The constrained equilibrium problem for the logarithmic potential was introduced by Rakhmanov (1996) as he realized that the asymptotic distribution of the zeros of discrete orthogonal polynomials in a compact interval could be described in terms of the equilibrium measure of this interval in a class of measures subject to a certain constraint. Other authors such as Dragnev and Saff (1997), and Kuijlaars and Van Assche (1999), extended this approach to more general settings. These problems have proven to be useful for describing asymptotic distributions in different settings. In the current paper, we consider constrained equilibrium problems in the Riesz setting, that is, for $s$-Riesz potentials in the hyperplane $\mathbb{R}^d,$ with $d\geq 1$ and $\max (0,d-2)<s<d$. Along with some general results, an illustrative example consisting of the solution of a constrained equilibrium problem in the unit ball is studied in detail. This is the main part of the paper. Finally, a number of numerical experiments show how the constrained equilibrium measure, the solution of the problem, may be discretized using the so-called'constrained Leja points'introduced by Coroian and Dragnev (2001).
In the present paper, we study the maximization problem of the $k$-Torsional rigidity under quermassintegral constraint, with particular emphasis on the role of the ball. Our main result shows a phenomenon which, as far as we know, has not previously been observed: the ball, rather than being an extremal shape, is a sa...
A. Masiello, Gloria Paoli, Francesco Salerno· 0 citations
We establish gradient potential estimates for solutions obtained as limits of approximations (SOLA) to the fractional $p$-Laplace equation with finite signed Radon measure data. Under the assumptions $n\ge2$, $p>2$, $0p-1$, every SOLA belongs to $W^{1,p-1}_{\mathrm{loc}}$, and its weak gradient satisfies a Wolff potent...
We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description...
Fernando Charro, M. del Mar González, María Soria-Carro· 0 citations
A classical idea in analysis going back at least to work of Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder type regularity estimates. The goal of this course is to describe some recent developments of this idea concerning the regularity of the complex Mong...
Yifan Chen, Hans-Joachim Hein, Ryan Mc Gowan· 0 citations
The huge amount of literature about the divergence equation in bounded Lipschitz domains of $\R^n$ ($n\ge2$) is fairly disconnected and even apparently simple problems remain unsolved. We go several steps further in the knowledge of this equation and of some related inequalities. We prove that among its infinitely many...
Filippo Gazzola, H. Grunau, Gianmarco Sperone· 2 citations· ⚡1
We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpr...
