Annales Academię Scientiarum Fennicę
Mathematica
Volumen 33, 2008, 439-473

AN L^p TWO WELL LIOUVILLE THEOREM

Andrew Lorent

Scuola Normale Superiore, Centro De Giorgi, Collegio Puteano
Piazza dei Cavalieri, 3, I-56100 Pisa, Italy; andrew.lorent 'at' sns.it

Abstract. We provide a different approach to and prove a (partial) generalisation of a recent theorem on the structure of low energy solutions of the compatible two well problem in two dimensions [Lor05], [CoSc06]. More specifically we will show that a "quantitative" two well Liouville theorem holds for the set of matrices K = SO(2) \cup SO(2)H where H = (\begin{smallmatrix} \sigma & 0\\ 0 & \sigma^-1 \end{smallmatrix}) under a constraint on the L^p norm of the second derivative. Our theorem is the following.

Let p \geq 1, q > 1. Let u \in W^2,p(B₁(0)) \cap W^1,q(B₁(0)). There exists positive constants C₁ << 1, C₂ >> 1 depending only on \sigma, p, q such that if u satisfies the following inequalities

\int_{B_{1/2}(0)} d^q(Du(z),K) dL²z \leq C₁\varepsilon, \int_{B_{1}(0)} |D²u(z)|^p dL²z \leq C₁\varepsilon^1-p

then there exist A \in K such that

(1) \int_{B_{1/2}(0)} |Du(z) - A|^qdL²z \leq C₂\varepsilon^1/2q.

We provide a proof of this result by use of a theorem related to the isoperimetric inequality, the approach is conceptually simpler than those previously used in [Lor05], [CoSc06], however it does not given the optimal c\varepsilon^1/q bound for (1) that has been proved (for the p = 1 case) in [CoSc06].

2000 Mathematics Subject Classification: Primary 74N15.

Key words: Two wells, Liouville.

Reference to this article: A. Lorent: An L^p two well Liouville theorem. Ann. Acad. Sci. Fenn. Math. 33 (2008), 439-473.

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