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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.

MSC:

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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