We study quasilinear elliptic equations of Leray Lions type in W 1, p (0), maximum principles, nonexistence and existence of solutions, the control of lower (upper) bound for essential supremum (essential infimum) of solutions, sign-changing solutions, local and global oscillation of solutions, geometry of domain, generating singularities of solutions, and lower bounds on constants appearing in Schauder, Agmon, Douglis, and Nirenberg estimates.

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We prove that the box dimension of the standard clothoid is equal to d = 4/3. Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. Oscillatory dimensions of component functions of the clothoid are also equal to 4/3. Fractals 2009.17:485-492. Downloaded from www.worldscientific.com by MCGILL UNIVERSITY on 02/05/15. For personal use only. Fractals 2009.17:485-492. Downloaded from www.worldscientific.com by MCGILL UNIVERSITY on 02/05/15. For personal use only. Theorem 4. (Minkowski content under bi-Lipschitz mappings) Let Ω and Ω be open sets in R N , and let H : Ω → Ω be a bi-Lipschitz mapping with lower and upper Lipschitz constants equal to C and C respectively, and its Jacobian defined by J H (x) := det H (x) a.e. Let A be a bounded set such that A ⊆ Ω. Then for all s ≥ 0 we have

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A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory (x,ẋ) in R 2 of a solution x = x(t), assuming that (x,ẋ) is a spiral converging to the origin. In this work, we study the phase dimension of the class of second-order nonautonomous differential equations with oscillatory solutions including the Bessel equation. We prove that the phase dimension of Bessel functions is equal to 4/3, and that the corresponding trajectory is a wavy spiral, exhibiting an interesting behavior. The phase dimension of a generalization of the Bessel equation has been also computed.