Related papers: Obstacle problems and free boundaries: an overview
In this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDEs. When the system is open, there are several mechanisms to couple the system with the…
This brief note addresses the free boundary problem arising from the steady two-dimensional seepage flow through a rectangular dam. The flow problem consists in finding the free boundary location, and the velocity and pressure fields. The…
We study a linear problem that arises in the study of dynamic boundaries, in particular in free boundary problems in connection with fluid dynamics. The equations are also very natural and of interest on their own.
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the…
In this survey we go through some of the recent results about the regularity of vectorial free boundary problems of Bernoulli type and free boundary systems. The aim is to illustrate the general methodologies as well as to outline a…
We study the isoperimetric problem in product spaces equipped with the uniform distance. Our main result is a characterization of isoperimetric inequalities which, when satisfied on a space, are still valid for the product spaces, up a to a…
The aim of this work is to point out that the class of free boundary problems governed by second order autonomous ordinary differential equations can be transformed to initial value problems. Interest in the numerical solution of free…
We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in various two dimensional settings. We assume that $V>0$ is a given function, $\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$ as…
Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of…
The aim of this paper is first to give necessary and sufficient condition of existence (of free boundaries) for both Laplacian and bi-Laplacian operators in the case where the overdetermined condition is not constant. second, by using some…
We study obstacle problems governed by two distinct types of diffusion operators involving interacting free boundaries. We obtain a somewhat surprising coupling property, leading to a comprehensive analysis of the free boundary. More…
In this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators.
We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, $\mbox{L}_{+}$ and $\mbox{L}_{-}$, within their positive and negative sets…
We study free boundary problems for incompressible inhomogeneous flows governed by the Navier--Stokes equations, focusing on the regularity and global-in-time well-posedness of solutions in critical functional frameworks for small initial…
In a wide class of the so called Obstacle Problems of parabolic type it is shown how to improve the optimal regularity of the solution and as a consequence how to obtain space-time regularity of the corresponding free boundary.
The non-transversal intersection of the free boundary with the fixed boundary is obtained for nonlinear uniformly elliptic operators when $\Omega = \{\nabla u \neq 0\} \cap \{x_n>0\}$ thereby solving a problem in elliptic theory that in the…
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type…
In this paper, we have defined the free boundary formulation for two extended Blasius problems. These problems are of interest in boundary layer theory and are deduced from the governing partial differential equations by using appropriate…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…