In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let $H=\left(\begin{array}{ccc}\sigma & 00& {\sigma }^{-1}\end{array}\right)$ for $\sigma >0$. Let $0<{\zeta }_1<1<{\zeta }_2<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in W^{2,1}\left(Q_1\left(0\right)\right)$ be a $$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<{\zeta }_2$, $\mathrm{Lip}\left(u^{-1}\right)<{\zeta }_1^{-1}$. 
There exists positive constants ${𝔠}_1<1$ and ${𝔠}_2>1$ depending only on , ${\zeta }_1$, ${\zeta }_2$ such that if ...

In this work we consider the magnetic NLS equation $${(\frac{\hslash }{i}\nabla -A\left(x\right))}^{2}u+V\left(x\right)u-{f\left(\right|u|}^2)u=0\text{in}{ℝ}^N(0.1)$$ where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ is a magnetic potential, possibly unbounded, $V:{ℝ}^N\to ℝ$ is a multi-well electric potential, which can vanish somewhere, is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u:{ℝ}^N\to ℂ$ to (0.1), under conditions on the nonlinearity which are nearly optimal.

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{u}_{\epsilon }\right)$ bounded in $L^p(\Omega ;{ℝ}^9),1<p<+\infty .$ Here it is shown that, up to a subsequence, $u_{\epsilon }$ may be decomposed as $w_{\epsilon }+z_{\epsilon },$ where $z_{\epsilon }$ carries all the concentration effects, $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{w}_{\epsilon }\right)\right|}^p\right\}$ is equi-integrable, and $w_{\epsilon }$ captures the oscillatory behavior, $z_{\epsilon }\to 0$ in measure. In addition, if $\left\{u_{\epsilon }\right\}$ is a recovering sequence then $z_{\epsilon }=z_{\epsilon }\left(x_{\alpha }\right)$ nearby $\partial \Omega .$

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure R and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis...