Remarks on the Grad-Shafranov equation. (English) Zbl 0772.35085

The paper presents an elegant existence proof for the following free boundary problem related to plasma confinement. Given a smooth bounded domain \(\Omega\subset\mathbb{R}^ N\) and a positive constant \(I\), find \(a\geq 0\), a subset \(\Omega_ p\subset\Omega\) and a function \(v\in C^ 1(r)\cap C^ 2(\Omega\backslash\partial\Omega_ p)\) such that \(-\Delta v=\sigma(v)\), \(v\geq 0\) on \(\Omega_ p\), \(v=0\) on \(\partial\Omega_ p\), \(-\Delta v=0\) on \(\Omega\backslash\Omega_ p\), \(v=-a\) on \(\partial\Omega\), \(-\int_{\partial\Omega}(\partial v/\partial n)=I\), \(n\) representing the outer normal. Typical assumptions on \(\sigma\) (besides monotonicity and regularity) are \(\sigma(s)\leq\alpha s+\beta\), \(\beta>0\), \(\alpha\) less than the first eigenvalue of \(-\Delta\) in \(H^ 1_ 0(\Omega)\); \(\sigma(0+)=b>0\).
Under some symmetry assumptions on \(\partial\Omega\) and supposing that \(b\cdot\text{meas}|\Omega|>I\), a solution is shown to exist such that \(\Omega_ p\) is strictly contained in \(\Omega\). The proof is obtained by means of a limiting procedure stemming from bifurcation techniques and topological arguments.


35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
82D10 Statistical mechanics of plasmas
35B32 Bifurcations in context of PDEs
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