Abstract
Talenti inequalities are a central feature in the qualitative analysis of PDE constrained optimal control as well as in calculus of variations. The classical parabolic Talenti inequality states that if we consider the parabolic equation \({\frac{\partial u}{\partial t}}-\Delta u=f=f(t,x)\) then, replacing, for any time t, \(f(t,\cdot )\) with its Schwarz rearrangement \(f^\#(t,\cdot )\) increases the concentration of the solution in the following sense: letting v be the solution of \({\frac{\partial v}{\partial t}}-\Delta v=f^\#\) in the ball, then the solution u is less concentrated than v. This property can be rephrased in terms of the existence of a maximal element for a certain order relationship. It is natural to try and rearrange the source term not only in space but also in time, and thus to investigate the existence of such a maximal element when we rearrange the function with respect to the two variables. In the present paper we prove that this is not possible.
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Notes
One could also observe that \(r_{\varphi }\) solves the differential equation \(dr_{\varphi }/dt=-\partial _tp/\partial _r{\varphi }\). Since \(r_{\varphi }\) is uniformly bounded away from 0, we also get the fact that \(r_{\varphi }\) is \(\mathscr {C}^1\).
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Acknowledgements
The author was partially supported by the French ANR Project ANR-18-CE40-0013 - SHAPO on Shape Optimization and by the Project “Analysis and simulation of optimal shapes - application to life sciences” of the Paris City Hall. The author would like to thank the anonymous referee for his or her numerous suggestions which helped improve the manuscript.
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Mazari, I. A note on the rearrangement of functions in time and on the parabolic Talenti inequality. Ann Univ Ferrara 68, 137–145 (2022). https://doi.org/10.1007/s11565-022-00392-y
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DOI: https://doi.org/10.1007/s11565-022-00392-y