Additional material

As discussed in the lecture on January 16, the Lavrentiev phenomenon is the somehow unexpected situation that the minimisation of a variational functional over the space of Lipschitz functions might yield a value that is strictly larger than minimisation over absolutely continuous functions (or similar function classes). That is, for integral functionals of the form \[ J(u) = \int_0^1 j(x,u(x),u'(x))\,dx \] it can happen that \[ \inf_{u \in W^{1,1}(0,1)} J(u) < \inf_{u \in W^{1,\infty}(0,1)} J(u). \] One of the first examples in this direction, namely the functional \[ J(u) = \int_0^1 (x-u(x)^3)^2 (u'(x))^6\,dx \] is due to Mania.¹⁾ Here it is easy to see that the function \(u(x) = x^{1/3}\) is the unique minimiser over \(W^{1,1}(0,1)\) satisfying the boundary conditions \(u(0) = 0\) and \(u(1) = 1\). However, one can show that \(J(u) \ge 7^2 3^5 2^{-18} 5^{-5}\) for all Lipschitz functions \(u\). A detailed proof of this estimate can be found in [Section 4.7, B. Dacorogna, Direct Methods in the Calculus of Variations, volume 78 of Applied Mathematical Sciences. Springer, New York, second edition, 2008].

In the lecture, we have briefly mentioned a different, more complicated example due to Ball and Mizel²⁾ of the form \[ J(u) = \int_{-1}^1 (x^4-u(x)^6)^2 (u'(x))^{28} + \varepsilon (u'(x))^2\,dx, \] which has the additional advantage, that the minimiser is actually an element of \(H^1(-1,1)\) and that the integrand is strongly convex in the \(u'\)-variable.³⁾ However, the analysis of this example is much more complicated.

One of the difficulties with such functionals is the fact that the minimiser in \(W^{1,1}\) or \(H^1\) cannot be found (or approximated) with standard finite element methods, since in these methods one (usually) works with ansatz functions that are Lipschitz continuous. Even more, there might be no actual possibility to detect the failure of the FE method, as it appears to converge - however, the limit is no actual solution of the optimisation problem.