Special Session 141: New trends and methods for differential problems

Multiscale analysis in composite materials with rough imperfect interfaces

Micol Amar Department of Basic and Applied Sciences for Engineering - Sapienza University of Rome Italy Co-Author(s):    Daniele Andreucci, Claudia Timofte

Abstract: In the last years, the improvements of industrial techniques has permitted to obtain more efficient materials constructed by assembling different constituents, whose physical properties are definitely superior than the ones of the single components. However, this bonding does not give rise in general to perfect contacts between the different components, so that discontinuities in the involved physical fields can appear. Also the study of boundary value problems in composites with rough boundaries or interfaces is a topic that has recently attracted a significant interest due to their applications in many areas of natural or engineering sciences, such as fluid mechanics, materials science, biology, etc. In this talk, we present some recent results concerning the study of problems where imperfect contact conditions couple with oscillating interfaces. More precisely, we consider a stationary heat diffusion problem in a two-component material, which exhibits an imperfect contact across an oscillating interface separating the two constituents. At the microscopic scale, it is mathematically described by an elliptic boundary value problem stated in each of two connected components. The heat potential is continuous across the interface, while its flux jumps according to suitable prescribed conditions, which lead to different macroscopic models, obtained through an asymptotic analysis.

The motion of a liquid drop

Domenico Angelo Angelo Universit\`a degli studi di Napoli Federico II Italy Co-Author(s):

Abstract: In this talk we summarize some new results obtained in the last few years regarding the evolution of a liquid drop. The main focus will be towards the existence of global solutions to the free boundary Euler equation describing the problem.

Nonlocal semilinear differential equations with superlinear growth

Irene Benedetti Department of Mathematics and Computer Science, University of Perugia Italy Co-Author(s):

Abstract: This talk is devoted to the study of existence results for semilinear parabolic partial differential equations with nonlocal initial conditions and superlinear growth. The class of nonlocal conditions considered includes, as special cases, multipoint Cauchy problems, weighted mean value conditions, and periodic problems. The analysis is carried out by applying a Leray-Schauder continuation principle, reformulating the problem as an ordinary differential equation in an abstract Banach space setting. In this framework, Lebesgue spaces provide the natural setting in which parabolic equations can be rewritten as ordinary differential equations. A major difficulty arises from the presence of superlinear nonlinearities, since the associated Nemytskii operator is not, in general, continuous on Lebesgue spaces unless sublinear growth conditions are satisfied, as highlighted by Vainberg`s theorem. This issue is addressed by exploiting the compactness and regularizing properties of the semigroup generated by the linear part of the equation, together with the construction of a suitable approximation scheme. These techniques allow us to establish existence results beyond the classical sublinear framework.

Solutions and algebraic properties of nonlinear equations of soliton type

Sandra Carillo University LA SAPIENZA Italy Co-Author(s):    Sandra Carillo

Abstract: Soliton equations play a central role from an applied perspective because they admit solutions whose shape remains unchanged during propagation and whose amplitude is conserved over time. Such equations frequently appear in applied mathematics, with applications spanning fluid dynamics, nonlinear optics, acoustics, theoretical physics and, more recently, biophysics. In the present work we consider solutions of the matrix modified Korteweg-de Vries (mKdV) equation. These solutions can properly be termed soliton solutions, since they manifest the characteristic behavior of solitons. In particular, two-soliton solutions of the $d\\times d$ matrix mKdV equation are constructed as in [1]. This work builds on a general explicit formula for $N$-soliton solutions in the infinite-dimensional (operator) case, as studied in [2], while the explicit finite-dimensional (matrix) case is treated in detail in [5]. Additionally, invariance properties of third-order KdV-type equations in non-commutative (matrix) contexts have been investigated more recently in [6]. Solutions of matrix mKdV equation are presented which can be termed soliton solutions since they exhibit the typical behaviour of solitons. Specifically, two-soliton solutions of the $d \\times d$-matrix modified Korteweg de-Vries equation are obtained in [1]. An explicit formula in the matrix case is studied in [3] while invariance properties of third order KdV-type equations are investigated in [6]. An overview on recent and perspective result is in [7]. [1] S. Carillo, C. Schiebold, Construction of soliton solutions of the matrix Korteweg-de Vries and modified Korteweg-de Vries equations, in Advances in Nonlinear Dynamics. NODYCON Conference Proceedings Series. Springer, Cham, ISBN 978-3-030-81169-3, W. Lacarbonara, et al. Ed.s, 481--491 (2022). doi: 10.1007/978-3-030-81170-9_42, arXiv: 2011.12677. [2] S.~Carillo and C.~Schiebold, Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Non-commutative soliton solutions, J. Math. Phys. 52 (2011), 053507. doi: 10.1063/1.3576185 27-37. [3] S. Carillo, M. Lo Schiavo, C. Schiebold, $N$-soliton matrix mKdV solutions: Some Special Solutions Revisited, Studies in Applied Mathematics, 2025; 154: e70061. doi: doi.org/10.1111/sapm.70061. [4] S.~Carillo and C.~Schiebold, On the asymptotical description of soliton solutions to the matrix modified Korteweg-de Vries equation, submitted 2023. Advances in Nonlinear Dynamics - Proc.s of the Third International Nonlinear Dynamics Conference (NODYCON 2023), Vol. 3, W. Lacarbonara, Ed. 565--575 (2024). doi: 10.1007/978-3-031-50635-2_43. [5]. S. Carillo, M. Lo Schiavo, C. Schiebold, Matrix solitons solutions of the modified Korteweg-de Vries equation, NODYCON 2019 Springer Proceedings, B. Balachandran, J. Ma, W. Lacarbonara, G.Quaranta, J. Machado, G. Stepan, Ed.s, (2020), 75-83. doi: 10.1007/978-3-030-34713-0_8. [6] S. Carillo, C. Schiebold, Soliton equations: admitted solutions and invariances via B\\cklund transformations, Open Commun. Nonlinear Math. Phys., Episciences, ISSN: 2802-9356, S.I. 1, 1--11 (2024). doi: 10.46298/ocnmp.12497. [7] S. Carillo, C. Schiebold, F. Zullo, Nonlinear evolution equations of soliton type: old and new results, Proceedings, MDPI, 2025; 123(1):9, 2025. doi: 10.3390/proceedings2025123009.

Uniqueness results for fractional Dirichlet problems in symmetruc domains

Isabella Ianni Sapienza University Italy Co-Author(s):    De Marchis, Dieb, Saldana

Abstract: We discuss recent uniqueness and nondegeneracy results obtained for non-negative solutions of some fractional semilinear problems in bounded domains with Dirichlet exterior condition. In particular we can consider least energy solutions in balls or in more general symmetric domains, for problems with power nonlinearities. The symmetry properties of the solutions of the associated linearized equation are also investigated.

Numerical analysis of a coupled system of parabolic equations with nonlinear and nonlocal drift

Nicklas J\\"{a}verg\\{aa}rd Karlstad University Sweden Co-Author(s):    Nicklas Javergard, Rainey Lyons, Adrian Muntean

Abstract: We study the formation and evolution of complex morphologies that arise in systems of equations governed by nonlinear and nonlocal interactions. Starting from the continuum formulation, which is derived from a particular hydrodynamic limit of the lattice-based Blume-Capel model with Kawasaki dynamics, we analyze a class of partial differential equations that capture the interplay between diffusion and interaction-driven transport in a ternary mixture. We propose a semi-discrete finite volume scheme to approximate the unique weak solution of the system. We prove that our scheme is well-posed and converge to the wanted solution. Furthermore, we show that the scheme is stable with respect to a parameter in the drift term.

A priori bounds for planar elliptic equations and systems

Gabriele Mancini University of Bari Aldo Moro Italy Co-Author(s):    Luca Battaglia, Laura Baldelli, Giulio Romani, Pierre-Damien Thizy

Abstract: I will discuss some recent results concerning uniform a priori bounds for positive solutions of elliptic equations and semilinear Hamiltonian elliptic systems involving exponential-type non-linearities in dimension two. These topics are presented in joint works in collaboration with Laura Baldelli (Karlsruhe Institute of Technology), Luca Battaglia (Universit\`{a} degli Studi Roma Tre), Giulio Romani (Universit\`{a} degli Studi di Udine) and Pierre-Damien Thizy (Universit\`{e} Claude Bernard Lyon 1). In the case of systems, we consider a broad class of coupled nonlinearities with asymptotic critical behaviour in the sense of Brezis-Merle. The approach we follow is based on a blow-up analysis combined with Liouville-type theorems and integral estimates. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of fixed point index theory.

Dirichlet problem on perturbed conical domains via converging generalized power series

Paolo Musolino Universita` degli Studi di Padova Italy Co-Author(s):    Martin Costabel, Matteo Dalla Riva, Monique Dauge

Abstract: We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $\mathbb{R}^{n}$ indexed by a small parameter $\varepsilon$. The domains depend on $\varepsilon$ only within a ball of radius proportional to $\varepsilon$ and, as $\varepsilon$ tends to zero, they converge in a self-similar way to a domain with a conical boundary singularity. We construct an expansion of the solution as a series of real positive powers of $\varepsilon$, and prove that it is not just an asymptotic expansion as $\varepsilon\to0$, but that, for small values of $\varepsilon$, it converges normally in the Sobolev space $H^{1}$. A planar version of such problem has been previously investigated by the authors with the so called {\it Functional Analytic Approach}, based on integral representations obtained through layer potentials. Here, instead, we choose a different technique that allows us to relax all regularity assumptions. We forgo boundary layer potentials and instead exploit expansions in terms of eigenfunctions of the Laplace-Beltrami operator on the intersection of the cone with the unit sphere. The basis for our analysis is a two-scale cross-cutoff ansatz for the solution that has similarities with the Maz'ya-Nazarov-Plamenevskij construction of a multiscale system for the asymptotic expansion of solutions of boundary value problems on domains singularly perturbed near singular points of the boundary.

The Monotonicity Principle in Quasilinear Inverse Problems

Gianpaolo Piscitelli Universita` degli studi di Napoli Parthenope Italy Co-Author(s):    A. Corbo Esposito, L. Faella, V. Mottola, R. Prakash, A. Tamburrino

Abstract: We discuss a framework for treating the inverse obstacle problem for nonlinear elliptic equations with nonlinear materials. The framework is based on two recent theoretical results for nonlinear partial elliptic PDEs: the p-Laplace Signature (p-LS) and the Monotonicity Principle (MP). The first result (p-LS) allows to model the solution of an elliptic PDE with nonlinear materials in terms of a proper p-Laplace equation, which captures the essence of the problem. The Monotonicity Principle (MP), recently extended to nonlinear materials, provides a monotonic relationship between the material property and the measured quantity (the average Dirichlet-to-Neumann map) that can be `inverted` to find the shape of anomalies. Numerical examples are provided to show and confirm the effectiveness of the strategy.

Uniqueness for Neumann problems for nonlinear elliptic equations with lower order terms

Maria Rosaria MR Posteraro University of Naples Federico II Italy Co-Author(s):    M.F.Betta, O. Guib\`e, A. Mercaldo

Abstract: We prove uniqueness results for weak solutions to a class of Neumann problems, whose prototype is \begin{gather*} \begin{cases} \null \lambda (1+ u^2)^{(p-2)/2}u-\diw((1+|\nabla u|^2)^{(p-2)/2} \nabla u) & \qquad\ - \diw(c(x) (1+|u|^2)^{(\tau+1)/2}) +b(x) (1+|\nabla u|^2)^{(\sigma+1)/2}=f &\text{ in } \Omega\ \left( (1+|\nabla u|^2)^{(p-2)/2} \nabla u + c(x) (1+|u|^2)^{(\tau+1)/2}) \right)\cdot\underline n=0 & \text{on}\ \partial \Omega \,, \end{cases} \end{gather*} where $\Omega$ is a bounded open subset of $\R^N$ $(N\ge 2)$ with Lipschitz boundary, $p$ is a real number $\frac{2N}{N+1}< p

Homogenization of reaction-diffusion equation with large nonlinear drift

Vishnu Raveendran University of Bonn Germany Co-Author(s):    I. de Bonis, E.N.M Cirillo , and A. Muntean.

Abstract: We discuss the periodic homogenization of a reaction-diffusion problem with nonlinear drift posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of an asymmetric simple exclusion process governing a population of interacting particles crossing a domain with obstacles. We are interested in deriving the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to a fast nonlinear drift.

Spectral clusters

Giorgio Saracco Universita` di Ferrara Italy Co-Author(s):    Giorgio Stefani

Abstract: Given any open, bounded set $\Omega$, we consider suitable combinations, via a reference function $\Phi$, of the first $p$-eigenvalue of the Dirichlet Laplacian of partitions of $\Omega$. We give two different formulations of the problem, one geometrical and one functional. We prove relations among the two formulations, existence and regularity of optimal partitions, convergence, and stability with respect to $p$ and to $\Phi$. Based on a joint work with G. Stefani (Padova).

On a new concept of controllability of second-order semilinear differential equations in Banach spaces

Valentina Taddei University of Modena and Reggio Emilia Italy Co-Author(s):    Martina Pavla\v{c}kov\'a

Abstract: In this talk, we investigate the controllability of second-order problems in Banach spaces, when the nonlinear term also depends on the first derivative. In the existing literature, the most commonly adopted notion of exact controllability for second-order equations ensures only that the state function attains the desired target value, while neglecting the behavior of its derivative at the final time, thereby violating the concept of controllability. The primary aim of this work is to introduce a new definition of controllability for second-order problems in Banach spaces, which simultaneously accounts for both the solution and its derivative at the final time, through a single control function. We then establish sufficient conditions guaranteeing this controllability. Our approach yields results under easily verifiable and non-restrictive assumptions on the cosine family generated by the associated linear operator, as well as on the nonlinear term, without requiring any compactness conditions. Finally, we illustrate the applicability of our results by considering a system governed by the one-dimensional Klein-Gordon equation.

Boundary regularity at characteristic points in the Heisenberg group

Giulio Tralli University of Ferrara Italy Co-Author(s):

Abstract: In this talk we focus on the boundary regularity of the solutions to Poisson-type problems with homogeneous boundary datum. We discuss the geometric case of the Heisenberg group, where we fix the subLaplacian as the relevant (degenerate-elliptic) operator and the characteristic half-space as the (scale-invariant) domain. We show an intrinsic second order expansion at the only characteristic point of the boundary when the source term belongs to an appropriate weighted $L^\infty$ space. The technique we adopt is inspired by Caffarelli approach to Krylov`s boundary $C^{1,\alpha}$-estimate for uniformly elliptic operators in nondivergence form. This is a joint work with F. Abedin.