The Second joint IMU-INdAM Conference in Analysis

The Second joint IMU-INdAM Conference in Analysis
September 16-20, 2019
Napoli

Abstracts

Special Hermitian metrics on complex manifolds
Daniele Angella, Umiversità di Firenze
In the tentative to move from the Kähler to the non-Kähler setting, we consider several problems concerning Hermitian metrics on complex manifolds with special curvature properties that can be translated and attacked as analytic pdes. In this context, we study an analogue of the Yamabe problem for Hermitian manifolds: more precisely, we prove the existence of Hermitian metrics having constant scalar curvature with respect to the Chern connection when the expected curvature is non-positive, and we point out the difficulties in the positive curvature case. This problem relates also to several notions of Chern-Einstein metrics. The plural here refers to the lack of symmetries of the curvature tensor of canonical connections of Hermitian manifolds. To the broad subject, the Chern-Ricci flow plays an useful role: the problem of uniform convergence of the normalized Chern-Ricci flow can be dealt with on Inoue-Bombieri surfaces with Gauduchon metrics.
The talk is based on joint collaborations with: Simone Calamai, Antonio Otal, Cristiano Spotti, Valentino Tosatti, Luis Ugarte, Raquel Villacampa. On the Sobolev quotient in sub-Riemannian geometry
Andrea Malchiodi, Scuola Normale Superiore di Pisa
We consider a class of three-dimensional CR manifoldswhich are modelled on the Heisenberg group. We introduce a natural concept of massand prove its positivity under the conditions that the Webster curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to the CR Yamabe problem, and we discuss the properties of Sobolev-type quotients, giving some counterexamples for Rossi spheres.
This is joint work with J.H.Cheng and P.Yang.
Bi-parameter potential theory
Nicola Arcozzi, Università di Bologna
Abstract. We develop the foundations of a potential theory for product kernels, for which we prove Capacitary Strong Inequality, Trace Inequalities and Wolff's Inequality. The main novelty comes from the fact that we work with kernels which are not reciprocal of distances, hence they fail to satisfy basic properties as the Maximum Principle for potentials of measures. Open problems will be discussed.
Work in collaboration with P. Mozolyako, K.M. Perfekt, G. Sarfatti. Multiplicity of Eigenvalues for the circular clamped plate problem
Dan Mangoubi, Hebrew University
A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. More precisely, Siegel shows that positive zeros of Bessel functions are transcendental. We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is uniformly bounded (by not more than six). Our method is based on new recursion formulas and Siegel-Shidlovskii theory. The talk is based on a joint work with Yuri Lvovsky.

An optimal transport viewpoint on Density Functional Theory: the strictly correlated regime
Simone Di Marino Ricercatore Indam, Scuola Normale Superiore
In the recent years P. Gori-Giorgi, M. Seidl and coauthors studied the so-called strictly correlated limit of DFT. They discovered in fact that this limit can be seen as a multimarginal optimal transport problem; this has been proven rigorously by Friesecke, Cotar and Kluppelberg. In this talk we will prove several properties of the limit functional such as duality, continuity, finiteness, with the sharpest assumptions. if time will permint we will talk also about the SGS conjectures, which may be regarded as the generalization of the classical Monge ansatz in this multimarginal case.
This is based on joint works with L. Nenna, A. Gerolin, P. Gori-Giorgi, M. Seidl, M. Colombo and F. Stra. Schrödinger equations with Hardy type potential and related non-linear problems
Moshe Marcus, Technion--I.I.T
Let $L_V=\Delta+ V$ where $V\in C^1(\Omega)$ is a positive potential of Hardy type, e.g. $\mu/ \delta_F^2$ where $\delta_F(x)$ is the distance of $x$ to the compact set $F \subset \partial\Omega$ and $\mu$ < $C_H(V)$ (= Hardy constant relative to $V$). We present estimates of the Green potential of a measure $\tau$ in $\Gw$ and of sub and super solutions of $L_V u=0$. We also dicuss the questions of existence }
Continuous and holomorphic semicocycles in Banach spaces
Mark Elin, ORT Braude College
Semicocycles play an important role in the study of non-autonomous dynamical systems. In this talk, we consider semicocycles whose elements are either continuous or holomorphic mappings on a domain in a real/complex Banach space and take values in a unital Banach algebra. We study some fundamental properties of semicocycles employing, in particular, their link with semigroups of non-linear mappings. On the other hand, we discover a dissimilarity of these two classes. One of our main aims is to establish conditions for differentiability of a semicocycle with respect the time parameter. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator. Simplest semicocycles are those independent of the spatial variable. So, another problem we consider is to establish whether a semicocycle is cohomologous to such independent one (i.e., is linearizable). Focusing on this problem, we provide some criteria for a holomorphic semicocycle to be linearizable as well as several easily verifiable sufficient conditions. These conditions are essential even for semicocycles over semigroups of linear operators.
The talk is based on joint work with F. Jacobzon and G. Katriel. The topology of constant mean curvature surfaces with convex boundary
Barbara Nelli, Università di L'Aquila
What is the topology of a constant mean curvature surface with boundary a convex curve in the plane? We give an overview of what is known about this problem in space forms. Moreover, adding some simple hypothesis, we answer to the previous question when the ambient manifold is ${\mathbb H}^2\times{\mathbb R}$. Here ${\mathbb H}^2$ is the hyperbolic plane.
This is a joint work with V. Moraru.
A representation formula fo the anisotropic total variation of a function in SBV
Fernando Farroni, Università degli studi di Napoli Federico II
Inspired by some new function space introduced by  J. Bourgain, H. Brezis and P. Mironescu, we provide the relation between certain BMO?type seminorms and the total variation of SBV functions in the anisotropic framework. Our result is motivated by the quest of a new characterization of the anisotropic perimeter which is independent of the theory of distributions.
This is a joint work with Nicola Fusco, Serena Guarino Lo Bianco and Roberta Schiattarella. Intrinsic sub-laplacians and function spaces: embeddings, algebra properties and applications
Marco Peloso, Università di Milano
In this talk we consider a class of natural sub-elliptic operators on a general Lie group. We study the Sobolev, Besov and spaces defined by such operators, prove embedding theorems and algebra properties.  We also provide some applications to nonlinear PDEs, obtaining existence and well-posedness.  These operators are a model for the intrinsic sub-Laplacian on a sub-Riemannian manifold of exponential growth.
Stability of the Riesz Potential Inequality
Nicola Fusco, Università degli studi di Napoli Federico II
Given a measurable set $E$ we consider the Riesz Potential
$$ \mathcal F(E)=\int_E\int_E{1 \over |x-y|^{n-\alpha}} \,dxdy\, $$ where $0$ < $\alpha$ < $n$. It is well known that the maximum of $\mathcal F(E)$ among all sets of the same volume of $E$ is achieved at a ball $B$, i.e. $$ \mathcal F(E)\leq\mathcal F(B)\,\,\,\,|E|=|B|. $$ Moreover, equality holds if and only if $E$ is a ball. We shall discuss the stability of this inequality.
This is a joint work with Aldo Pratelli. Minimizers of a variational problem for nematic liquid crystals with variable degree of orientation in two dimensions
Itai Shafrir, Technion--I.I.T
We study the asymptotic behavior, when $k\to\infty$, of the minimizers of the energy \begin{equation*} G_k(u)=\int_{\Omega}\Big((k-1)|\nabla|u||^2+|\nabla u|^2\Big)\,, \end{equation*} over the class of maps $u\in H^1(\Omega,{\mathbb R}^2)$ satisfying the boundary condition $u=g$ on $\partial\Omega$, where $\Omega$ is a smooth, bounded and simply connected domain in ${\mathbb R}^2$ and $g:\partial\Omega\to S^1$. The motivation comes from a simplified version of Ericksen model for nematic liquid crystals. We will present similarities and differences with respect to the analog problem for the Ginzburg-Landau energy.
This is a joint work with Dmitry Golovaty.
Mappings with integrally controlled moduli
Anatoly Golberg, Holon Institute of Technology
In the talk we discuss the differential properties of multidimensional homeomorphic/open discrete mappings. Such mappings ssentially generalize the well-known customarily investigated classes of mappings as quasiregular, quasiisometric, Lipschitzian, etc. But in contrast to these known classes, the definition of our mapping class does not involve any analytic restrictions. We also illustrate the regularity properties by several examples and present a collection of open related problems.
The talk is based on joint works with R. Salimov (Institute of Mathematics, Kyiv, Ukraine). Nonlinear resolvent of holomorphic generators
David Shoikhet, Holon Institute of Technology
Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t>0}$ of holomorphic self-mappings of the open unit disk, i.e., $f=\lim_{t\rightarrow 0}\frac{1}{t}\left( I-F_{t}\right) .$ In this work, we study properties of the resolvent family $R=\left\{ \left( I+rf\right) ^{-1}\right\} _{r>0}$ in the spirit of geometric function theory. We have discovered, in particular, that $R$ forms an inverse Loewner chain and consists of starlike functions of order $\alpha >1/2$. Moreover, each element of $R$ satisfies the Noshiro-Warshawskii condition $\left( \func{Re}\left( I+rf\right) ^{-1}\left( z\right) >0\right).$ This, in turn, implies that all elements of $R$ are also holomorphic generators. Finally, we study the existence of repelling fixed points of this family.
This talk is based on joint work with Mark Elin and Toshiyuki Sugawa.
Linear representations of random groups
Gady Kozma, Weizmann Institute of Science
We will survey Gromov random groups and their fascinating phase transitions, with particular emphesis on results that pertain to the sharpness of the transition at density 1/2.
Joint work with Alex Lubotzky. Isoperimetric inequalities for Steklov-Laplacian eigenvalues
Cristina Trombetti, Università di Napoli Federico II
Abstract
Boundaries, conformal maps, and sub-Riemannian geometry
Enrico Le Donne, Università Pisa and University of Jyväskylä
In this talk there will be a mix of complex analysis, metric geometry, and differential geometry. The objective is to give a new point of view for the validity of Fefferman's mapping theorem from 1974. This result states that a biholomorphism between two smoothly bounded strictly pseudoconvex domains in C^n extends as a smooth diffeomorphism between their closures. Following ideas from Gromov, Mostow, and Pansu, we discuss a method of proof in the context of quasi-conformal geometry. In particular, we show that every isometry between smoothly bounded strictly pseudoconvex domains is 1-quasi-conformal with respect to the sub-Riemannian distance defined by the Levi form on the boundaries. Subsequently, a PDE argument shows that such maps are smooth.
This method was proposed by M. Cowling, and it has been implemented in collaboration with L. Capogna, G. Citti, and A. Ottazzi. On holomorphic motions
Genadi Levin, Hebrew University of Jerusalem
Holomorphic motion is a holomorphic family of injections of a subset of the plane. This notion was originated in holomorphic dynamics and turned out to be very useful in complex analysis and applications, with tight connections to quasi-conformal mappings. In the talk, I discuss some problems related to holomorphic motions which are also originated in holomorphic dynamics.
Based on joint work with Weixiao Shen and Sebastian van Strien.
Generating functionals on quantum groups
Ami Viselter, University of Haifa
We will discuss generating functionals on locally compact quantum groups. One type of examples comes from probability: the family of distributions of a Lévy process forms a convolution semigroup, which in turn admits a natural generating functional. Another type of examples comes from (locally compact) group theory, involving semigroups of positive-definite functions and conditionally negative-definite functions, which provide important information about the group's geometry. We will explain how these notions are related and how all this extends to the quantum world; see how generating functionals may be (re)constructed and study their domains; and indicate how our results can be used to study cocycles.
Based on joint work with Adam Skalski. Is the Gilbert-Steiner problem convex?
Gershon Wolansky, Technion
I will introduce the notion of a conditional Wasserstein metric on the set of probability measures, and apply it to the Gilbert-Steiner problem of minimal graphs. A convexification of the finite Gilbert problem for optimal networks is introduced. It is a convex functional on the set of probability measures subject to the Wasserstein $p-$ metric. The minimizer of this convex functional is a one dimensional current measure supported in a directed graph. If this graph is a tree (i.e. contains no cycles) then this tree is also a minimum of the corresponding Gilbert problem. The convexification of the Steiner problem is the limit of these convexified Gilbert's problems for p=infinity. I will discuss a numerical algorithm for the implementation of the convexified Gilbert-mailing problem is also suggested, based on entropic regularization.

Special Hermitian metrics on complex manifolds Daniele Angella, Umiversità di Firenze In the tentative to move from the Kähler to the non-Kähler setting, we consider several problems concerning Hermitian metrics on complex manifolds with special curvature properties that can be translated and attacked as analytic pdes. In this context, we study an analogue of the Yamabe problem for Hermitian manifolds: more precisely, we prove the existence of Hermitian metrics having constant scalar curvature with respect to the Chern connection when the expected curvature is non-positive, and we point out the difficulties in the positive curvature case. This problem relates also to several notions of Chern-Einstein metrics. The plural here refers to the lack of symmetries of the curvature tensor of canonical connections of Hermitian manifolds. To the broad subject, the Chern-Ricci flow plays an useful role: the problem of uniform convergence of the normalized Chern-Ricci flow can be dealt with on Inoue-Bombieri surfaces with Gauduchon metrics. The talk is based on joint collaborations with: Simone Calamai, Antonio Otal, Cristiano Spotti, Valentino Tosatti, Luis Ugarte, Raquel Villacampa.	On the Sobolev quotient in sub-Riemannian geometry Andrea Malchiodi, Scuola Normale Superiore di Pisa We consider a class of three-dimensional CR manifoldswhich are modelled on the Heisenberg group. We introduce a natural concept of massand prove its positivity under the conditions that the Webster curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to the CR Yamabe problem, and we discuss the properties of Sobolev-type quotients, giving some counterexamples for Rossi spheres. This is joint work with J.H.Cheng and P.Yang.
Bi-parameter potential theory Nicola Arcozzi, Università di Bologna Abstract. We develop the foundations of a potential theory for product kernels, for which we prove Capacitary Strong Inequality, Trace Inequalities and Wolff's Inequality. The main novelty comes from the fact that we work with kernels which are not reciprocal of distances, hence they fail to satisfy basic properties as the Maximum Principle for potentials of measures. Open problems will be discussed. Work in collaboration with P. Mozolyako, K.M. Perfekt, G. Sarfatti.	Multiplicity of Eigenvalues for the circular clamped plate problem Dan Mangoubi, Hebrew University A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. More precisely, Siegel shows that positive zeros of Bessel functions are transcendental. We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is uniformly bounded (by not more than six). Our method is based on new recursion formulas and Siegel-Shidlovskii theory. The talk is based on a joint work with Yuri Lvovsky.
An optimal transport viewpoint on Density Functional Theory: the strictly correlated regime Simone Di Marino Ricercatore Indam, Scuola Normale Superiore In the recent years P. Gori-Giorgi, M. Seidl and coauthors studied the so-called strictly correlated limit of DFT. They discovered in fact that this limit can be seen as a multimarginal optimal transport problem; this has been proven rigorously by Friesecke, Cotar and Kluppelberg. In this talk we will prove several properties of the limit functional such as duality, continuity, finiteness, with the sharpest assumptions. if time will permint we will talk also about the SGS conjectures, which may be regarded as the generalization of the classical Monge ansatz in this multimarginal case. This is based on joint works with L. Nenna, A. Gerolin, P. Gori-Giorgi, M. Seidl, M. Colombo and F. Stra.	Schrödinger equations with Hardy type potential and related non-linear problems Moshe Marcus, Technion--I.I.T Let $L_V=\Delta+ V$ where $V\in C^1(\Omega)$ is a positive potential of Hardy type, e.g. $\mu/ \delta_F^2$ where $\delta_F(x)$ is the distance of $x$ to the compact set $F \subset \partial\Omega$ and $\mu$ < $C_H(V)$ (= Hardy constant relative to $V$). We present estimates of the Green potential of a measure $\tau$ in $\Gw$ and of sub and super solutions of $L_V u=0$. We also dicuss the questions of existence }
Continuous and holomorphic semicocycles in Banach spaces Mark Elin, ORT Braude College Semicocycles play an important role in the study of non-autonomous dynamical systems. In this talk, we consider semicocycles whose elements are either continuous or holomorphic mappings on a domain in a real/complex Banach space and take values in a unital Banach algebra. We study some fundamental properties of semicocycles employing, in particular, their link with semigroups of non-linear mappings. On the other hand, we discover a dissimilarity of these two classes. One of our main aims is to establish conditions for differentiability of a semicocycle with respect the time parameter. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator. Simplest semicocycles are those independent of the spatial variable. So, another problem we consider is to establish whether a semicocycle is cohomologous to such independent one (i.e., is linearizable). Focusing on this problem, we provide some criteria for a holomorphic semicocycle to be linearizable as well as several easily verifiable sufficient conditions. These conditions are essential even for semicocycles over semigroups of linear operators. The talk is based on joint work with F. Jacobzon and G. Katriel.	The topology of constant mean curvature surfaces with convex boundary Barbara Nelli, Università di L'Aquila What is the topology of a constant mean curvature surface with boundary a convex curve in the plane? We give an overview of what is known about this problem in space forms. Moreover, adding some simple hypothesis, we answer to the previous question when the ambient manifold is ${\mathbb H}^2\times{\mathbb R}$. Here ${\mathbb H}^2$ is the hyperbolic plane. This is a joint work with V. Moraru.
A representation formula fo the anisotropic total variation of a function in SBV Fernando Farroni, Università degli studi di Napoli Federico II Inspired by some new function space introduced by J. Bourgain, H. Brezis and P. Mironescu, we provide the relation between certain BMO?type seminorms and the total variation of SBV functions in the anisotropic framework. Our result is motivated by the quest of a new characterization of the anisotropic perimeter which is independent of the theory of distributions. This is a joint work with Nicola Fusco, Serena Guarino Lo Bianco and Roberta Schiattarella.	Intrinsic sub-laplacians and function spaces: embeddings, algebra properties and applications Marco Peloso, Università di Milano In this talk we consider a class of natural sub-elliptic operators on a general Lie group. We study the Sobolev, Besov and spaces defined by such operators, prove embedding theorems and algebra properties. We also provide some applications to nonlinear PDEs, obtaining existence and well-posedness. These operators are a model for the intrinsic sub-Laplacian on a sub-Riemannian manifold of exponential growth.
Stability of the Riesz Potential Inequality Nicola Fusco, Università degli studi di Napoli Federico II Given a measurable set $E$ we consider the Riesz Potential $$ \mathcal F(E)=\int_E\int_E{1 \over \|x-y\|^{n-\alpha}} \,dxdy\, $$ where $0$ < $\alpha$ < $n$. It is well known that the maximum of $\mathcal F(E)$ among all sets of the same volume of $E$ is achieved at a ball $B$, i.e. $$ \mathcal F(E)\leq\mathcal F(B)\,\,\,\,\|E\|=\|B\|. $$ Moreover, equality holds if and only if $E$ is a ball. We shall discuss the stability of this inequality. This is a joint work with Aldo Pratelli.	Minimizers of a variational problem for nematic liquid crystals with variable degree of orientation in two dimensions Itai Shafrir, Technion--I.I.T We study the asymptotic behavior, when $k\to\infty$, of the minimizers of the energy \begin{equation} G_k(u)=\int_{\Omega}\Big((k-1)\|\nabla\|u\|\|^2+\|\nabla u\|^2\Big)\,, \end{equation} over the class of maps $u\in H^1(\Omega,{\mathbb R}^2)$ satisfying the boundary condition $u=g$ on $\partial\Omega$, where $\Omega$ is a smooth, bounded and simply connected domain in ${\mathbb R}^2$ and $g:\partial\Omega\to S^1$. The motivation comes from a simplified version of Ericksen model for nematic liquid crystals. We will present similarities and differences with respect to the analog problem for the Ginzburg-Landau energy. This is a joint work with Dmitry Golovaty.
Mappings with integrally controlled moduli Anatoly Golberg, Holon Institute of Technology In the talk we discuss the differential properties of multidimensional homeomorphic/open discrete mappings. Such mappings ssentially generalize the well-known customarily investigated classes of mappings as quasiregular, quasiisometric, Lipschitzian, etc. But in contrast to these known classes, the definition of our mapping class does not involve any analytic restrictions. We also illustrate the regularity properties by several examples and present a collection of open related problems. The talk is based on joint works with R. Salimov (Institute of Mathematics, Kyiv, Ukraine).	Nonlinear resolvent of holomorphic generators David Shoikhet, Holon Institute of Technology Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t>0}$ of holomorphic self-mappings of the open unit disk, i.e., $f=\lim_{t\rightarrow 0}\frac{1}{t}\left( I-F_{t}\right) .$ In this work, we study properties of the resolvent family $R=\left\{ \left( I+rf\right) ^{-1}\right\} _{r>0}$ in the spirit of geometric function theory. We have discovered, in particular, that $R$ forms an inverse Loewner chain and consists of starlike functions of order $\alpha >1/2$. Moreover, each element of $R$ satisfies the Noshiro-Warshawskii condition $\left( \func{Re}\left( I+rf\right) ^{-1}\left( z\right) >0\right).$ This, in turn, implies that all elements of $R$ are also holomorphic generators. Finally, we study the existence of repelling fixed points of this family. This talk is based on joint work with Mark Elin and Toshiyuki Sugawa.
Linear representations of random groups Gady Kozma, Weizmann Institute of Science We will survey Gromov random groups and their fascinating phase transitions, with particular emphesis on results that pertain to the sharpness of the transition at density 1/2. Joint work with Alex Lubotzky.	Isoperimetric inequalities for Steklov-Laplacian eigenvalues Cristina Trombetti, Università di Napoli Federico II Abstract
Boundaries, conformal maps, and sub-Riemannian geometry Enrico Le Donne, Università Pisa and University of Jyväskylä In this talk there will be a mix of complex analysis, metric geometry, and differential geometry. The objective is to give a new point of view for the validity of Fefferman's mapping theorem from 1974. This result states that a biholomorphism between two smoothly bounded strictly pseudoconvex domains in C^n extends as a smooth diffeomorphism between their closures. Following ideas from Gromov, Mostow, and Pansu, we discuss a method of proof in the context of quasi-conformal geometry. In particular, we show that every isometry between smoothly bounded strictly pseudoconvex domains is 1-quasi-conformal with respect to the sub-Riemannian distance defined by the Levi form on the boundaries. Subsequently, a PDE argument shows that such maps are smooth. This method was proposed by M. Cowling, and it has been implemented in collaboration with L. Capogna, G. Citti, and A. Ottazzi.	On holomorphic motions Genadi Levin, Hebrew University of Jerusalem Holomorphic motion is a holomorphic family of injections of a subset of the plane. This notion was originated in holomorphic dynamics and turned out to be very useful in complex analysis and applications, with tight connections to quasi-conformal mappings. In the talk, I discuss some problems related to holomorphic motions which are also originated in holomorphic dynamics. Based on joint work with Weixiao Shen and Sebastian van Strien.
Generating functionals on quantum groups Ami Viselter, University of Haifa We will discuss generating functionals on locally compact quantum groups. One type of examples comes from probability: the family of distributions of a Lévy process forms a convolution semigroup, which in turn admits a natural generating functional. Another type of examples comes from (locally compact) group theory, involving semigroups of positive-definite functions and conditionally negative-definite functions, which provide important information about the group's geometry. We will explain how these notions are related and how all this extends to the quantum world; see how generating functionals may be (re)constructed and study their domains; and indicate how our results can be used to study cocycles. Based on joint work with Adam Skalski.	Is the Gilbert-Steiner problem convex? Gershon Wolansky, Technion I will introduce the notion of a conditional Wasserstein metric on the set of probability measures, and apply it to the Gilbert-Steiner problem of minimal graphs. A convexification of the finite Gilbert problem for optimal networks is introduced. It is a convex functional on the set of probability measures subject to the Wasserstein $p-$ metric. The minimizer of this convex functional is a one dimensional current measure supported in a directed graph. If this graph is a tree (i.e. contains no cycles) then this tree is also a minimum of the corresponding Gilbert problem. The convexification of the Steiner problem is the limit of these convexified Gilbert's problems for p=infinity. I will discuss a numerical algorithm for the implementation of the convexified Gilbert-mailing problem is also suggested, based on entropic regularization.