ 0 such that {λ ∈ C : Re λ > ω} ⊆ ρ(A) and
kλR(λ, A)k ≤ M
Then T (t) =
1
2πi
R
Γe
λt R(λ, A) dλ
(Re λ > ω).
defines a hol. semigroup on a sector Σθ .
It is a C0 -semigroup if A is densely defined.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
3 / total
Dirichlet regular
Let Ω ⊆ Rd be open and bounded.
Definition
Ω is Dirichlet regular if, for all g ∈ C (∂Ω), there exists
u ∈ C 2 (Ω) ∩ C (Ω) such that
(
∆u
= 0 in Ω,
u|∂Ω = g .
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Non-local boundary conditions
April 2017
4 / total
Dirichlet regular
Let Ω ⊆ Rd be open and bounded.
Definition
Ω is Dirichlet regular if, for all g ∈ C (∂Ω), there exists
u ∈ C 2 (Ω) ∩ C (Ω) such that
(
∆u
= 0 in Ω,
u|∂Ω = g .
Examples
(a) Lipschitz boundary implies Dirichlet regular.
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Non-local boundary conditions
April 2017
4 / total
Dirichlet regular
Let Ω ⊆ Rd be open and bounded.
Definition
Ω is Dirichlet regular if, for all g ∈ C (∂Ω), there exists
u ∈ C 2 (Ω) ∩ C (Ω) such that
(
∆u
= 0 in Ω,
u|∂Ω = g .
Examples
(a) Lipschitz boundary implies Dirichlet regular.
(b) d = 2: simply connected implies Dirichlet regular.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
4 / total
Measures defining non-local Dirichlet boundary conditions
For each z ∈ ∂Ω let µ(z) be a Borel measure on Ω such that
(a) µ(z)(Ω) ≤ 1,
(b) ∂Ω 3 z 7→ hµ(z), f i ∈ R is continuous for each f ∈ B b (Ω) (bounded
and measurable).
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
5 / total
Measures defining non-local Dirichlet boundary conditions
For each z ∈ ∂Ω let µ(z) be a Borel measure on Ω such that
(a) µ(z)(Ω) ≤ 1,
(b) ∂Ω 3 z 7→ hµ(z), f i ∈ R is continuous for each f ∈ B b (Ω) (bounded
and measurable).
Non-local Dirichlet boundary condition:
Z
u(z) = hµ(z), ui =
u(y ) dµ(z)(y )
∀z ∈ ∂Ω.
Ω
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
5 / total
Measures defining non-local Dirichlet boundary conditions
For each z ∈ ∂Ω let µ(z) be a Borel measure on Ω such that
(a) µ(z)(Ω) ≤ 1,
(b) ∂Ω 3 z 7→ hµ(z), f i ∈ R is continuous for each f ∈ B b (Ω) (bounded
and measurable).
Non-local Dirichlet boundary condition:
Z
u(z) = hµ(z), ui =
u(y ) dµ(z)(y )
∀z ∈ ∂Ω.
Ω
Remark
Cµ := {u ∈ C (Ω) : u(z) = hµ(z), ui for all z ∈ Ω} is a closed proper
subspace of C (Ω)!
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
5 / total
The Laplacian with non-local Dirichlet b.c.
Ω ⊆ Rd open, bounded, Dirichlet regular.
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Non-local boundary conditions
April 2017
6 / total
The Laplacian with non-local Dirichlet b.c.
Ω ⊆ Rd open, bounded, Dirichlet regular.
the space: L∞ (Ω)
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Non-local boundary conditions
April 2017
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The Laplacian with non-local Dirichlet b.c.
Ω ⊆ Rd open, bounded, Dirichlet regular.
the space: L∞ (Ω)
the operator:
D(∆µ ) := {u ∈ C (Ω) : ∆u ∈ L∞ (Ω),
u(z) = hµ(z), ui for all z ∈ ∂Ω},
∆µ := ∆u.
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Non-local boundary conditions
April 2017
6 / total
The generation theorem for non-local Dirichlet b.c.
Ω Dirichlet regular.
Theorem
∆µ generates a holomorphic semigroup Tµ on L∞ (Ω).
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
7 / total
The generation theorem for non-local Dirichlet b.c.
Ω Dirichlet regular.
Theorem
∆µ generates a holomorphic semigroup Tµ on L∞ (Ω).
Moreover:
(a) Tµ (t) ≥ 0 [i.e. f ≥ 0 ⇒ Tµ (t)f ≥ 0]
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
7 / total
The generation theorem for non-local Dirichlet b.c.
Ω Dirichlet regular.
Theorem
∆µ generates a holomorphic semigroup Tµ on L∞ (Ω).
Moreover:
(a) Tµ (t) ≥ 0 [i.e. f ≥ 0 ⇒ Tµ (t)f ≥ 0]
(b) kTµ (t)k ≤ 1.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
7 / total
The generation theorem for non-local Dirichlet b.c.
Ω Dirichlet regular.
Theorem
∆µ generates a holomorphic semigroup Tµ on L∞ (Ω).
Moreover:
(a) Tµ (t) ≥ 0 [i.e. f ≥ 0 ⇒ Tµ (t)f ≥ 0]
(b) kTµ (t)k ≤ 1.
(c) Tµ (t) is strong Feller for all t > 0.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
7 / total
Strong Feller
0 ≤ S ∈ L(L∞ (Ω)) is strong Feller if
(a) SL∞ (Ω) ⊆ C (Ω) and
(b) fn ↓ 0 a.e. ⇒ Sfn ↓ 0 in C (Ω).
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
8 / total
Strong Feller
0 ≤ S ∈ L(L∞ (Ω)) is strong Feller if
(a) SL∞ (Ω) ⊆ C (Ω) and
(b) fn ↓ 0 a.e. ⇒ Sfn ↓ 0 in C (Ω).
⇔
Sf (x) =
R
k(x, y )f (y ) dy where
k(x, · ) ∈ L1 (Ω) for all x ∈ Ω and
x 7→ k(x, · ) is σ(L1 (Ω), L∞ (Ω)) continuous.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
8 / total
Strong Feller
0 ≤ S ∈ L(L∞ (Ω)) is strong Feller if
(a) SL∞ (Ω) ⊆ C (Ω) and
(b) fn ↓ 0 a.e. ⇒ Sfn ↓ 0 in C (Ω).
⇔
Sf (x) =
R
k(x, y )f (y ) dy where
k(x, · ) ∈ L1 (Ω) for all x ∈ Ω and
x 7→ k(x, · ) is σ(L1 (Ω), L∞ (Ω)) continuous.
⇒
S 2 is compact.
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Non-local boundary conditions
April 2017
8 / total
Perron–Frobenius theory
Let E be a Banach lattice (e.g. E = L∞ (Ω)).
Let T : (0, ∞) → L(E ) be a semigroup (i.e. T (t + s) = T (t)T (s), no
topological condition!)
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Non-local boundary conditions
April 2017
9 / total
Perron–Frobenius theory
Let E be a Banach lattice (e.g. E = L∞ (Ω)).
Let T : (0, ∞) → L(E ) be a semigroup (i.e. T (t + s) = T (t)T (s), no
topological condition!)
Theorem (Lotz)
Assume
(a) T (t0 ) is compact for at least one t0 > 0.
(b) T (t) ≥ 0 and kT (t)k ≤ M for all t > 0.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
9 / total
Perron–Frobenius theory
Let E be a Banach lattice (e.g. E = L∞ (Ω)).
Let T : (0, ∞) → L(E ) be a semigroup (i.e. T (t + s) = T (t)T (s), no
topological condition!)
Theorem (Lotz)
Assume
(a) T (t0 ) is compact for at least one t0 > 0.
(b) T (t) ≥ 0 and kT (t)k ≤ M for all t > 0.
Then P := limt→∞ T (t) exists in L(E ).
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
9 / total
Perron–Frobenius theory
Let E be a Banach lattice (e.g. E = L∞ (Ω)).
Let T : (0, ∞) → L(E ) be a semigroup (i.e. T (t + s) = T (t)T (s), no
topological condition!)
Theorem (Lotz)
Assume
(a) T (t0 ) is compact for at least one t0 > 0.
(b) T (t) ≥ 0 and kT (t)k ≤ M for all t > 0.
Then P := limt→∞ T (t) exists in L(E ).
P is a projection of finite rank onto the fixed space of T .
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
9 / total
Asymptotic behaviour for non-local Dirichlet boundary conditions.
Assume that Ω is connected.
Theorem
(a) If µ(z0 )(Ω) < 1 for some z0 ∈ ∂Ω, then limt→∞ kTµ (t)kL∞ (Ω) = 0.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
10 / total
Asymptotic behaviour for non-local Dirichlet boundary conditions.
Assume that Ω is connected.
Theorem
(a) If µ(z0 )(Ω) < 1 for some z0 ∈ ∂Ω, then limt→∞ kTµ (t)kL∞ (Ω) = 0.
(b) If µ(z0 )(Ω) = 1 for all z ∈ ∂Ω, then limt→∞ Tµ (t) = P in L(L∞ (Ω)),
where
Z
(f ∈ L∞ (Ω))
Pf =
f · h dx · 1Ω
Ω
for some 0 ≤ h ∈ L1 (Ω).
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Non-local boundary conditions
April 2017
10 / total
Proof.
Remark
µ(z)(Ω) = 1 for all z ∈ ∂Ω
⇒ hµ(z)(Ω), 1Ω i = 1 for all z ∈ ∂Ω
⇒
1Ω ∈ D(∆µ ).
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Non-local boundary conditions
April 2017
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Proof.
Remark
µ(z)(Ω) = 1 for all z ∈ ∂Ω
⇒ hµ(z)(Ω), 1Ω i = 1 for all z ∈ ∂Ω
⇒
1Ω ∈ D(∆µ ).
Anti-maximum principle
Let u ∈ C (Ω) such that u(z) = hµ(z), ui for all z ∈ ∂Ω and let
C := maxx∈Ω u(x) > 0.
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Non-local boundary conditions
April 2017
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Proof.
Remark
µ(z)(Ω) = 1 for all z ∈ ∂Ω
⇒ hµ(z)(Ω), 1Ω i = 1 for all z ∈ ∂Ω
⇒
1Ω ∈ D(∆µ ).
Anti-maximum principle
Let u ∈ C (Ω) such that u(z) = hµ(z), ui for all z ∈ ∂Ω and let
C := maxx∈Ω u(x) > 0.
Then there exists x0 ∈ Ω such that u(x0 ) = C (i.e. the maximum is
attained in the interior).
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Non-local boundary conditions
April 2017
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Proof.
Remark
µ(z)(Ω) = 1 for all z ∈ ∂Ω
⇒ hµ(z)(Ω), 1Ω i = 1 for all z ∈ ∂Ω
⇒
1Ω ∈ D(∆µ ).
Anti-maximum principle
Let u ∈ C (Ω) such that u(z) = hµ(z), ui for all z ∈ ∂Ω and let
C := maxx∈Ω u(x) > 0.
Then there exists x0 ∈ Ω such that u(x0 ) = C (i.e. the maximum is
attained in the interior).
Let u ∈ ker(∆µ ). Then ∆u = 0 and the maximum is attained in the
interior.
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Non-local boundary conditions
April 2017
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Proof.
Remark
µ(z)(Ω) = 1 for all z ∈ ∂Ω
⇒ hµ(z)(Ω), 1Ω i = 1 for all z ∈ ∂Ω
⇒
1Ω ∈ D(∆µ ).
Anti-maximum principle
Let u ∈ C (Ω) such that u(z) = hµ(z), ui for all z ∈ ∂Ω and let
C := maxx∈Ω u(x) > 0.
Then there exists x0 ∈ Ω such that u(x0 ) = C (i.e. the maximum is
attained in the interior).
Let u ∈ ker(∆µ ). Then ∆u = 0 and the maximum is attained in the
interior.
⇒ u is constant.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
11 / total
Extensions.
∆ may be replaced with an elliptic operator in non-divergence form
Au :=
d
X
i,j=1
aij Di Dj u +
d
X
bj D j u + c 0 u
j=1
with aij Dini continuous.
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Non-local boundary conditions
April 2017
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Previous results.
P. Gurevich, W. Jäger, A. Skubachevskii (SIAM J. Math. Anal., 2009)
E. I. Galakhov, A. Skubachevskii (J. Diff. Equ., 2001)
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Non-local boundary conditions
April 2017
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Part 2: Robin boundary conditions
Normal derivative.
Ω ⊆ Rd open, bounded with Lipschitz boundary,
σ = Hd−1 surface measure on ∂Ω.
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Non-local boundary conditions
April 2017
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Part 2: Robin boundary conditions
Normal derivative.
Ω ⊆ Rd open, bounded with Lipschitz boundary,
σ = Hd−1 surface measure on ∂Ω.
Definition
Let u ∈ H 1 (Ω) such that ∆u ∈ L2 (Ω).
(a) Let h ∈ L2 (∂Ω).
Z
∂ν u = h
Z
:⇔
Wolfgang Arendt (Ulm University)
∇u∇v =
∆uv +
Ω
Z
Ω
Non-local boundary conditions
hv
∀v ∈ C 1 (Ω).
∂Ω
April 2017
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Part 2: Robin boundary conditions
Normal derivative.
Ω ⊆ Rd open, bounded with Lipschitz boundary,
σ = Hd−1 surface measure on ∂Ω.
Definition
Let u ∈ H 1 (Ω) such that ∆u ∈ L2 (Ω).
(a) Let h ∈ L2 (∂Ω).
Z
∂ν u = h
Z
:⇔
∇u∇v =
∆uv +
Ω
Z
Ω
hv
∀v ∈ C 1 (Ω).
∂Ω
(b) Let E ⊆ L2 (∂Ω).
∂ν u ∈ E
Wolfgang Arendt (Ulm University)
:⇔
∃h ∈ E : ∂ν u = h.
Non-local boundary conditions
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Local Robin boundary conditions
Let 0 ≤ β ∈ L∞ (∂Ω).
Define the Robin Laplacian on C (Ω) by
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
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Non-local boundary conditions
April 2017
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Local Robin boundary conditions
Let 0 ≤ β ∈ L∞ (∂Ω).
Define the Robin Laplacian on C (Ω) by
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
Theorem (R. Nittka, J.D.E. 2014)
∆β generates a holomorphic C0 -semigroup on C (Ω).
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Non-local boundary conditions
April 2017
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Non-local Robin boundary conditions
Ω ⊆ Rd bounded, open with Lipschitz boundary.
For each z ∈ ∂Ω we are given a Borel measure µ(z) on Ω such that for
each f ∈ B b (Ω) (bounded measurable)
∂Ω 3 z 7→ hµ(z), f i ∈ R
is continuous.
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Non-local boundary conditions
April 2017
16 / total
Non-local Robin boundary conditions
Ω ⊆ Rd bounded, open with Lipschitz boundary.
For each z ∈ ∂Ω we are given a Borel measure µ(z) on Ω such that for
each f ∈ B b (Ω) (bounded measurable)
∂Ω 3 z 7→ hµ(z), f i ∈ R
is continuous.
Let 0 ≤ β ∈ L∞ (∂Ω).
D(∆β,µ ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν + βu|∂Ω = hµ( · ), ui a.e. on ∂Ω}
∆β,µ u := ∆u.
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Non-local boundary conditions
April 2017
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Generation Theorem.
Theorem
∆β,µ generates a holomorphic C0 -semigroup Tβ,µ on C (Ω). Moreover,
(a) Tβ,µ (t) ≥ 0,
(b) Tβ,µ (t) is compact.
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Non-local boundary conditions
April 2017
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Generation Theorem.
Theorem
∆β,µ generates a holomorphic C0 -semigroup Tβ,µ on C (Ω). Moreover,
(a) Tβ,µ (t) ≥ 0,
(b) Tβ,µ (t) is compact.
Method: Greiner’s boundary perturbation.
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Non-local boundary conditions
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Generation Theorem.
Theorem
∆β,µ generates a holomorphic C0 -semigroup Tβ,µ on C (Ω). Moreover,
(a) Tβ,µ (t) ≥ 0,
(b) Tβ,µ (t) is compact.
Method: Greiner’s boundary perturbation.
We also have a generation result on L∞ (Ω) with strong Feller property.
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Non-local boundary conditions
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Monotonicity.
Tβ,µ (t) decreases if β increases.
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Non-local boundary conditions
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Monotonicity.
Tβ,µ (t) decreases if β increases.
Tβ,µ (t) increases if µ increases.
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Non-local boundary conditions
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Monotonicity.
Tβ,µ (t) decreases if β increases.
Tβ,µ (t) increases if µ increases.
µ(z)(Ω) ≤ β(z) a.e. ⇒ kTβ,µ (t)k ≤ 1.
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Non-local boundary conditions
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Asymptotics.
Theorem
(a) If β(z) = µ(z)(Ω) a.e., then limt→∞ Tβ,µ (t) = P in L(C (Ω)), where
Z
Pf =
f dρ · 1Ω
Ω
for some finite Borel measure ρ on Ω.
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Non-local boundary conditions
April 2017
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Asymptotics.
Theorem
(a) If β(z) = µ(z)(Ω) a.e., then limt→∞ Tβ,µ (t) = P in L(C (Ω)), where
Z
Pf =
f dρ · 1Ω
Ω
for some finite Borel measure ρ on Ω.
(b) If β(z) ≤ µ(z)(Ω) a.e. but not = a.e., then limt→∞ Tβ,µ (t) = 0 in
L(C (Ω)).
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Non-local boundary conditions
April 2017
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Extensions.
Replace ∆ with an elliptic operator A in divergence form with real
measurable coefficients. Then the generation theorem holds.
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Non-local boundary conditions
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Extensions.
Replace ∆ with an elliptic operator A in divergence form with real
measurable coefficients. Then the generation theorem holds.
If A is symmetric, then the theorem about the asymptotic behaviour
remains true, but the condition (a) becomes
0 ≤ µ( · )(Ω) = β +
d
X
tr bj · νj .
j=1
So the coefficients come into play.
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Non-local boundary conditions
April 2017
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Continuous kernels.
Let T : (0, ∞) → L(L2 (Ω)) be a selfadjoint semigroup such that
T (t)L2 (Ω) ⊆ C (Ω)
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Non-local boundary conditions
April 2017
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Continuous kernels.
Let T : (0, ∞) → L(L2 (Ω)) be a selfadjoint semigroup such that
T (t)L2 (Ω) ⊆ C (Ω)
(this is stronger than strong Feller).
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Non-local boundary conditions
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Continuous kernels.
Let T : (0, ∞) → L(L2 (Ω)) be a selfadjoint semigroup such that
T (t)L2 (Ω) ⊆ C (Ω)
(this is stronger than strong Feller).
Theorem (A.F.M. ter Elst, W. A.)
Then there exist continuous functions kt : Ω × Ω → R such that
Z
(T (t)f )(x) = kt (x, y )f (y ) dy
for all f ∈ L2 (Ω), x ∈ Ω, t > 0.
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Non-local boundary conditions
April 2017
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The Robin Laplacian on C (Ω).
Let β ∈ L∞ (∂Ω) (not necessarily ≥ 0).
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
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Non-local boundary conditions
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The Robin Laplacian on C (Ω).
Let β ∈ L∞ (∂Ω) (not necessarily ≥ 0).
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
Theorem (A.F.M. ter Elst, W. A.)
The operator ∆β generates a positive C0 -semigroup Tβ on C (Ω).
Moreover,
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
22 / total
The Robin Laplacian on C (Ω).
Let β ∈ L∞ (∂Ω) (not necessarily ≥ 0).
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
Theorem (A.F.M. ter Elst, W. A.)
The operator ∆β generates a positive C0 -semigroup Tβ on C (Ω).
Moreover,
(a) Tβ is given by a continuous kernel.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
22 / total
The Robin Laplacian on C (Ω).
Let β ∈ L∞ (∂Ω) (not necessarily ≥ 0).
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
Theorem (A.F.M. ter Elst, W. A.)
The operator ∆β generates a positive C0 -semigroup Tβ on C (Ω).
Moreover,
(a) Tβ is given by a continuous kernel.
(b) Tβ is irreducible. In particular,
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
22 / total
The Robin Laplacian on C (Ω).
Let β ∈ L∞ (∂Ω) (not necessarily ≥ 0).
D(∆β ) := {u ∈ C (Ω) ∩ H 1 (Ω) : ∆u ∈ C (Ω),
∂ν u + βu|∂Ω = 0}
∆β u := ∆u.
Theorem (A.F.M. ter Elst, W. A.)
The operator ∆β generates a positive C0 -semigroup Tβ on C (Ω).
Moreover,
(a) Tβ is given by a continuous kernel.
(b) Tβ is irreducible. In particular,
(c) the first eigenfunction u1 ∈ C (Ω) satisfies minx∈Ω u1 (x) > 0.
Wolfgang Arendt (Ulm University)
Non-local boundary conditions
April 2017
22 / total
Strict positivity.
Irreducible implies in this case:
6 0, then
If f ∈ C (Ω), f ≥ 0 but f =
(T (t)f (x) > 0
Wolfgang Arendt (Ulm University)
for all x ∈ Ω, t > 0.
Non-local boundary conditions
April 2017
23 / total