Nonconvex Variational Problems

A variational problem for a functional $\mathcal {E}$ : $\mathcal {A}$ $\rightarrow$ $\mathbb {R}$ $\cup$ { + $\infty$} is showing the existence of a minimizer u^* $\in$ $\mathcal {A}$ where the infimum of $\mathcal {E}$[u] over u $\in$ $\mathcal {A}$ is attained. Here $\mathcal {A}$ is the admissible set (usually a subset of an appropriate function space) and $\mathcal {E}$ is typically an integral of a local energy density W(x, u, Du) over the domain $\mathcal {S}$ :

$$\mathcal {E} \int_{{\mathcal{S}}}^{}$$ (VP)

A natural approach is to consider a minimizing sequence u_k $\in$ $\mathcal {A}$ (a sequence such that $\mathcal {E}$[u_k] $\rightarrow$ $\inf_{{u \in \mathcal{A}}}^{}$$\mathcal {E}$[u]). If $\mathcal {E}$ (or equivalently W) is convex, u_k (or a subsequence) converges to a minimizer u^* [Eva98,Dac89]. However, for many interesting variational problems that arise in physics and geometry, W is nonconvex. In this case, u_k may only converge weakly, and the variational problem (VP) need not have a minimizer. Instead, the minimizing sequence might exhibit oscillations and/or concentration.

Models for real physical systems cannot exhibit arbitrarily fine scale structures. So, we consider the regularized functional $\mathcal {E}$^$\epsilon$ obtained by perturbing (VP),

$$\mathcal {E} \epsilon \mathcal {E} \epsilon^{2}_{} \int_{{\mathcal{S}}}^{}$$ (R-VP)

where F is convex in D²u. For $\epsilon$ = 0, we recover the original functional $\mathcal {E}$. For any nonzero $\epsilon$ however, the second term penalizes structures on very fine scales. This effectively introduces a small scale cutoff and regularizes the original nonconvex variational problem (VP). For every $\epsilon$ $\neq$ 0, $\mathcal {E}$^$\epsilon$ has a minimizer u^$\epsilon$. The following questions arise in studying the $\epsilon$ $\rightarrow$ 0 limit of the problems (R-VP):

Morphology

Do the minimizers u^$\epsilon$ display oscillations or concentration effects? Describe the oscillatory/singular regions in the solutions.

Scaling

How do $\mathcal {E}$^$\epsilon$(u^$\epsilon$), and the length scales associated with the oscillations/singular regions in the minimizer u^$\epsilon$ depend on $\epsilon$?

Variational Convergence

Does $\lim_{{\epsilon \rightarrow 0}}^{}$u^$\epsilon$ exist, in some appropriate sense? If so, is it possible to define a suitable limiting process for the energy functionals, so that $\lim_{{\epsilon \rightarrow 0}}^{}$$\mathcal {E}$^$\epsilon$ $\rightarrow$ $\bar{{\mathcal{E}}}$, implies that $\bar{{\mathcal{E}}}$ has a minimizer $\lim_{{\epsilon \rightarrow 0}}^{}$u^$\epsilon$?

The answers to these questions depend on the precise functional forms of W and F. As $\epsilon$ $\rightarrow$ 0, we expect that Du^$\epsilon$ converges to K, the zero set of W(.) in some fashion. In particular we are looking for gradients v = Du such that v $\in$ K. Whether such gradient fields v exist depend on a subtle interplay between the differential constraint Dv = Dv^T and the geometry of K.

If we ignore the fact that v is a gradient, Eq. (R-VP) is related to the harmonic map v : $\mathcal {S}$ $\rightarrow$ K, given by minimizing $\int$F(v, Dv)dx. The variational problem (R-VP) can thus be interpreted as a Harmonic map problem with an additional differential constraint. The variational problem for (R-VP) is also related to the recent work by Müller and Sverák on obtaining counterexamples to regularity using convex integration.

Another motivation for this study is condensed matter physics. I am interested in the general principles that underlie the spontaneous formation of structures in the minimizers for variational problems. Many physical systems are governed by the minimization of an appropriate free energy. This energy usually has many scales reflecting the multiplicity of the relevant physical processes. Energy minimization can then lead to a variety of multiple scale structures including point and line singularities (vortices), microstructure, domain wall branching and multiple scale oscillations, and this behavior can be analyzed through the model problem (R-VP).

Energy Concentration and Minimal Obstructions

Consider energy functionals of the type $\mathcal {E}$^$\epsilon$ = $\int$Wdx + $\epsilon^{2}_{}$$\int$Fdx. We will assume that W is nonconvex, F is convex in a higher order derivative, W, F $\geq$ 0 by definition and $\lim_{{\epsilon \rightarrow 0}}^{}$$\mathcal {E}$^$\epsilon$(u^$\epsilon$) = 0. These assumptions are satisfied, for instance, by the generalized crumpling problem.

If $\epsilon$ is small, then W for the minimizer u^$\epsilon$, is small over a large part of the domain. This motivates the consideration of the set

$$\mathcal {A}$$' = $$\left\{\vphantom{ u \in \mathcal{A}, \int_{\mathcal{S}} F dx < \infty \mbox{ and } W = 0 \mbox{ a.e.} }\right.$$u $$\in$$ $$\mathcal {A}$$,$$\int_{{\mathcal{S}}}^{}$$Fdx < $$\infty$$ and W = 0 a.e.$$\left.\vphantom{ u \in \mathcal{A}, \int_{\mathcal{S}} F dx < \infty \mbox{ and } W = 0 \mbox{ a.e.} }\right\}$$.

Conjecture 1   If $\mathcal {A}$' $\neq$ $\emptyset$, then, as $\epsilon$ $\rightarrow$ 0, the solutions u^$\epsilon$ converge to the minimizer of the convex variational problem - Minimize $\int$Fdx for u $\in$ $\mathcal {A}$'.

If $\mathcal {A}$' = $\emptyset$ and the minimizing sequence u^$\epsilon$ does not contain fine scale oscillations, then u^$\epsilon$ and some of its derivatives still converge pointwise a.e. to the set W = 0. However, there is now an exceptional set of points, the singular set $\mathcal {R}$, where some of the higher derivatives diverge in a manner such that

$$\limsup_{{\epsilon \rightarrow 0}}^{}$$$$\int_{O}^{}$$Fdx = + $$\infty$$,

for any open set O $\supset$ $\mathcal {R}$. This leads to energy concentration on the set $\mathcal {R}$ as $\epsilon$ $\rightarrow$ 0. Since points in $\mathcal {S}$ - $\mathcal {R}$ are regular, the previous argument restricted to $\mathcal {S}$ - $\mathcal {R}$ gives

$$\mathcal {A}$$'_loc($$\mathcal {R}$$) = $$\left\{\vphantom{ u \in \mathcal{A}, \int_{\mathcal{S} - O} F dx ... ...y \quad \forall O \supset \mathcal{R}, \mbox{ and } W = 0 \mbox{ a.e.} }\right.$$u $$\in$$ $$\mathcal {A}$$,$$\int_{{\mathcal{S} - O}}^{}$$Fdx < $$\infty$$    $$\forall$$O $$\supset$$ $$\mathcal {R}$$, and W = 0 a.e.$$\left.\vphantom{ u \in \mathcal{A}, \int_{\mathcal{S} - O} F dx <... ... \quad \forall O \supset \mathcal{R}, \mbox{ and } W = 0 \mbox{ a.e.} }\right\}$$ $$\neq$$ $$\emptyset$$.

We define $\mathcal {R}$ to be a minimal obstruction, if $\mathcal {A}$'_loc($\mathcal {R}$) $\neq$ $\emptyset$, and no proper subset of $\mathcal {R}$ has this property. These arguments suggest the following conjecture :

Conjecture 2   The minimizing sequence does not have oscillations/concentration effects iff $\mathcal {A}$' is nonempty. Also, if $\mathcal {A}$' = $\emptyset$ and the minimizing sequence does not have oscillations, the energy concentrates on a (finite union of) minimal obstruction(s).

This conjecture addresses the issue of Morphology in the list above. It also relates analytical questions about the family of variational problems (R-VP) to geometric/topological questions about the set of functions {u  |  W = 0,$\int$Fdx < $\infty$}.