Nonlocal operators and partial differential equations
Bȩdlewo, June 27th –July 3th , 2010
Tadeusz Kulczycki
Polish Academy of Sciences and
Wrocław University of Technology
[email protected]
Spectral theory for square
root of Laplacian
We study the following eigenvalue problem for (−∆)1/2 with Dirichlet outer condition on a bounded Lipschitz domain D ⊂ Rd
(−∆)1/2 ϕn (x) = λn ϕn (x),
x ∈ D,
(1)
ϕn (x)
= 0,
x ∈ Dc .
where (−∆)1/2 is the pseudodifferential operator given by
Z
f (x) − f (y)
(−∆)1/2 f (x) = Cd lim
dy.
|x − y|d+1
ε→0+
|y|>ε
It is known that there exists an orthonormal basis in L2 (D) consisting of eigen∞
functions {ϕn }∞
n=1 and a sequence of eigenvalues {λn }n=1 : 0 < λ1 < λ2 ≤ λ3 ≤
. . . → ∞. We establish a connection between this spectral problem and a boundary value problem for the Laplacian in one dimension higher, known as the Steklov
problem. The connection with the Steklov problem allows us to use new methods.
Some of these methods were developed in hydrodynamics where a similar Steklov
problem is considered. Using these methods we obtain many detailed properties of
the eigenvalues and eigenfunctions for the spectral problem (1) inspired by those
for the Dirichlet Laplacian.
In particular we obtain spectral gap estimates for some bounded convex planar
domains.
Our results are new even in the simplest geometric setting of the interval D =
(−1, 1) where we obtain more precise information on eigenvalues and eigenfunctions. In the case when D = (−1, 1) we show that the first eigenfunction ϕ1 is
convex, all eigenvalues λn are simple and
nπ π
λn =
− +O
2
8
1
.
n
Such results, although trivial for the spectral problem for the Dirichlet Laplacian
on the interval take considerable work to prove for the spectral problem (1).
We also study the following eigenvalue problem for (−∆)1/2 with Dirichlet outer
condition on the halfline (0, ∞)
(−∆)1/2 ϕλ (x) = λϕλ (x),
x ∈ (0, ∞),
(2)
ϕλ (x)
= 0,
x ∈ (−∞, 0].
For the eigenvalue problem (2) we compute explicit formulas for generalized eigenfunctions ϕλ :
√ Z ∞
Z
π
2
t
1 ∞ log(t + s)
ϕλ (x) = sin(λx + ) −
exp −
ds e−tλx dt.
8
2π 0 1 + t2
π 0
1 + s2
This allows us to obtain an explicit formula for the density p(0,∞) (t, x, y) of the
1-dimensional Cauchy process (which infinitesimal generator is −(−∆)1/2 ) killed
on exiting (0, ∞). In the analytic language p(0,∞) (t, x, y) is the heat kernel of
−(−∆)1/2 on (0, ∞). Namely we obtain
Z t f w f t−w
x
y
1
t
1
p(0,∞) (t, x, y) =
−
dw,
w
t−w
2
2
π t + (x − y)
xy 0
x + y
where
1 w
exp
f (w) =
π 1 + w2
Z ∞
1
log(w + s)
.
π 0
1 + s2
The talk is based on my papers with R. Bañuelos [1], [2], [3] and my recent work
[4] with M. Kwaśnicki, J. Małecki and A. Stos.
References
[1] R. Bañuelos, T. Kulczycki The Cauchy process and the Steklov problem, J.
Funct. Anal. 211 (2004), 355-423.
[2] R. Bañuelos, T. Kulczycki Eigenvalue gaps for the Cauchy process and a Poincare
inequality, J. Funct. Anal. 234 (2006), 199–225.
[3] R. Bañuelos, T. Kulczycki Spectral gap for the Cauchy process on convex, symmetric domains, Comm. in Partial Diff. Equations 31 (2006), 1841–1878.
[4] T. Kulczycki, M. Kwaśnicki, J. Małecki, A. Stos Spectral properties of the Cauchy
process on half-line and interval, Proc. Lond. Math. Soc. (2010), to appear.