Workshop information
1. Dates: 2. Conference venue: 3. Accommodation & Sign-in Venue: 4. Catering Arrangements: 5. Scientific Committee: 6. Financed and Organized by: |
Time | June 3^{rd}, Monday | June 4^{th}, Tuesday | June 5^{th}, Wednesday | June 6^{th}, Thursday | June 7^{th}, Friday |
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8:50-9:00 | Opening ceremony | ||||
9:00-9:40 | Daomin Cao (Xi-Nan Ma) |
Cristina Trombetti (Xiaoping Yang) |
Berardino Sciunzi (Jianfu Yang) |
Giuseppe Buttazzo (Jun Wang) |
Carlo Nitsch (Wei Cheng) |
9:45-10:25 | Paolo Salani (Xi-Nan Ma) |
Yujin Guo (Xiaoping Yang) |
Nicola Gigli (Jianfu Yang) |
Giulio Ciraolo (Jun Wang) |
Yi Zhang (Wei Cheng) |
10:30-11:00 | Break | ||||
11:00-11:40 | Andrea Cianchi ( ) |
Graziano Crasta (Yong Liu) |
Ugo Gianazza (Zhizhang Wang) |
Matteo Focardi (Qianzhong Ou) |
Bangxian Han (Dongsheng Li) |
11:45-12:25 | Tianling Jin ( ) |
Anna Mercaldo (Yong Liu) |
Antonio Iannizzotto (Zhizhang Wang) |
Nunzia Gavitone (Qianzhong Ou) |
Luigi Montoro (Dongsheng Li) |
12:30-13:00 | Lunch | ||||
14:30-15:10 | Robert McCann (Shibing Chen) |
Wenming Zou (Lili Du) |
Jun Geng (Guohuan Qiu) |
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15:15-15:55 | Zhitao Zhang (Shibing Chen) |
Elisa Francini (Lili Du) |
Angela Alberico (Guohuan Qiu) |
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16:00-16:20 | Break | ||||
16:20-17:00 | Lorenzo Cavallina (Genggeng Huang) |
Cale Rankin (Shuangjian Zhang) |
Giorgio Poggesi (Jiakun Liu) |
Titles and Abstracts
On the Existence and Stability of Traveling Vortex Pairs for the gSQG Equation Daomin Cao (Academy of Mathematics and Systems Science, CAS) Abstract: For the 2D incompressible Euler equation, there is a well-known traveling wave solution, which is called Lamb dipole. The generalized surface quasi-geostrophic (gSQG) equation is the generalization of 2D Euler equation. Is there a traveling wave solution for gSQG equation similar to Lamb dipole? In this talk the speaker will talk about this topic and introduce some results obtained in joint papers with Shanfai Lai, Guolin Qin, Weicheng Zhan and Changjun Zou. Log-concavity of the First Eigenfunction and Brunn-Minkowski Inequality of the First Eigenvalue of the Ornstein-Uhlenbeck Operator Paolo Salani (Università di Firenze) Abstract: Several functionals from Calculus of Variations satisfy a Brunn-Minkowski type inequality, which means that they are \(\alpha\)-concave as functions of the domain, for a suitable \(\alpha\). In particular, the first Dirichlet eigenvalue of the Laplacian satisfies the following inequality: \[\lambda((1-t)\Omega_0+t\Omega_1)^{-\frac 1 2}\geqslant(1-t)\lambda(\Omega_0)^{-\frac 1 2}+t\lambda(\Omega_1)^{-\frac 1 2}\] for \(\Omega_0,\Omega_1\) sufficiently regular bounded open sets and \(t\in(0,1)\). I will discuss the same type of inequality for the first Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator \[L(u)=\Delta u-\langle x,\nabla u\rangle,\] that is analogous to the Laplacian when the Lebesgue measure in \(\mathbb R^n\) is substituted by the Gaussian measure. I will also discuss the log-concavity of the associated eigenfunction for a convex domain, a property intimately connected to such inequality. This talk is based on a joint work with A. Colesanti, E. Francini and G. Livshyts. Sobolev Regularity of the Stress-field in Nonlinear Elliptic Problems Andrea Cianchi (Università di Firenze) Abstract: Second-order regularity results are established for solutions to second-order elliptic equations and systems, in divergence form, with principal part having Uhlenbeck structure and square-integrable right-hand sides. Such a regularity amounts to the membership in a Sobolev space of the so-called stress-field associated with the elliptic operator. In the case of equations, differential operators depending on anisotropic norms of the gradient are also included. Both local and global estimates are obtained. Global estimates concern solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, no regularity of its boundary is needed if the domain is convex. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on diverse joint investigations with C.A. Antonini, A.Kh. Balci, G. Ciraolo, L. Diening, A. Farina and V. Maz'ya. Stability of the Separable Solutions for a Nonlinear Boundary Diffusion Problem Tianling Jin (The Hong Kong University of Science and Technology) Abstract: We consider a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We prove the stability of its separable solutions, and obtain the sharp convergence rates as well as the higher order asymptotics. This talk is based on the joint work with Jingang Xiong and Xuzhou Yang. An Elliptic Proof of the Lorentzian Splitting Theorems Robert McCann (University of Toronto) Abstract: Splitting theorems play a vital role in both Riemannian and Lorentzian geometry. Under the strong energy condition from general relativity, Yau conjectured in 1982 that a timelike geodesically complete spacetime ought to be exceptional: if even one of its complete geodesics is timelike and maximizing, then the space is a stationary, static, geometric product. Although Yau's conjecture was proved by Newman(1990) following works by Eschenberg (1988) and Galloway (1999), the proof is complicated relative to the Riemannian case by the fact that the Lorentzian Laplacian is not elliptic. We describe a new proof of the Lorentzian splitting theorems, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity p-d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry. Our proof relies on a p-d'Alembert comparison result obtained with Beran, Braun, Calisto, Gigli, Ohanyan, Rott, Saemann. We anticipate that work in progress will confirm that our method can be used to lower the regularity requirements on the Lorentzian metric tensor for the splitting to occur. On Normalized Solutions of Schrödinger Equations and Systems Zhitao Zhang (CAS, AMSS & Jiangsu University) Abstract: We introduce some new results on normalized solutions, especially for normalized solutions of mass subcritical Schrodinger equations in exterior domains; normalized solutions to p-Laplacian equations with combined nonlinearities; normalized solutions to Schrodinger systems etc. A Characterization of Radial Symmetry for Composite Media by Overdetermined Level Sets Lorenzo Cavallina (Tohoku University) Abstract: In this talk, we introduce the concept of an "overdetermined level set" (that is, a set where both a given function and the absolute value of its gradient are constant). As a corollary of J. Serrin's celebrated symmetry theorem for overdetermined elliptic problems, we know that the ball is the only domain such that the solution to some elliptic problem for the Dirichlet Laplacian, called the torsion problem, admits at least one overdetermined level set. Moving beyond the classical case, we study how this symmetry result generalizes to a multi-phase setting (that is, when the Laplacian is replaced with an elliptic operator in divergence form with piece-wise constant coefficients that take finitely many values). Notice that, in a multi-phase setting, various types of overdeterminations are possible, depending on the number and relative position of the overdetermined level sets. In this talk, we give a complete characterization in the two-phase setting by means of overdetermined level sets. The content of this talk is based on a joint work with Giorgio Poggesi (Univ. of Western Australia). Some Isoperimetric Estimates for Robin Eigenvalues Cristina Trombetti (Università Federico II, Napoli) Abstract: In this seminar, I discuss the application of the web functions technique to estimate the first Robin eigenvalue with a negative parameter \(\beta\). As is well known, in this scenario, a Faber-Krahn-type inequality does not exist. However, with some additional restrictions, a weaker form of Faber-Krahn still holds true, and here is where web functions play a role. Variational Problems of Quantum Many-Body Systems with Interactions Yujin Guo (Central China Normal University) Abstract: This talk is focused on variational problems of quantum many-body systems with interactions, including ultracold Bose gases, ultracold Fermi gases, pseudo-relativistic boson stars, and so on. The interactions among these quantum systems represent a major difficulty in mathematically understanding. The investigations on the corresponding variational problems can be traced back as early as the celebrated works of P. L. Lions and E. Lieb around 1980s. In this talk, we introduce and review the recent main progresses, as well as many unsolved problems, of several typical variational problems arising from these quantum systems with interactions. Variational Worn Stones Graziano Crasta (Sapienza Università di Roma) Abstract: We introduce an evolution model `a la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball. On a Class of Nonlinear Elliptic Equations With General Growth in the Gradient Anna Mercaldo (Università Federico II, Napoli) Abstract: Let \(\Omega\) be a bounded open set of \(\mathbb R^N\), \(N\geqslant2\). Let us consider the class of Dirichlet boundary value problems as \[\begin{cases} -\operatorname{div}(a(x,u,\nabla u))=H(x,\nabla u)+g(x,u)&\text{ in }\Omega,\\ u=0&\text{ on }\partial\Omega, \end{cases}\] where the principal part is a coercive, pseudomonotone operator of Leray-Lions type in Sobolev space \(W^{1,p}_0(\Omega)\), \(1 < p < N\), the functions \(H\) and \(g\) growth at most like \(|\nabla u|^q+f(x)\) with \(p − 1 < q < p\) and \(c(x)|u|^p\), respectively. By assuming that \(f(x)\) and \(c(x)\) belong to optimal Lorentz spaces, existence, uniqueness and pointwise comparison results are presented by discussing several ranges of values of \(q\) and \(p\). The main difficulties which arise in studying this type of problems are due to the "superlinear" character of the first order terms and the presence of the zero order terms. The Classification of Critical Points of Heat Kernel on Tori and Applications to Elliptic Equation and Minimization Problem on Lattice Wenming Zou (Tsinghua University) Abstract: In this paper, we investigate the critical points of the Heat kernel on two-dimensional flat tori. Using methods related to theta functions, we determine that the Heat kernel exhibits four and six critical points on rectangular and hexagonal tori, respectively. Furthermore, on a rhombic torus, the number of critical points of the Heat kernel depends on the geometry of torus. We have also established a connection between the Heat kernel, linear elliptic equations with singularity, and particle energy. This connection allows us to recover partial results of the Green function and provides a partial answer to the Luo-Wei's conjecture (ARMA,2022) regarding the Mueller-Ho Conjecture. An intriguing finding of our study is that all three functions exhibit uniform critical points on rectangular and hexagonal tori. Jointly with C. G. Long and J. C. Wei. Quantitative Unique Continuation Estimates and Their Application to the Study of Stability in Inverse Boundary Value Problems Elisa Francini (Università di Firenze) Abstract: Inverse boundary value problems involve the retrieval of unknown parameters of a partial differential equation (PDE) from boundary data. In practical scenarios, this entails reconstructing internal properties of a medium (e.g., conduction, stiffness, density) based on observations made at its boundary. Typically, parameter estimation problems exhibit high nonlinearity and are ill-posed according to Hadamard's definition: small errors in the data may result in uncontrollable errors in the unknowns. In view of the many applications, this leads to the search for appropriate methods to contain such instability. In this talk I will introduce the conductivity problem, a prototypical example of ill-posed nonlinear inverse problem at the basis of Electrical Impedence Tomography (EIT). I will show that by introducing, mathematically suitable but physically relevant, a-priori assumptions on the unknown parameters one can mitigate the ill-posedeness. Quantitative Unique Continuation Estimates will emerge as a crucial tool in this endeavor. The Monopolist's Problem: Regularity and Structure Results for a New Free Boundary Problem Cale Rankin (Monash University) Abstract: The Monopolist's problem is a simple model from economics which displays rich mathematical behaviour and lies at the intersection of optimal transport, free boundary problems, and convex analysis. Mathematically one aims to minimize a uniformly convex Lagrangian, however restricted to the space of convex functions. The requirement that the minimization take place over the space of convex functions leads to a free boundary between the regions of strict and nonstrict convexity and in these regions the solution displays qualitatively different behaviour. In this talk I discuss joint work-in-progress with Robert McCann and Kelvin Shuangjian Zhang in which we prove results on the configuration of the different domains, regularity of the free boundary, and completely describe the solution in the case of most interest to applications. Classification of Solutions to Hardy-Sobolev Doubly Critical Systems Berardino Sciunzi (Università della Calabria, Cosenza) Abstract: I will talk about a recent paper in collaboration with Francesco Esposito and Rafael L´opez-Soriano regarding the classification of solutions to a family of Hardy-Sobolev doubly critical system defined in \(\mathbb R^n\): \[(S^*)\begin{cases} -\Delta u=\gamma\dfrac{u}{|x|^2}+u^{2^*-1}+\nu\alpha u^{\alpha-1}v^\beta&\text{ in }\mathbb R^n\\ -\Delta v=\gamma\dfrac{v}{|x|^2}+v^{2^*-1}+\nu\alpha u^\alpha v^{\beta-1}&\text{ in }\mathbb R^n\\ u,v>0&\text{ in }\mathbb R^n\backslash\{0\} \end{cases}\] where \(\gamma\in[0,\Lambda_n)\) with \(\Lambda_n=\left(\dfrac{n-2}{2}\right)^2\) the best constant in the Hardy’s inequality for \(n\geqslant 3\), \(\nu > 0\) is the coupling parameter and \(\alpha,\beta > 1\) are real parameters satisfying \(\alpha+\beta=2^*\). We provide the classification of the positive solutions, whose expressions comprise multiplies of solutions of the decoupled scalar equation. Our strategy is based on the study of symmetry properties of the solutions, deduced via a suitable version of the moving planes technique, joint with the study of the asymptotic behavior of the solutions. Trading linearity for ellipticity - a novel approach to global Lorentzian geometry Nicola Gigli (SISSA, Trieste) Abstract: The concepts of Sobolev functions, elliptic operators and Banach spaces are central in modern geometric analysis. In the setting of Lorentzian geometry, however, unless one restricts the attention to Cauchy hypersurfaces these do not have a clear analogue, due to the signature of the metric tensor. Aim of the talk is to discuss some recent observations in this direction centered around the fact that for \(p < 1\) the \(p\)-D'Alambertian is elliptic on the space of time functions. The talk is mostly based on joint project with Beran, Braun, Calisti, McCann, Ohanyan, Rott, Saemann. Boundary Behavior of Functions in De Giorgi Classes Defined in Non-smooth Domains Ugo Gianazza (Università di Pavia) Abstract: The homogeneous De Giorgi classes \([DG]_p^{\pm}(\Omega)\) in the interior of \(\Omega\) consist of functions \(u\in W^{1,p}_{\operatorname{loc}}(\Omega)\) which satisfy inequality of the type \[\int_{B_\rho(y)}|D(u-k)_{\pm}|^p\mathrm{d}x\leqslant\frac{\gamma}{(R-\rho)^p}\int_{B_\rho(y)}|(u-k)_{\pm}|^p\mathrm{d}x,\quad\forall B_\rho(y)\subset\Omega,\ 0 < \rho < R,\ k\in\mathbb R,\] where \(\gamma\) is a positive constant, \(A^{\pm}_{k,R}\equiv\{x\in B_R(y):(u-k)_{\pm}>0\}\), and \(|\Sigma|\) denotes the Lebesgue measure of a measurable set \(\Sigma\). We further define \([DG]_p(\Omega)\equiv[DG]_p^+(\Omega)\cap[DG]_p^-(\Omega)\). It is well-known that functions in \([DG]_p(\Omega)\) are locally Hölder continuous in \(\Omega\) and non-negative functions in \([DG]_p(\Omega)\) satisfy the Harnack's inequality. Let \(h\in W^{1,p}(\mathbb R^N)\cap C(\mathbb R^N)\). The homogeneous De Giorgi classes \([DG]_p^+(h;\overline\Omega)\) in the closure of \(\Omega\) consist of functions \(u\in[DG]_p^+(\Omega)\) such that \((u-h)\in W^{1,p}_0(\Omega)\) and satisfying in addition inequalities of the type \[\int_{B_\rho(y)\cap\Omega}|D(u-k)_+|^p\mathrm{d}x\leqslant\frac{\gamma}{(R-\rho)^p}\int_{B_\rho(y)\cap\Omega}|(u-k)_+|^p\mathrm{d}x,\quad\forall y\in\partial\Omega,\ B_\rho(y),\ 0 < \rho < R,\ k\geqslant\sup_{B_\rho(y)\cap\partial\Omega}h,\] The constant \(\gamma\) is as above and \(A^+_{k,R}\equiv\{x\in B_R(y):(u-k)_+>0\}\). The classes \([DG]^-_p(h;\overline\Omega)\) are defined analogously by replacing \((u-k)_+\) with \((u-k)_-\) provided that \(k\leqslant\inf\limits_{B_\rho(y)\cap\partial\Omega}h\). As before, we let \([DG]_p(h;\overline\Omega)\equiv[DG]_p^+(h;\overline\Omega)\cap[DG]_p^-(h;\overline\Omega)\). Again, it is known that functions in \([DG]_p(h;\overline\Omega)\) are Hölder continuous up to points \(y\in\partial\Omega\), provided \(\Omega\) satisfies a positive geometric density condition at \(y\). What happens to the regularity of \(u\in [DG]_p(h;\overline\Omega)\) when the boundary of \(\Omega\) is not smooth? I will present some results in properly defined irregular domains, and I will briefly discuss what is still open. Fine Boundary Regularity for the Fractional \(p\)-Laplacian Antonio Iannizzotto (Università di Cagliari) Abstract: The fractional \(p\)-Laplacian is a nonlinear, nonlocal operator with fractional order \(0 < s < 1\) and homogeneity exponent \(p>1\), arising in game theory and extending (in some sense) both the classical \(p\)-Laplacian and the linear fractional Laplacian. Combining nonlocal and nonlinear features, such operator requires an ad hoc approach especially in regularity theory. In this talk, we are going to discuss a special form of fine boundary regularity for the weak solutions, namely, Hölder continuity of the quotient between the solution and the \(s\)-power of the distance from the boundary. Such type of regularity, already known in the degenerate case \(p>2\), was recently achieved even in the singular case \(1 < p < 2\), and it bears interesting applications in comparison principles, Hopf's type lemmas, Harnack's inequalities, and an equivalence principle between Sobolev and Hölder minimizers of the associated energy functional. The talk is based on some works in collaboration with S. Mosconi and M. Squassina. Optimal Coefficients for Elliptic PDEs Giuseppe Buttazzo (Università di Pisa) Abstract: The goal is to present an optimization problem related to elliptic PDEs of the form \(-\operatorname{div}(a(x)\nabla u)=f\) with Dirichlet boundary condition on a given domain \(\Omega\). The coefficient \(a(x)\) has to be determined, in a suitable given class of admissible choices, in order to optimize a given criterion. The first deal with the case when the cost is the so-called elastic compliance, and then we discuss the more general case when the problem is written as an optimal control problem. Quantitative Stability Results via the Method of Moving Planes Giulio Ciraolo (Università di Milano) Abstract: The method of the moving planes is a classical tool to prove symmetry properties for overdetermined PDE's boundary value problems and for rigidity problems in geometric analysis. In this talk we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained. On the Free Boundary for Thin Obstacle Problems with Lipschitz Coefficients Matteo Focardi (Università di Firenze) Abstract: Recently the regularity of free boundaries corresponding to minimizers of thin obstacle problems has attracted a lot of attention after the seminal paper by Athanasopoulos, Caffarelli, and Salsa (Amer. J. Math. '08). In the classical case of the Dirichlet energy, many deep results about the fine structure of free boundaries are nowadays available. Several efforts have been also devoted to extending such regularity theory to the variable coefficient case. We will review the results known in the literature for quadratic energies driven by Lipschitz matrix fields, and finally discuss the rectifiability of the related free boundaries. This is joint work with G. Andreucci and E. Spadaro. Eigenvalue Problems in Doubly Connected Domains Nunzia Gavitone (Università degli Studi di Napoli Federico II) Abstract: In this talk, I will speak about some eigenvalue problems in doubly connected domains for the Laplace operator. In this problems, two boundary conditions appear, one for each component of the boundary and they could be different. I will describe some results obtained in collaboration with Gloria Paoli, Gianpaolo Piscitelli and Rossano Sannipoli. The Neumann Problem for the Stokes System on Convex Domains Jun Geng (Lanzhou University) Abstract: We show that the Neumann problem for Stokes system on convex domain \(\Omega\) with boundary data in \(L^p(\partial\Omega)\) is uniquely solvable for \(\begin{cases} 1 < p < +\infty&\text{ if }d=2,\\ 1 < p < 4+e&\text{ if }d=3,\\ \dfrac{2(d-1)}{d+1}-e< p < \dfrac{2(d-1)}{d-2}+e&\text{ if }d\geqslant4, \end{cases}\) and the \(W^{1,p}\) estimate for the Poisson problem is true for . The ranges of \(p\) are sharp for \(d = 2\) and these intervals are larger than the known interval on Lipschitz domain. Fractional Orlicz-Sobolev Spaces and Embedding Theorems Angela Alberico (CNR, IAC) Abstract: The optimal target space is exhibited for embeddings of fractional order Orlicz-Sobolev spaces. Both the subcritical and the supercritical regimes are considered. In the former case, the smallest possible Orlicz target space is detected. In the latter, the relevant Orlicz-Sobolev spaces are shown to be embedded into the space of bounded continuous functions in \(\mathbb R^n\). Moreover, their optimal modulus of continuity is exhibited. These results are the subject of a series of joint papers with Andrea Cianchi, Luboˇs Pick and Lenka Slav´ıkov´a. Soap Bubbles, Overdetermined Problems, and Related Questions Giorgio Poggesi (The University of Western Australia) Abstract: The talk will be focused on recent research works related to Alexandrov-type, Serrin-type, and Gidas-Ni-Nirenberg-type symmetry results. A Talenti Result for the Neumann Boundary Value Problem Carlo Nitsch (Università Federico II, Napoli) Abstract: In light of a recent paper that extended Talenti's a priori estimate for solutions to elliptic problems from Dirichlet to Robin boundary conditions, we demonstrate that the Neumann case can also be addressed. This leads to the derivation of a novel type of optimal bounds that appear to be previously unknown. John domains in variational problems Yi Zhang (CAS, AMSS) Abstract: The notion of a John domain was initially introduced in 1961 by Fritz John, and later named after him by Martio and Sarvas. Typically, its study is motivated by its connections to the properties of quasiconformal and quasisymmetric mappings. Moreover, John domains find extensive applications in the theory of Sobolev functions in metric measure spaces and functional analysis, as they represent essentially the sole class of domains that uphold the Sobolev-Poincare inequality. In this presentation, I will introduce several recent applications of John domains in the theory of the calculus of variations. Barycenter-curvature-dimension condition on metric measure spaces Bang-Xian Han (Shandong University) Abstract: Based on some recent results about the Wasserstein barycenter problem and lower Ricci curvature bound, I will introduce a notion called barycenter-curvature-dimension condition and some applications. This is a joint work with Dengyu Liu and Zhuonan Zhu. Classification of all weak solutions to \(-\Delta u=u^{-\gamma}\) in the half-space Luigi Montoro (Università della Calabria, Cosenza) Abstract: We provide a classification result for positive solutions to \(-\Delta u=u^{-\gamma}\) in the half space, under zero Dirichlet boundary condition. |