Variational Problems with Concentration

approximation assume asymptotic BANDLE Bernoulli's free-boundary problem boundary boundary element method calculus of variations concentration compactness alternative concentration limit concentration point conformal radius defined denotes Differential Equations dimensions Dirichlet problem domain elliptic entire extremals Euler Lagrange equation exterior ball condition extremal functions F(ue F(uɛ F(we Figure finite Floc FLUCHER follows formula G₂ harmonic center hydrodynamic Green's function hydrodynamic Robin function hyperbolic solutions isoperimetric inequality L¹(N Lagrange multiplier Lemma linear Liouville's equation low energy extremals lower bound Math maximizing sequence method minimal motion nonlinear obtain optimal p-capacity potential p-Green's p-harmonic radius p-harmonic transplantation p-Robin Partial Differential Equations pcap positive solutions Proof properties radial renormalized satisfies semilinear single vortex singularity small balls Sobolev inequality stream function subconformal symmetrization Theorem 9.2 Trudinger-Moser inequality variational problem volume integrand vortex vortex centers vortices yields