Abstracts

Camilla Brizzi

Abstract In contrast with the classical one, the Optimal Transport problem in \(L^{\infty}\), i.e. the problem of minimizing the essential supremum of the cost function among all the transport plans, is a nonconvex and presumably much harder problem. Due to the success of entropic approximation of the Monge-Kantorovich problem and of Sinkhorn's algorithm, seeking an analogue for the infinity case seems natural in order to get a better understanding.

In my talk I will show the \(\Gamma\) - convergence of the regularized functionals to the one related to the transport problem in \(L^{\infty}\). An interesting result is that every cluster point of the minimizers is a so- called ∞-cyclically monotone transport plan which is for particular cost functions a solution of the Monge problem.

Leon Bungert

Abstract I will characterize the \(\mathrm{L}^\infty\) eigenvalue problem associated to the Rayleigh quotient \(\|\nabla u\|_{\mathrm{L}^\infty}/\|u\|_\infty\) and relate it to a divergence-form PDE, similarly to what is known for \(\mathrm{L}^p\) eigenvalue problems and the p-Laplacian for \(p \lt \infty\). Contrary to existing methods, which study \(\mathrm{L}^\infty\)-problems as limits of \(\mathrm{L}^p\)-problems for \(p\to\infty\), I shall present a novel framework for analyzing the limiting problem directly using convex analysis and measure theory. The eigenvalue problem takes the form \(\lambda \nu u =-\operatorname{div}(\tau\nabla_\tau u)\), where \(\nu\) and \(\tau\) are non-negative measures concentrated where \(|u|\) respectively \(|\nabla u|\) are maximal, and \(\nabla_\tau u\) is the tangential gradient of \(u\) with respect to \(\tau\). The proof relies on a novel fine characterization of the subdifferential of the Lipschitz-constant-functional \(u\mapsto\|\nabla u\|_{\mathrm{L}^\infty}\). I also study a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich–Rubinstein norm.

This is joint work with Yury Korolev and based on the article (https://arxiv.org/abs/2107.12117).

Ed Clark

Abstract We discuss minimisation problems in \(L^\infty\) for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. There are numerous examples of admissible operators, including pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, jacobian and null Lagrangian constraints. Via the method of \(L^p\) approximations as \(p\to \infty\), we conclude the existence of a special \(L^\infty\) minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This talk is based on joint work with Nikos Katzourakis.

Luigi De Pascale

Abstract I will consider the \(L^\infty\)-optimal mass transportation problem \[ \min_{\Pi(\mu, \nu)} \gamma-\mathop{\mathrm{ess\, sup}} c(x,y), \] for a wide class of costs \(c(x,y)\). I will first discuss classical integral optimal transport problems and the relevant notions in that setting, then I will introduce a tentative notion of twist condition for this \(L^\infty\) version. In particular I will analyze the conditions under which the ∞-monotone transport plans are induced by a transportation map.

If time permits also uniqueness will be examined.

From a joint work with Camilla Brizzi and Anna Kausamo.

Ed Gallagher

Abstract In this talk, based on the preprint of the same name (https://arxiv.org/abs/2202.07407), we consider the problem of minimising the \(L^\infty\) norm of the curvature of curves, satisfying fixed boundary conditions and a length constraint, which live on a given complete Riemannian manifold \((M,g)\). This builds on a paper by Moser considering the same problem in Euclidean space \(\mathbb{R}^n\). Using the method of \(L^p\) approximation we show that solutions to our problem, as well as a more general class of curves called ∞-elastica, must satisfy a second order ODE system derived as the limit as \(p \rightarrow \infty\) of the Euler-Lagrange equations for the \(L^p\) approximations. This system gives us some geometric information about ∞-elastica and in particular implies that their curvature takes on at most two values.

Maria Stella Gelli

Abstract I will present some work in progress regarding the notion of supremal functionals on metric measure spaces. I will treat more in details the case of euclidean spaces, introducing the notion of \(W^{1,\infty}\) with respect a given measure and showing the consistency of this approach with the relaxation of supremal energies on smooth functions. This general framework allows to deal with variational problems whose reference domains are non-regular subsets of \(\mathbb{R}^N\).

Anna Kausamo

Abstract c-cyclical monotonicity is the most important optimality condition for a transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. In this talk I will present a new approach developed in collaboration with Luigi De Pascale. I will show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

Juan Manfredi

Abstract We study the regularity of viscosity solutions to equations governed by the ∞-Laplace operator relative to a frame of vector fields: \[ \sum_{i,j=1}^n X_i X_j u(x) X_iu(x) X_ju(x) = f (x, u(x), X_1u(x), \ldots X_n u(x)), \] where \(f\) is a real valued continuous functions and \(X_1, X_2, \ldots X_n\) are linearly independent smooth vector fields in \(\mathbb{R}^n\). We proved Lipschitz and Hölder estimates in the intrinsic Riemannian metric defined by the vector fields.
We also consider the case of the Heisenberg group.

This is joint work with Fausto Ferrari (Bologna).

Roger Moser

Abstract We consider infinity-harmonic functions, and in particular their streamlines, in a two-dimensional domain. If an infinity-harmonic function \(u\) is smooth with non-vanishing gradient, then we may consider the function \(f = -\log |\nabla u|\). It is well-known that the level sets of \(f\) correspond to the streamlines of \(u\). Less well-known is the fact that they solve a geometric evolution problem known as the inverse mean curvature flow.

In general, the above regularity assumptions are not satisfied. We explore to what extent the connection between the two problems persists.

Fa Peng

Abstract In the seminal paper[ARAM, 2005], Savin proved that if the gradient of the infinity harmonic function \(u\) is bounded in the plane, then \(u\) must be a linear function. His proof heavily relies on the \(C^1\)-regularity of the planar infinity harmonic function \(u\). Here we give a new and simple proof for this result. Our method is based on the \(W^{1,2}_{\mathrm{loc}}\)-regularity of \(|Du|\) and reversed Poincare type inequality for planar infinity harmonic function. This is based on joint works with Hongjie Dong, Yi Ru-Ya Zhang and Yuan Zhou.

Francesca Prinari

Abstract In this talk we give some results concerning the optimisation problems for the scaling free functional \[ F_{p,\beta}(\Omega)=\frac{\lambda_p(\Omega)T_p^\beta(\Omega)}{|\Omega|^{\alpha(p,\beta,N)}} \] and the limit shape functional \[ F_{\infty,\beta}(\Omega)= \lim_{p\to \infty}F^{1/p}_{p,\beta}(\Omega) \] over the class of open sets \(\Omega\subset\mathbb{R}^N\) with \(0 \lt |\Omega| \lt \infty\) and in that of bounded convex open sets. Here \(1 \lt p \lt \infty\), \(\beta \gt 0\), \(\lambda_p(\Omega)\) is the principal eigenvalue relative to the p-Laplace operator, \(T_p(\Omega)\) is the p-torsional rigidity and \(\alpha(p,\beta,N):=\beta(p-1)+\frac{p(\beta-1)}{N}\). The study of the functionals \(F_{p,\beta}\) has been already considered in the literature in case when \(p=2, \beta=1\). The last part of the talk is devoted to the minimisation and maximisation problems of the functional \[ J_{p,\beta}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)}. \]

Anna Chiara Zagati

Abstract We introduce the generalized principal frequencies and their connection with the Lane-Emden equation for the p−Laplacian. Then we prove, throughout a comparison principle, a variety of results, as the uniqueness of solutions under minimal assumptions on the set, sharp pointwise two-sided estimates for positive solutions in convex sets, a hierarchy of sign-changing solutions of the equation and a sharp geometric estimate on the generalized principal frequencies of convex sets. The results are obtained in collaboration with Lorenzo Brasco (Ferrara) and Francesca Prinari (Pisa).

Elvira Zappale

Abstract I will present some results, obtained in collaboration with Carolin Kreisbeck and Antonella Ritorto, devoted to detect suitable convexity notions ensuring lower semicontinuity of non local supremal functionals.

Yuan Zhou

Abstract In plane, by adapting Savin's approach for \(C^1\)-regularity of ∞-harmonic functions, for convex Hamiltonian \(H\) whose level set \(H^{-1}(c)\) does not include any line-segment, we prove the \(C^1\)-regularity of absolute minimizers. Recall that Katzourakis' example shows that the condition \(H^{-1}(c)\) does not include any line-segment is necessary to get such \(C^1\)-regularity.