Global estimates for a first-order operator on the line #
For symmetric coefficients, positivity of the potential A(s)² - A'(s) outside a
compact interval gives the global estimate
∫ ‖u'‖² + γ ∫ ‖u‖² ≤ ∫ ‖u' + A u‖² + C ∫_{[-R,R]} ‖u‖².
The constants are independent of the compactly supported C¹ path u, including
paths crossing the entire line. For finite-dimensional targets, the interior term
is the error supplied by compact restriction from W^{1,2} to L² on a bounded
interval in Peetre's Fredholm criterion.
We obtain the tail positivity from uniformly bounded-below limits of A and
vanishing limits of A'. The target need not be finite dimensional or complete.
The support-restricted coercivity estimate follows by making the interior error vanish.
Without symmetry or a derivative of A, the same kind of estimate holds when A is continuous
and its limits A± at ±∞ have Lyapunov forms (ContinuousLinearMap.HasLyapunovForm):
∫ ‖u'‖² + ∫ ‖u‖² ≤ C ∫ ‖u' + A u‖² + C ∫_{[-R,R]} ‖u‖²
(exists_pos_nonneg_forall_integral_norm_deriv_sq_add_integral_norm_sq_le_of_hasLyapunovForm).
Its proof integrates the quadratic form of a C¹ family of symmetric operators interpolating
between Lyapunov forms of A- and A+ (TauCeti.half_mul_integral_norm_sq_le).
These are estimates on classical test paths. Their extension to Sobolev paths and the Fredholm
conclusion are in TauCeti.Analysis.ODE.EnergyEstimate.Sobolev and
TauCeti.Analysis.ODE.EnergyEstimate.Lyapunov.
References #
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., Appendix A.1.
- M. Schwarz, Morse Homology, Chapter 2 (linearized trajectory operators).
Positivity of A² - A' outside a compact interval gives a global energy
estimate with an interior L² error. No sign restriction on γ is needed.
Global coercivity with an interior error. Symmetric coefficients with a
common positive lower bound at both limits and derivative tending to zero satisfy
a uniform estimate for every compactly supported C¹ path on the whole line.
Coercivity at infinity. If the symmetric coefficients tend in operator norm to operators
with a common positive lower bound, and their derivative tends to zero at both ends, then
d/ds + A(s) controls the L² value and derivative of every compactly supported C¹ path
supported outside one fixed compact interval. In finite dimension, invertible limits supply the
required lower bound.
Hyperbolic limits #
The trajectory estimate for hyperbolic limits on test paths. Let A be a continuous
family of operators whose limits A± at ±∞ have Lyapunov forms. Then there are R > 0 and
C ≥ 0 with
∫ ‖u'‖² + ∫ ‖u‖² ≤ C ∫ ‖u' + A u‖² + C ∫_{[-R, R]} ‖u‖²
for every compactly supported C¹ path u.