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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 $< \pi/2$.

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    $\begingroup$ 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)". $\endgroup$
    – Jochen Glueck
    Commented 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
    Commented Dec 28, 2019 at 22:46
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    $\begingroup$ Good question... Are you interested in real coefficients only, or also in PDEs with complex coefficients? $\endgroup$
    – Jochen Glueck
    Commented Dec 29, 2019 at 8:56
  • $\begingroup$ Yes I'm interested in real coefficients, but if necessary no problem with complex ones. $\endgroup$
    – Sigma
    Commented Dec 29, 2019 at 9:34
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    $\begingroup$ 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. $\endgroup$
    – Jochen Glueck
    Commented Dec 29, 2019 at 14:22

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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.

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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 $b$.

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