Post on 10-Jun-2021
Essential Uses of Probability in Analysis
Krzysztof BurdzyUniversity of Washington
Part I. Brownian Couplings and Neumann Eigenfunctions
Hot Spots Conjecture
Rauch (1974) : In Euclidean domains, the second Neumann Laplacianeigenfunction attains its maximum at the boundary.
0 π
Multidimensional domains
CounterexamplesB and Werner (1999)Bass and B (2000)B (2005)
Positive directionKawohl (1985)Bañuelos and B (1999)Pascu (2002) : Conjecture holds forplanar convex domains with a line of symmetry.Atar and B (2004) : Conjecture holdsfor all lip domains.Jerison and Nadirashvili (2000)Atar (2001), …
Symmetric domains
Lip domains
We say that a set is a “lip domain”if it lies between the graphs of twoLipschitz functions with the Lipschitzconstant 1.
Open problems
(i) Hot spots conjecture for convex domains.
(ii) Hot spots conjecture for simply connected planar domains.
(iii) Hot spots conjecture for triangles.
A partial order
yyx ≤x
temperature at at time (Neumannboundary conditions)
Heat equation and reflected Brownian motion
DA
)()(),( AXPXfEtxu tx
tx ∈==
=),( txu
tX)(1)()0,( xxfxu A==
x t
- reflected Brownian motion in D
x
Neumann eigenfunctionsK++= − texctxu λϕ )(),(
)),(),(()()( tyutxueyx t −≈− λϕϕ
))()(()()( AXPAXPeyx ty
txt ∈−∈≈− λϕϕ
))()(()()( AYPAXPeyx ty
txt ∈−∈≈− λϕϕ
Synchronous couplings
xX =0
yY =0
tYXyx tt ∀≤⇒≤
B and Kendall (2000)
tX tY
(monotonicity)
Effect of reflection
tX
tY
tt YX ≤
Eigenfunction monotonicity
x AtY
tX
y
)()( AYPAXP ty
tx ∈≤∈
)()( yx ϕϕ ≤
tYXyx tt ∀≤⇒≤
Mirror couplings (free BM)
x
y
tX
TtYX tt ≥=
tY
TT YX =
T - coupling time
Mirror couplings for reflected BM
x ytX
Wang (1994)B and Kendall (2000) : reflection on a single lineAtar and B (2004) : piecewise smooth domainsB (2005) : simultaneous reflections
tY
hinge
D∂
Symmetric domains
x
Neumann
Dirichlet
Bañuelos and B (1999)Jerison and Nadirashvili (2000)Pascu (2002)
tX
Scale couplings – Pascu (2002)
D
D
- semidisc
Eigenvalue multiplicity
Maximum eigenvalue multiplicity
Nadirashvili (1986, 1988) : The maximum multiplicity of the second Neumann eigenvalue for a simply connected planar domain is 2.
Long convex domainsBañuelos and B (1999) : If is convexand diameter/width is greater than 3.07then the second eigenvalue is simple.
D
D
diameter
width
Convex domains - conjectureBañuelos and B (1999) : (Conjecture)If is convex and diameter/width is greater than 1.41 then the second eigenvalue is simple.
D
width = 1
diameter = 2
Lip domainsAtar and B (2004) : Second Neumanneigenvalue is simple in lip domains(except squares).
Eigenvalue multiplicity- lip domains
Parabolic boundary Harnackprinciple
mirror
Points outside nodal lines
Bañuelos and B (1999) : If for some point the nodal (zero) line for any second eigenfunction does not pass through this point, then the second eigenvalue is simple.
Bottleneck domains
Nodal line
This is not a nodal line
Bottleneck domains (idea of proof)
Neumann
Dirichlet
Nodal line locationOld results
(i) Rectangles, ellipses, etc.(ii) Domains with symmetry
(Melas (1992), Alessandrini (1994) : Dirichlet nodal lines)
(iii) Jerison (2000) Long and thin domains
Nodal lines and couplingsAtar and B (2002) : Consider a coupling of reflected Brownian motions. If the particles cannot couple in a subset of the domain then this subset is too small to contain a nodal domain.Atar and B (in preparation) : If the particles can couple only in a subset of the domain then this subset must contain the nodal set.
Nodal lines and couplings(idea of proof)
Suppose that the coupling point must lie between the green lines.
This might be the nodal line
Obtuse triangles
a a
Possible nodal line
0 10.63
Synchronous couplings viaunique strong solutionsto Skorohod equation
Xt
t
stt dLXNBxX )(0∫++=
Yt
t
stt dLYNByY )(0∫++=
Lions and Sznitman (1984) : - domains 2C
Skorohod equation:unique strong solutions in
Bass, B and Chen (2004) : Unique strong solutions exist in planar Lipschitz domains with Lipschitzconstant less than 1.Bass and B (2005)
Xt
t
stt dLXNBxX )(0∫++=
2R
Higher dimensional domains
We say that is a -domain if its boundary is represented by a function
and
γ,1CD
Φγ|||)()(| yxyx −≤Φ∇−Φ∇
Skorohod equation:unique strong solutions in
Bass and B (in preparation) : (i) Unique strong solutions exist in
-domains if (ii) Counterexample (?) for some
γ,1C 2/1>
Xt
t
stt dLXNBxX )(0∫++=
3, ≥nRn
γ0>γ