Relaxation and Young measures

On the interval $[0,1]$, take functions in $W_0^{1,4}$, meaning the function and weak derivative have integrable fourth powers and the endpoint traces are zero. Use the nonnegative energy density $(v^2-1)^2+u^2$. Its gradient part is zero at slopes plus or minus one, but costs one at slope zero. A triangular shape with alternating slopes can satisfy the endpoint conditions while spending no gradient energy. Refining the triangles makes its displacement term small without moving the derivative into the higher-cost zero-slope state.

Divide the interval into equally sized cells and make one positive triangular tent in each cell, rising with slope one for half the cell and falling with slope minus one for the rest. The heights tend uniformly to zero as the number of cells grows. The derivatives oscillate with zero mean and converge weakly to zero in the fourth-power integrable space; this follows by cancellation against continuous tests and density for the dual test space. Yet their squared values stay one almost everywhere, illustrating that nonlinear functions of weakly convergent quantities need not converge to the function of the weak limit.

For this sequence, the local gradient Young measure is the probability measure putting equal mass at minus one and plus one. Its barycenter is zero, matching the weak derivative of the limiting shape. Averaging the nonlinear gradient density against the measure gives zero, while evaluating that density at the barycenter gives one. The measure describes spatial proportions of fine oscillations, not measurement uncertainty or a claim that the material randomly chooses a slope at each point. It retains information lost by the weak limit alone.

On this bounded interval, a sequence bounded in $W^{1,4}(0,1)$ has a weakly convergent subsequence and, after further extraction, its derivatives generate a measurable family of probability measures whose barycenters equal the weak derivative almost everywhere. For bounded continuous gradient test functions, their compositions converge weak-star in $L^\infty$ to the corresponding measure averages. Passing an unbounded nonlinear integrand to its Young-measure average additionally needs conditions such as uniform integrability of the composed sequence; the fourth-power bound does not justify every nonlinear limit. Concentration can require generalized measures beyond the classical oscillation description. In the explicit tent sequence the gradients take only two bounded values, so averaging any continuous gradient-only function is directly controlled without these extra concentration difficulties.

On a bounded interval with fixed endpoint traces, a continuous scalar density $W$ satisfying $c|v|^4-C\le W(v)\le C(1+|v|^4)$ for constants $c,C>0$ has weak sequential relaxation on $W^{1,4}$ given by integrating its convex lower-semicontinuous envelope. This polynomial meets those growth bounds. Its envelope is zero between the two wells and agrees with the original density outside that interval. The displacement-square term remains, since bounded fourth-power Sobolev sequences on this interval have the required strong compactness for that lower-order term. The relaxed problem therefore admits the zero shape with energy zero, representing a limit of finer original competitors.

The original problem has infimum zero but no attaining function: zero total energy would force the displacement to vanish almost everywhere, hence its weak derivative to vanish, contradicting the requirement that gradient density also be zero. The relaxed minimizer represents unresolved microstructure rather than an original zero-energy shape. In vector-valued higher-dimensional problems, compatibility of gradients makes quasiconvexity central; simply convexifying every matrix argument can give the wrong relaxation. The elementary scalar construction should not be promoted to a general vectorial relaxation formula.