Annales Academię Scientiarum Fennicę

Mathematica

Volumen 33, 2008, 439-473

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}(*B*_{1}(0))
\cap *W*^{1,q}(*B*_{1}(0)).
There exists positive constants *C*_{1} << *C*_{2} >> 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)
*dL*^{2}*z* \leq
*C*_{1}\varepsilon,
\int_{B_{1}(0)}
|*D*^{2}*u*(z)|^{p}
*dL*^{2}*z* \leq
*C*_{1}\varepsilon^{1-p}

then there exist *A* \in *K* such that

(1) \int_{B_{1/2}(0)}
|*Du*(*z*) -
*A*|^{q}*dL*^{2}*z*
\leq *C*_{2}\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.