Blow-up analysis for the prescribed mean curvature equation on \(\mathbb R^2\)
The \(H\)-system, for a given function \(H:\mathbb R^3\to\mathbb R\), is given by \(\Delta u=2H(u)u_x\wedge u_y\) for \(u:\mathbb R^3\to\mathbb R^3\). If \(u\) is conformal, it describes a surface in \(\mathbb R^3\) which has prescribed mean curvature \(H(u)\) in every point. In the current paper, sequences of approximate solutions to the system (without the requirement of \(u\) being conformal) are considered, that is sequences with uniformly bounded Dirichlet integrals and \(\Delta u^n-2H(u^n)u^n_x\wedge u^n_y\to 0\) in the dual of some weighted Sobolev space natural for the problem. For such a sequence, the following is proven: Either \(\nabla u^n\to 0\) strongly in \(L^2\), or the sequence can be decomposed (using a different sequence of conformal transformations of the plane for each bubble) in the limit to give a finite set of \(\widetilde{H}_i\)-bubbles, i.e.\ solutions of the \(\widetilde{H}_i\)-system (for suitable \(\widetilde{H}_i\) derived from \(H\)) on \(\mathbb R^2\). The set of bubbles is complete in the sense that their Dirichlet energies add up to the limit energy of the sequence.
scientific article; zbMATH DE number 5577358
scientific article