Related papers: On one phase free boundary problem in $\mathbb{R}^…
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…
The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained…
In this paper, we consider the following free boundary problem $$ (P)\left\{\begin{array}{ll} \Delta u = \lambda \phi(x)\Sum_{i=1}^n H(u-\mu_i )& \quad \mbox{ in }\ \Omega=\Omega_2\setminus \overline{\Omega}_1, \\[0.3cm]u =0 &\quad \mbox{…
In this article we establish the exact growth of the solution to the singular quasilinear p-parabolic free boundary problem in non-divergence form near the free boundary from which follows its porosity.
Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and…
In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…
In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…
The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it…
We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in $C^1$ and $C^{k,\alpha}$ domains, providing that the quotient of two…
In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the "vanishing order" of a solution at a free boundary point is close to $3/2$ or an…
In this paper we first introduce an innovative equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we then present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann…
Let (M,g) be a smooth compact, n dimensional Riemannian manifold,with smooth n-1 dimensional boundary. We prove that the stable critical points of the mean curvature of the boundary generates solutions for a singularly perturbed elliptic…
We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$. The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha}$-regular free boundaries by…
For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion…
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 paper we propose a Hamiltonian approach to gapped topological phases on an open surface with boundary. Our setting is an extension of the Levin-Wen model to a 2d graph on the open surface, whose boundary is part of the graph. We…
We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region {\Omega} for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth…
We develop a numerical method to estimate the average speed of the free boundary in a Hele-Shaw problem with periodic coefficients in both space and time. We test the accuracy of the method and present a few examples. We show numerical…
This paper investigates the regularity of solutions and structural properties of the free boundary for a class of fourth-order elliptic problems with Neumann-type boundary conditions. The singular and degenerate elliptic operators studied…
Interpolating scaling functions give a faithful representation of a localized charge distribution by its values on a grid. For such charge distributions, using a Fast Fourier method, we obtain highly accurate electrostatic potentials for…