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Maria Rosaria Lancia

Publications and source records attributed to Maria Rosaria Lancia.

9 recordsLinked to original sources
arXiv2025

Non-autonomous semilinear fractional evolution equations: well-posedness and ultracontractivity results

We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of the problem at hand. Then we prove, under suitable assumptions on the initial datum, that the solution is also global in time. This is achieved by proving ultracontractivity estimates for the fractional evolution families associated with the operator $A(t)$.

math.AP↗
arXiv2025

Identification problems for anisotropic time-fractional subdiffusion equations

We investigate the inverse problem consisting in the identification of constant coefficients for a fractional-in-time partial differential equation governed by a finite sum of positive self-adjoint operators on a Hilbert space under energy-type overdeterminating conditions. We prove the uniqueness of the solution to the inverse problem when the fractional order $α$ of the derivative is in $(0,1)$. A conditioned existence result is also provided, complemented with a suitable selection of numerical calculations. In addition, we prove that, as $α\to 1^{-}$, the solution corresponding to $α$ tends to the classical one ($α=1$). Applications to examples of heat diffusion and elasticity are presented.

math.AP↗
arXiv2025

Non-local boundary energy forms for quasidiscs: Codimension gap and approximation

We consider non-local energy forms of fractional Laplace type on quasicircles and prove that they can be approximated by similar energy forms on polygonal curves. The approximation is in terms of generalized Mosco convergence along a sequence of varying Hilbert spaces. The domains of the energy forms are the natural trace spaces, and we focus on the case of quasicircles of Hausdorff dimension greater than one. The jump in Hausdorff dimension results in a mismatch of fractional orders, which we compensate by a suitable choice of kernels. We provide approximations of quasidiscs by polygonal $(\varepsilon,\infty)$-domains with common parameter $\varepsilon>0$ and show convergence results for superpositions of Dirichlet integrals and non-local boundary energy forms.

math.FA↗
arXiv2023

Dynamic boundary conditions for time dependent fractional operators on extension domains

We consider a parabolic semilinear non-autonomous problem $(\tilde P)$ for a fractional time dependent operator $\mathcal{B}^{s,t}_Ω$ with Wentzell-type boundary conditions in a possibly non-smooth domain $Ω\subset\mathbb{R}^N$. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem $(P)$ via an evolution family $U(t,τ)$. We then prove that the mild solution of the abstract problem $(P)$ actually solves problem $(\tilde P)$ via a generalized fractional Green formula.

math.AP↗
arXiv2018

Magnetostatic problems in fractal domains

We consider a magnetostatic problem in a 3D "cylindrical" domain of Koch type. We prove existence and uniqueness results for both the fractal and pre-fractal problems and we investigate the convergence of the pre-fractal solutions to the limit fractal one. We consider the numerical approximation of the pre-fractal problems via FEM and we prove a priori error estimates. Some numerical simulations are also shown. Our long term motivation includes studying problems that appear in quantum physics in fractal domains.

math-ph↗
arXiv2016

Fractal snowflake domain diffusion with boundary and interior drifts

We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial Ventsell problem. As a second result we prove that functions that are intrinsically Lipschitz on the snowflake boundary admit Euclidean Lipschitz extensions to the closure of the entire domain. Our methods combine the fractal membrane analysis, the vector analysis for local Dirichlet forms and PDE on fractals, coercive closed forms, and the analysis of Lipschitz functions.

math.AP↗