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In models of $N$ interacting particles in $\R^d$ as in Density Functional Theory or crowd motion, the repulsive cost is usually described by a two-point function $c_\e(x,y) =\ell\Big(\frac{|x-y|}{\e}\Big)$ where $\ell: \R_+ \to [0,\infty]$ is decreasing to zero at infinity and parameter $\e>0$ scales the interaction distance. In this paper we identify the mean-field energy of such a model in the short-range regime $\e\ll 1$ under the sole assumption that $\exists r_0>0 \ : \ \int_{r_0}^\infty \ell(r) r^{d-1}\, dr <+\infty$. This extends recent results \cite{hardin2021, HardSerfLebl, Lewin} obtained in the homogeneous case $\ell(r) = r^{-s}$ where $s>d$.

It is known that the torsional rigidity for a punctured ball, with the puncture having the shape of a ball, is minimum when the balls are concentric and the first eigenvalue for the Dirichlet Laplacian for such domains is also a maximum in this case. These results have been obtained by Ashbaugh and Chatelain (private communication), Harrell et. al., by Kesavan and, by Ramm and Shivakumar. In this paper we extend these results to the case of $p$-Laplacian for $1 < p < \infty$. For proving these results, we follow the same line of ideas as in the aforementioned articles, namely, study the sign of the shape derivative using the moving plane method and comparison principles. In the process, we obtain some interesting new side results such as the Hadamard perturbation formula for the torsional rigidity functional for the Dirichlet $p$-Laplacian, the existence and uniqueness result for a nonlinear pde and some extensions of known comparison results for nonlinear pdes.

The aim of this article is to study the asymptotic behaviour of some low-cost control problems. These problems motivate the study of H-convergence with weakly convergingdata. An improved lower bound for the limit of energy functionals correspondingto weak data is established, in the periodic case. This fact is used to prove the Gamma-convergence of a low-cost problem with Dirichlet-type integral. Finally, we study the asymptotic behaviour of a low-cost problem with controls converging to measures.

In this note we will present an extension of the Krein-Rutman theorem for an abstract nonlinear, compact, positively 1-homogeneous, monotone non-decreasing operators on a Banach space and apply the result to many nonlinear elliptic partial differential operators.