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Bianca Stroffolini

Publications and source records attributed to Bianca Stroffolini.

At least 19 recordsLinked to original sources
arXiv2026

Lipschitz regularity for orthotropic functionals with general growth

We study the local Lipschitz regularity of local minimizers for a class of degenerate orthotropic functionals with $\varphi$-growth, where $\varphi$ is a general N-function. Unlike standard isotropic functionals, the ellipticity of the associated Euler-Lagrange equation degenerates separately in each coordinate direction, presenting significant anisotropic difficulties. Furthermore, the general N-function setting lacks the algebraic scale invariance available in the classical orthotropic $p$-Laplacian case. Despite these structural difficulties, we prove that local minimizers are locally Lipschitz continuous. Our approach relies on a regularized approximation scheme, mixed-direction Caccioppoli inequalities, and a carefully designed Moser-type iteration that incorporates an interpolation argument to bridge the gaps between consecutive integrability exponents.

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arXiv2026

The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~{II}: Refined structure of the energy-concentration set

In this paper, we continue our study, started in~\cite{CDS1}, of a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\Q$-tensor for the liquid crystal component and a magnetisation vector field~$\M$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\Q$ and~$\M$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\eps$ tends to zero. While in~\cite{CDS1} we showed that the (rescaled) energy density for the~$\Q$-component concentrates, to leading order, on a finite number of singular points, in this paper we prove the energy density for the~$\M$-component concentrates along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component. Crucial to our arguments will be the energy estimates and compactness results proved in~\cite{CDS1}.

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arXiv2026

Liftings of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets, with an application to ferronematics

We consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We establish a sharp lower bound on the jump length of the lifting, expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities. As an application, we analyse minimisers of a two-dimensional model of ferronematics under ``mixed'' boundary conditions -- that is, Dirichlet conditions for the liquid crystal order parameter and Neumann conditions for the magnetisation vector.

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arXiv2025

Partial regularity for parabolic systems of double phase type

We study partial regularity for nondegenerate parabolic systems of double phase type, where the growth function is given by $H(z,s)=s^p+a(z)s^q$, $z=(x,t)\in\Omega_T$, with $\tfrac{2n}{n+2}<p\le q$ and $a(z)$ a nonnegative $C^{0,\alpha,\frac{\alpha}{2}}$-continuous function for some $\alpha\in(0,1]$. As the main result we prove that if $q< \min \{p+\tfrac{\alpha p }{n+2}, p+1 \}$ the spatial gradient of any weak solution is locally H\"older continuous, except on a set of measure zero.

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arXiv2025

The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~I: Energy estimates and compactness results

We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\Q$-tensor for the liquid crystal component and a magnetisation vector field~$\M$, both of them governed by a Ginzburg-Landau-type energy. The energy, the largest part of which is carried by the $\Q$-component, includes a singular coupling term favouring alignment between~$\Q$ and~$\M$. In this article and in the companion paper~\cite{CDS2}, we analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\eps$ tends to zero. In this paper, we prove that the (rescaled) energy density for the $\Q$-component, concentrates, to leading order, on a finite number of singular points. Moreover, we prove energy estimates and compactness results that will be crucially used in~\cite{CDS2} to determine the structure of the energy concentration set for the $\M$-component as well as the relationship between the two singular sets.

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arXiv2024

Partial regularity for degenerate systems of double phase type

We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by $H(x,t)=t^p+a(x)t^q$ with $1<p\leq q$ and $a(x)$ a nonnegative $C^{0,\alpha}$-continuous function. Our main result proves that if $\frac{q}{p}\leq 1+\frac{\alpha}{n}$, the gradient of any weak solution is locally H\"older continuous, except on a set of measure zero.

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arXiv2024

Variational Dual Solutions for Incompressible Fluids

We consider a construction proposed in \cite{acharyaQAM} that builds on the notion of weak solutions for incompressible fluids to provide a scheme that generates variationally a certain type of dual solutions. If these dual solutions are regular enough one can use them to recover standard solutions. The scheme provides a generalisation of a construction of Y$.$Brenier for the Euler equations. We rigorously analyze the scheme, extending the work of Y$.$Brenier for Euler, and also provide an extension of it to the case of the Navier-Stokes equations. Furthermore we obtain the inviscid limit of Navier-Stokes to Euler as a $\Gamma$-limit.

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arXiv2024

Quasistatic evolution of Orlicz-Sobolev nematic elastomers

We investigate the variational model for nematic elastomer proposed by Barchiesi and DeSimone with the director field defined on the deformed configuration under general growth conditions on the elastic density. This leads us to consider deformations in Orlicz-Sobolev spaces. Our work builds upon a previous paper by Henao and the Second Author, and extends their analysis to the quasistatic setting. The overall strategy parallels the one devised by the First author in the case of Sobolev deformations for a similar model in magnetoelasticity. We prove two existence results for energetic solutions in the rate-independent setting. The first result concerns quasistatic evolutions driven by time-dependent applied loads. For this problem, we establish suitable Poincar\'{e} and trace inequalities in modular form to recover the coercivity of the total energy. The second result ensures the existence of quasistatic evolution for both time-depend applied loads and boundary conditions under physical confinement. In its proof, we follow the approach advanced by Francfort and Mielke based on a multiplicative decomposition of the deformation gradient and we implement it for energies comprising terms defined on the deformed configuration. Both existence results rely on a compactness theorem for sequences of admissible states with uniformly bounded energy which yields the strong convergence of the composition of the nematic fields with the corresponding deformations. While proving it, we show the regular approximate differentiability of Orlicz-Sobolev maps with suitable integrability, thus generalizing a classical result for Sobolev maps due to Goffman and Ziemer.

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arXiv2023

Partial regularity for degenerate parabolic systems with general growth via caloric approximations

We establish a partial regularity result for solutions of parabolic systems with general $\varphi$-growth, where $\varphi$ is an Orlicz function. In this setting we can develop a unified approach that is independent of the degeneracy of system and relies on two caloric approximation results: the $\varphi$-caloric approximation, which was introduced in Diening, Schwarzacher, Stroffolini and Verde (2017) (arXiv:1606.01706), and an improved version of the \mathcal{A}-caloric approximation, which we prove without using the classical compactness method.

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arXiv2023

Manifold-constrained free discontinuity problems and Sobolev approximation

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\Omega \subset \R^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano \cite{CFI-ARMA, CFI-AIHP} for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.

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arXiv2023

Internal Schauder estimates for H\"ormander type equations with Dini continuous source

We study the regularity properties of a general second order H\"ormander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the H\"ormander condition, we consider the linear operator in $\mathbb{R}^{N}$, with $N>m+1$: \begin{equation*} \mathcal{L} u := \sum_{i, j= 1}^{m} a_{ij} X_{i}X_{j} u - X_0 u. \end{equation*} The vector field $X_0$ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathcal{L} u = f$ are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.

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arXiv2023

KFP operators with coefficients measurable in time and Dini continuous in space

We consider degenerate KFP operators \[ Lu=\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u\equiv\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+Yu \] ($(x,t)\in\mathbb{R}^{N+1}$, $1\leq m_{0}\leq N$) s.t. the model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a Lie group in $\mathbb{R}^{N+1}$ and $2$-homogeneous w.r.t. a family of dilations; $(a_{ij})_{i,j=1}^{m_{0}}$ is symmetric and uniformly positive on $\mathbb{R}^{m_{0}}$; $a_{ij}$ are bounded and Dini continuous in space, bounded measurable in time, i.e.: setting \[ S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) , \] \[ \omega_{f,S_{T}}(r)=\sup_{\substack{(x,t),(y,t)\in S_{T}\\\Vert x-y\Vert\leq r}}|f(x,t)-f(y,t)| \] \[ \Vert f\Vert_{\mathcal{D}(S_{T})}=\int_{0}^{1}\frac{\omega_{f,S_{T}}(r)}% {r}dr+\Vert f\Vert_{L^{\infty}\left( S_{T}\right) } \] we ask $\Vert a_{ij}\Vert_{\mathcal{D}(S_{T})}<\infty$. We bound $\omega_{u_{x_{i}x_{j}},S_{T}}$, $\left\Vert u_{x_{i}x_{j}}\right\Vert _{L^{\infty}(S_{T})}$ ($i,j=1,2,...,m_{0}$), $\omega_{Yu,S_{T}}$, $\Vert Yu\Vert_{L^{\infty}(S_{T})}$ in terms of $\omega_{\mathcal{L}u,S_{T}}$, $\Vert Lu\Vert_{L^{\infty}(S_{T})}$ and $\Vert u\Vert_{L^{\infty}\left( S_{T}\right) }$, getting a control on the uniform continuity in space of $u_{x_{i}x_{j}},Yu$ if $Lu$ is bounded and Dini-continuous in space. Under the additional assumption that $a_{ij}$ and $\mathcal{L}u$ are log-Dini continuous, meaning the finiteness of the quantity% \[ \int_{0}^{1}\frac{\omega_{f,S_{T}}\left( r\right) }{r}\left\vert \log r\right\vert dr, \] we prove that $u_{x_{i}x_{j}}$ and $Yu$ are Dini continuous; moreover, in this case, the derivatives $u_{x_{i}x_{j}}$ are locally uniformly continuous in space and time.

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arXiv2023

Regularity theory for parabolic systems with Uhlenbeck structure

We establish local regularity theory for parabolic systems of Uhlenbeck type with $\varphi$-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local H\"older continuity of the gradients, providing suitable assumptions on the growth function $\varphi$. Our approach, being independent of the degeneracy of the system, allows for a unified treatment of both the degenerate and the singular case.

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arXiv2022

A free discontinuity model for smectic thin films

We attempt to describe surface defects in smectic A thin films by formulating a free discontinuity problem - that is, a variational problem in which the order parameter is allowed to have jump discontinuities on some (unknown) set. The free energy functional contains an interfacial energy which penalizes dislocations of the smectic layers at the jump. We discuss mathematical issues related to the existence of minimizers and provide examples of minimizers in some simplified settings.

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arXiv2022

Two-dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections

We study a variational model for ferronematics in two-dimensional domains, in the "super-dilute" regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg-Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a suitable asymptotic regime, we prove that the nematic order parameter converges to a canonical harmonic map with non-orientable point defects, while the magnetisation converges to a singular vector field, with line defects that connect the non-orientable point defects in pairs, along a minimal connection.

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arXiv2022

Integral representation and $\Gamma$-convergence for free-discontinuity problems with $p(\cdot)$-growth

An integral representation result for free-discontinuity energies defined on the space $GSBV^{p(\cdot)}$ of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-H\"older continuity for the variable exponent $p(x)$. Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitt\`e, Fonseca, Leoni and Mascarenhas (2002) for a constant exponent. We prove $\Gamma$-convergence of sequences of energies of the same type, we identify the limit integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions.

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arXiv2022

Singular multiple integrals and nonlinear potentials

We prove sharp partial regularity criteria of nonlinear potential theoretic nature for the Lebesgue-Serrin-Marcellini extension of nonhomogeneous singular multiple integrals featuring $(p,q)$-growth conditions.

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