Weak limit and blowup of approximate solutions to H-systems

Abstract

Let H : R 3 → R be a continuous function such that H (p) → H 0 ∈ R as |p| → +∞. Fixing a domain Ω in R 2 we study the behaviour of a sequence (u n ) of approximate solutions to the H -system u = 2H (u)u x ∧ u y in Ω. Assuming that sup p∈R 3 |(H (p) − H 0 )p| < 1, we show that the weak limit of the sequence (u n ) solves the H -system and u n → u strongly in H 1 apart from a countable set S made by isolated points. Moreover, if in addition H (p) = H 0 + o(1/|p|) as |p| → +∞, then in correspondence of each point of S we prove that the sequence (u n ) blows either an H -bubble or an H 0 -sphere.

Blow-up analysis for the prescribed mean curvature equation onR2

“…In view of the Poincaré inequality, (v n ) is bounded inĤ 1 (R 2 , R 3 ) and then, up to a subsequence, converges weakly to some U ∈Ĥ 1 ( [10] where, in agreement with the previous claim, a single blow-up phenomenon is described for mappings H : R 3 → R of the form…”

“…A proof of Theorem 2.3 can be found in [10]. In particular the case (p n ) bounded corresponds to Lemma 2.2 in the above mentioned paper.…”

“…In particular the case (p n ) bounded corresponds to Lemma 2.2 in the above mentioned paper. The case |p n | → +∞ follows from Lemma 3.3, steps 1 and 2 in the proof of Lemma 3.4, and Remark 2.3 in [10].…”

“…Under the assumptions made on H an H -bubble is in fact bounded (see, e.g., [10] or [16]). As soon as the prescribed mapping H is slightly more regular than continuous, e.g.…”

“…In Section 2 we state some useful results concerning H -bubbles and H -systems; in particular we recall a local "ε-compactness" theorem for sequences of approximate solutions, already proved in [10], which constitutes one of the main tools in the argument. Section 3 contains the proof of Theorem 0.1.…”