Related papers: Sharp estimates, uniqueness and spikes condensatio…
We prove that spike sequences related to a nonlinear problem of Grad-Shafranov type are always simple and always converge toward interior points of the domain. This sharpens the blow-up analysis carried out by Bartolucci-Jevnikar-Wu [Calc.…
We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the…
We classify the singular limits relative to a free boundary problem arising in plasma physics in dimension $d=2$, under suitable natural integral bounds. It turns out that one of the asymptotic behaviors allowed corresponds to the…
In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set…
For $\Omega\subset \mathbb{R}^2$ a smooth and bounded domain, we derive a sharp universal energy estimate for non-negative solutions of free boundary problems on $\Omega$ arising in plasma physics. As a consequence, we are able to deduce…
In this paper we generalize an equation studied by Mossino and Temam, to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of…
For any $\Omega\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $\Omega$ in a neat interval depending only by the best constant of the Sobolev…
The problem of equilibrium of a plasma in a Tokamak is a free boundary problemdescribed by the Grad-Shafranov equation in axisymmetric configurations. The right hand side of this equation is a non linear source, which represents the…
The Grad-Shafranov (GS) equation is a nonlinear elliptic partial differential equation that governs the ideal magnetohydrodynamic equilibrium of a tokamak plasma. Previous studies have demonstrated the existence of multiple solutions to the…
We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy…
Equilibria in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for…
We consider a class of Fokker--Planck equations with linear diffusion and superlinear drift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite…
The reconstruction of the equilibrium of a plasma in a Tokamak is a free boundary problem described by the Grad-Shafranov equation in axisymmetric configuration. The right-hand side of this equation is a nonlinear source, which represents…
The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for…
We discuss a new family of solutions of the Grad--Shafranov (GS) equation that describe D-shaped toroidal plasma equilibria with sharp gradients at the plasma edge. These solutions have been derived by exploiting the continuous Lie symmetry…
A class of nonlinear Fokker-Planck equations with superlinear drift is investigated in the $L^1$-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex…
We study the free boundary in the supercooled Stefan problem, a classical model for the solidification of water below its freezing temperature. In contrast with the melting problem, physical experiments and heuristics indicate that the…
We are concerned with the global bifurcation analysis of positive solutions to free boundary problems arising in plasma physics. We show that in general, in the sense of domain variations, the following alternative holds: either the shape…
We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega \subset…
A class of different schemes for the numerical solving of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of…