Symmetrisation
The theory of rearrangement and symmetrisation allows us to obtain a priori estimates for different problems. The most basic case is the following. Let $u$ be the solution of \begin{equation} \nonumber \begin{cases} - \Delta u = f & \Omega \newline u = 0 & \partial \Omega \end{cases} \end{equation}
Then the symmetric decreasing rearrangement of $u$, which we call $u^*$, is a solution of $-\Delta u ^ * \le f ^ *$, with the same boundary condition. This allows to use comparison principle with the explicit (radial) solution of $-\Delta v = f ^ *$. This implies, among other things, that $| u | _ {L^p} \le | v | _ {L^p}$. This technique is called Schwarz rearrangent. For a nice introduction see, e.g., [1].
The technique can be extended to partial symmetrisation (only with respect to some coordinates), a method that is sometimes called Steiner symmetrisation. This allows to apply the method, for example, to cylindrical domain $\Omega\prime \times \Omega\prime\prime$ with the solution on any other domain $B \times \Omega\prime\prime$, where $B$ is a ball such that $|B|=|\Omega\prime|$. This had been successfully in the linear setting in [2]. In this direction I published two papers with J. I. Díaz during my PhD on semilinear equations $-\Delta u + g(u) = f$ in [3,4] which extend to the non-linear setting previous results by P.L. Lions (Fields Medal 1994).
Then, jointly with A. Mercaldo, A. Ferone, and F. Brock, we became asked during my viva if this kind of results could be extended to the rearrangement of \begin{equation} \nonumber -\mathrm{div} (a (|\nabla_x u|) \nabla_x u) - \Delta_y u = f \end{equation} where $a$ is of $p$-Laplacian type, and the rearrangement is performed in $x$. We proved a general result with respect to $a$ in [5]. This problem had been open for a number of years, and we solved it by introducing a discretisation technique in the $y$ variable.
We continue to work in this line. Lately, we are interested in the extension of our technique to the non-local setting.
References
- Talenti, Giorgio. (2016) The art of rearranging. Milan J. Math.. Link
- Alvino, Angelo and Trombetti, Guido and Díaz, J I and Lions, P L. (1996) Elliptic equations and Steiner symmetrization. Commun. Pure Appl. Math.. Link
- Gómez-Castro, D. and Díaz, J. I.. (2015) Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. in Dynamical Systems and Differential Equations, Proceedings of the 10th AIMS International Conference (Madrid, Spain). Link
- Díaz, J. I. and Gómez-Castro, D.. (2016) On the Effectiveness of Wastewater Cylindrical Reactors: an Analysis Through Steiner Symmetrization. Pure and Applied Geophysics. Link
- Brock, Friedemann and Díaz, J. I. and Ferone, Adele and Gómez-Castro, D. and Mercaldo, Anna. (2021) Steiner symmetrization for anisotropic quasilinear equations via partial discretization. Annales de l’Institut Henri Poincaré C, Analyse non linéaire. Link