Related papers: Analysis of a free boundary problem modeling the g…
Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of…
We study the optimal control problem of a free boundary PDE model describing the growth of multilayered tumor tissue in vitro. We seek the optimal amount of tumor growth inhibitor that simultaneously minimizes the thickness of the tumor…
Growing experimental evidence highlights the relevant role of mechanics in the physiology of solid tumours, even in their early stages. While most of the mathematical models describe tumour growth as a volumetric increase of mass in the…
A novel numerical technique has been proposed to solve a two-phase tumour growth model in one spatial dimension without needing to account for the boundary dynamics explicitly. The equivalence to the standard definition of a weak solution…
We derive a thermodynamically consistent model, which describes the time evolution of a two-phase flow in an evolving domain. The movement of the free boundary of the domain is driven by the velocity field of the mixture in the bulk, which…
We study the two-phase Muskat--Verigin free-boundary problem for elliptic equations with nonlinear sources. The existence of a smooth solution and a smooth free boundary is proved locally in time by applying the parabolic regularization of…
We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of…
A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely…
Tumor growth has a number of features in common with a physical process known as molecular beam epitaxy. Both growth processes are characterized by the constraint of growth development to the body border, and surface diffusion of…
We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface…
We study a free boundary problem modeling tumor growth with a T-periodic supply $\Phi(t)$ of external nutrients. The model contains two parameters $\mu$ and $\widetilde{\sigma}$. We first show that (i) zero radially symmetric solution is…
The work presents the analysis of the free boundary value problem related to the invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of…
In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the…
We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model derived from taking the incompressible limit of a cell density model. We analyze the boundary stability/instability in two…
The inviscid barotropic quasi-geostrophic equation with a free surface is considered. The free surface mandates a non-standard boundary condition. The global existence existence and uniqueness of a weak solution is established, thanks to…
A well-known question in classical differential geometry and geometric analysis asks for a description of possible boundaries of $K$-surfaces, which are smooth, compact hypersurfaces in $\mathbb{R}^d$ having constant Gauss curvature equal…
We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric…
We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone…
In this thesis, we present various contributions to the study of free boundary minimal surfaces. After introducing some basic tools and discussing some delicate aspects related to the definition of Morse index when allowing for a contact…
Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free stream turbulence are considered in the present work. Within an input-output model where non-linear terms of the…