MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

An $L^p$ two well Liouville Theorem

Andrew Lorent


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, proved first by Lorent (for bilipschitz invertible mapping with control of order $\varepsilon^{\frac{1}{800}}$) then later by by Conti, Schweizer in full generality with optimal control. More specifically we will show that a "quantitative" two well Liouville theorem holds for the set of matrices $K=SO\left(2\right)\cup SO\left(2\right)H$ where $H$ is a diagonal matrix with determinant equal to $1$, under a constraint on the $L^p$ norm of the second derivative. Our theorem is the following.

Let $p\geq 1$, $q> 1$. Let $u\in W^{2,p}\left(B_1\left(0\right)\right)\cap W^{1,q}\left(B_1\left(0\right)\right)$. There exists positive constants $\mathcal{C}_1<1,\mathcal{C}_2>>1$ depending only on $\sigma$, $p$, $q$ such that if $u$ satisfies the following inequalities $$\label{aa1} \int_{B_{1}\left(0\right)} d^q\left(Du\left(z\right),K\right) dL^2 z\leq \mathcal{C}_1\varepsilon,\;\;\;\int_{B_{1}\left(0\right)} \left|D^2 u\left(z\right)\right|^p dL^2 z\leq \mathcal{C}_1 \varepsilon^{1-p} $$ then there exist $A\in K$ such that \begin{equation} \label{aa3} \int_{B_{\frac{1}{2}}\left(0\right)} \left|Du\left(z\right)-A\right|^q dL^2 z\leq \mathcal{C}_2\varepsilon^{\frac{1}{2q}}. \end{equation} 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, however it does not given the optimal $c\varepsilon^{\frac{1}{q}}$ bound for (\ref{aa3}) that has been proved (for the $p=1$ case) by Conti, Schweizer.

MSC Codes:
two wells, Liouviille

Related publications

2008 Journal Open Access
Andrew Lorent

An L-p two well Liouville theorem

In: Annales Academiae Scientiarum Fennicae / Mathematica, 33 (2008) 2, pp. 439-473