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We revisit the question of whether the functions defined on the real $$m\times n$$ matrices that are convex along rank-one directions are also quasi-convex in the sense of Morrey. Using the linearity of the map $$f\to \int_{\TT^n} f(\nabla u(x))\,dx$$, we propose to study the question as a problem in convex optimization. This might be useful when trying to resolve the open cases, such as the case $m=2$, or various cases with symmetries.
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We consider the equation q_1+qq_x=q_{xx} for quaternionic-valued functions and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. The equation is studied both on the real line and on the circle.more » « less
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