4
$\begingroup$

Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ?

For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. I'm wondering if there is a known example where the angle is e.g., $\pi/4$ or other value $< \pi/2$.

$\endgroup$
5
  • 1
    $\begingroup$ Yes, such examples are known. For some references, see for instance Secion 7.2.6 in "W. Arendt: Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates (2004)". $\endgroup$ Dec 28, 2019 at 21:00
  • $\begingroup$ @JochenGlueck Thank you. The example given is about the realization of the semigroup in $L^p$. Do you know any example in Hilbert space, say $L^2$ for example? $\endgroup$
    – Sigma
    Dec 28, 2019 at 22:46
  • 1
    $\begingroup$ Good question... Are you interested in real coefficients only, or also in PDEs with complex coefficients? $\endgroup$ Dec 29, 2019 at 8:56
  • $\begingroup$ Yes I'm interested in real coefficients, but if necessary no problem with complex ones. $\endgroup$
    – Sigma
    Dec 29, 2019 at 9:34
  • 1
    $\begingroup$ With complex coefficients it's quite easy: just consider the PDE $\dot u = e^{i\frac{\pi}{4}} \frac{\partial^2}{\partial x^2} u$ on, say, $L^2(\mathbb{R})$ (though one might argue whether this is really a "parabolic" PDE). I don't know an example with real coefficients at the moment, but I would suspect that such examples exist. $\endgroup$ Dec 29, 2019 at 14:22

2 Answers 2

6
$\begingroup$

Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $\Delta+Bx \cdot \nabla$, where $B$ is a matrix. Assuming that all eigenvalues of $B$ have negative real parts, then an invariant measure $\mu$ exists (and is given by a Gaussian density). It turns out that the angle of analiticity in $L^p$ of the invariant measure can be computed exactly and can be smaller than $\pi/2$, even for $p=2$. This can be found in a paper by Chill, Fasangova, Pallara and myself.

Chill, R.; Fašangová, E.; Metafune, G.; Pallara, D., The sector of analyticity of the Ornstein-Uhlenbeck semigroup on (L^p) spaces with respect to invariant measure, J. Lond. Math. Soc., II. Ser. 71, No. 3, 703-722 (2005). ZBL1123.35030.

To obtain similar examples in unweighted spaces one has only (but patiently) to compensate the weight thus obtaining an operator with a linear drift and a quadratic potential. This works however only from dimension $2$ on; in the one dimensional case the operator is always self-adjont but still the angle of analitycity is different from $\pi/2$ in $L^p$, for $p$ different from $2$. There is also a paper by E. Priola and myself dealing with non-analytic Markov semigroups where one finds other examples.

Metafune, G.; Priola, E., Some classes of non-analytic Markov semigroups, J. Math. Anal. Appl. 294, No. 2, 596-613 (2004). ZBL1067.47055.

It would be nice to have a direct approach, avoiding the detour.

$\endgroup$
0
3
$\begingroup$

There are examples but you need singularities or unbounded coefficients; the uniformly parabolic case with regular coefficients is indeed a perturbation of the laplacian. In 1D, if you perturb the harmonic oscillator $D^2-x^2$ by a linear drift $bxD$, the angle of analyticity depends on $b$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.