On isocline lines for functions and convex stratifications of two variables |
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Authors: | Marco Longinetti Paolo Manselli Adriana Venturi |
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Institution: | 1. Dipartimento di Ingegneria Agraria e Forestale , Università degli Studi di Firenze , P.le delle Cascine 15, 50139, Firenze, Italy marco.longinetti@unifi.it;3. Dipartimento di Matematica e Applicazioni per l'Architettura , Università degli Studi di Firenze , P.zza Ghiberti 27, 50122 Firenze, Italy;4. Dipartimento di Ingegneria Agraria e Forestale , Università degli Studi di Firenze , P.le delle Cascine 15, 50139, Firenze, Italy |
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Abstract: | Let the isoclines of a function u be the level lines of the function θ = arg(Du). Formulas for the curvature and the length of isocline lines in terms of the curvatures k, j of the level curves and of the steepest descent lines of u are given. The special case when all isoclines are straight lines is studied: in this case the steepest descent lines bend proportionally to the level lines; the support function of the level lines is linear function on the isoclines parameterized by the level values, possibly changing them. This characterization gives a new proof of a property of the developable surfaces found in A. Fialkow, Geometric characterization of invariant partial differential equations, Amer. J. Math. 59(4) (1937), pp. 833–844]. When u is in the class of quasi convex functions, the L p norm of the length function I θ of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have arbitrarily large length and arbitrarily large L 1 norm of I θ. |
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Keywords: | isoclines curvatures developable surfaces support function convexity of level sets elliptic equations |
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