Gamma-star-shapedness for semilinear elliptic equations
Antonio Greco Marcello Lucia
A generic semilinear equation in a star-shaped ring is considered. Any solution bounded between its boundary values is shown to be decreasing along rays starting from the origin, provided that a structural condition is satisfied. A corresponding property for the product between the solution and a (positive) power of $|x|$ is also investigated. Applications to the Emden-Fowler and the Liouville equation are developed.
keywords: Comparison principle Star-shapedness Monotone iteration scheme.
Optimization problems for the energy integral of p-Laplace equations
Antonio Greco Giovanni Porru
We study maximization and minimization problems for the energy integral of a sub-linear $p$-Laplace equation in a domain $\Omega$, with weight $\chi_D$, where $D\subset\Omega$ is a variable subset with a fixed measure $\alpha$. We prove Lipschitz continuity for the energy integral of a maximizer and differentiability for the energy integral of the minimizer with respect to $\alpha$.
keywords: optimization regularity. Energy integral rearrangements
Existence and convexity of solutions of the fractional heat equation
Antonio Greco Antonio Iannizzotto

We prove that the initial-value problem for the fractional heat equation admits an entire solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. The result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al.[1] for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established.

keywords: Heat equation fractional Laplacian convexity

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