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Riccardo Molinarolo

Publications and source records attributed to Riccardo Molinarolo.

13 recordsLinked to original sources
arXiv2026

Shape analysis in Schauder spaces of the energy of heat problems in perturbed annular domains

This paper is devoted to the shape analysis of the energy of a caloric family of Schauder functions defined on a bounded perforated domain $Ω^o \setminus \overline{Ω^i[ϕ]}$ of $\mathbb{R}^n$, where the outer boundary is fixed, and the inner boundary is obtained by a $C^{1,α}$-perturbation $ϕ$ of the boundary of a reference cavity $Ω^i$. Without imposing any boundary conditions, we prove that in a suitable neighborhood of the identity $ϕ_0$, the domain-to-energy map is of class $C^{\infty}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $ϕ$, from the reference annulus onto the perturbed one and on suitable regularity and smoothness assumptions on the pull-back family onto the reference domain. We then apply our main result to two boundary value problems: a nonlinear mixed Robin-type problem and a linear Dirichlet problem. After recalling some known existence and shape analysis results for the solutions, we prove that the corresponding domain-to-energy map is of class $C^{\infty}$. The proof is based on a decomposition of the fixed domain into near, intermediate, and far regions relative to the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.

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arXiv2026

On fractional semilinear wave equations in non-cylindrical domains

In this paper, we investigate a class of semilinear wave equations in non-cylindrical time-dependent domains, subject to exterior homogeneous Dirichlet conditions. Under mild regularity and monotonicity assumptions on the evolving spatial domains, we establish existence of weak solutions by two different methods: a constructive time-discretization scheme and a penalty approach. The analysis applies to nonlocal fractional Laplacians and potentials with Lipschitz continuous gradient, and to vector-valued maps.

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arXiv2025

On the eigenvalues of the biharmonic operator on annuli

We show that the fundamental tone of the bilaplacian with Dirichlet or Navier boundary conditions on radially symmetric domains is always simple in dimension $N\ge3$. In dimension $N=2$ we show that it is simple if the inner radius is big enough.

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arXiv2025

On the Serrin's problem with Robin boundary conditions

Let $Ω\subset \mathbb{R}^N$, $N\ge 2$, be an open, connected, bounded set with $C^2$ boundary. In this paper we consider the torsion problem with Robin boundary conditions and we study the symmetry of the solutions when suitable extra conditions are imposed on the boundary of $Ω$. In particular, we prove the Serrin's rigidity result under suitable assumptions on the domain and on the Robin parameter.

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arXiv2024

Shape perturbation of a nonlinear mixed problem for the heat equation

We consider the heat equation in a domain that has a hole in its interior. We impose a Neumann condition on the exterior boundary and a nonlinear Robin condition on the boundary of the hole. The shape of the hole is determined by a suitable diffeomorphism $ϕ$ defined on the boundary of a reference domain. Assuming that the problem has a solution $u_0$ when $ϕ$ is the identity map, we demonstrate that a solution $u_ϕ$ continues to exist for $ϕ$ close to the identity map and that the "domain-to-solution" map $ϕ\mapsto u_ϕ$ is of class $C^\infty$. Moreover, we show that the family of solutions $\{u_ϕ\}_ϕ$ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks a the linear case complete the paper.

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arXiv2024

Existence result for a nonlinear mixed boundary value problem for the heat equation

In this paper we study the existence of solutions in parabolic Schauder space of a nonlinear mixed boundary value problem for the heat equation in a perforated domain. From a given regular open set $Ω\subseteq\mathbb{R}^n$ we remove a cavity $ω\subseteq Ω$. On the exterior boundary of $Ω\setminus\overlineω$ we prescribe a Neumann boundary condition, while on the interior boundary we set a nonlinear Robin-type condition. Under suitable assumptions on the data and by means of Leray Schauder Fixed-Point Theorem, we prove the existence of (at least) one solution $u \in C_{0}^{\frac{1+α}{2}; 1+α}([0,T] \times (\overlineΩ \setminus ω))$.

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arXiv2024

A general integral identity with applications to a reverse Serrin problem

We prove a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that the first and third authors previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As an application, we prove a quantitative symmetry result for the reverse Serrin problem, which we introduce for the first time in this paper. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition.

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arXiv2023

Multi-parameter perturbations for the space-periodic heat equation

This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of the results from the first part. Specifically, we consider a transmission problem for the heat equation in a periodic two-phase composite material and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and conductivity parameters. Finally, in the last part of the paper, we fix all parameters except for the contrast parameter and outline a strategy to deduce an explicit expansion of the solution using a Neumann-type series.

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arXiv2022

Existence results for a nonlinear nonautonomus transmission problem via domain perturbation

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $ϕ$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $ϕ$.

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arXiv2022

Existence of solutions for a singularly perturbed nonlinear non-autonomous transmission problem

In this paper we analyse a boundary value problem for the Laplace equation with a nonlinear non-autonomous transmission conditions on the boundary of a small inclusion of size $ε$. We show that the problem has solutions for $ε$ small enough and we investigate the dependence of a specific family of solutions upon $ε$. By adopting a functional analytic approach we prove that the map which takes $ε$ to (suitable restrictions of) the corresponding solution can be represented in terms of real analytic functions.

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arXiv2022

On the wave equation on moving domains: regularity, energy balance and application to dynamic debonding

We revisit some issues about existence and regularity for the wave equation in noncylindrical domains. Using a method of diffeomorphisms, we show how, through increasing regularity assumptions, the existence of weak solutions, their improved regularity and an energy balance can be derived. As an application, we give a rigorous definition of dynamic energy release rate density for some problems of debonding, and we formulate a proper notion of solution for such problems. We discuss the consistence of such formulation with previous ones, given in literature for particular cases.

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arXiv2020

Radial solutions for a dynamic debonding model in dimension two

In this paper we deal with a debonding model for a thin film in dimension two, where the wave equation on a time-dependent domain is coupled with a flow rule (Griffith's principle) for the evolution of the domain. We propose a general definition of energy release rate, which is central in the formulation of Griffith's criterion. Next, by means of an existence result, we show that such definition is well posed in the special case of radial solutions, which allows us to employ representation formulas typical of one-dimensional models.

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