Related papers: Obstacle problems and free boundaries: an overview
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…
Real physical systems are only understood, experimentally or theoretically, to a finite resolution so in their analysis there is generally an ignorance of possible short-range phenomena. It is also well-known that the boundary conditions of…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
Free bondary value problem for elliptic differential-operator equations with variable coefficients is studied. The uniform maximal regularity properties and Fredholmness of this problem are obtained in vector-valued Holder spaces.
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
The semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove existence of solutions of the incompressible semi-geostrophic equations in a fully three-dimensional domain with a free…
The boundary conditions at the deformable interface between two contacting fluids are derived for the general case of the large-amplitude perturbations. The interface is modeled as perturbed free boundary that evolves in time, and the…
A new boundary value problem for partial differential equations is discussed. We consider an arbitrary solution of an elliptic or parabolic equation in a given domain and no boundary conditions are assumed. We study which restrictions the…
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between…
One-dimensional free boundary problem for a nonlinear diffusion - convection equation with a Dirichlet condition at fixed face $x=0$, variable in time, is considered. Throught several transformations the problem is reduced to a free…
The Oxygen Depletion problem is an implicit free boundary value problem. The dynamics allow topological changes in the free boundary. We show several mathematical formulations of this model from the literature and give a new formulation…
We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the…
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured…
State-dependent parameter identification, where unknown model parameters depend on one or more state variables in partial differential equations (PDEs) or coupled PDE systems, is fundamental to a wide range of problems in physics,…
In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the…
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in…
We consider the problem of bounding large deviations for non-i.i.d. random variables that are allowed to have arbitrary dependencies. Previous works typically assumed a specific dependence structure, namely the existence of independent…
In this paper, we study a free boundary problem for a class of nonlinear nonautonomous size structured population model. Using the comparison principle and upper lower solution methods, we establish the existence of the solution for such…
We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more…
The interior free boundary theory for linear elliptic operators in higher dimensions was developed by Caffarelli in the low regularity context. In these notes, the up-to-the boundary free boundary regularity is discussed for nonlinear…