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We give a soft proof of Albertis Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the $C^2$-rectifiability problem are also discussed.
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Delta^2 u=|u|^{p-1}u {in} R^n,$$ where $ p>1$ and $nge1$. We give a complete classification of stable and finite Morse index solutions (whether posit
In his monograph Lec{c}ons sur les syst`emes orthogonaux et les coordonnees curvilignes. Principes de geometrie analytique, 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type [
We give an elementary probabilistic proof of Veraverbekes Theorem for the asymptotic distribution of the maximum of a random walk with negative drift and heavy-tailed increments. The proof gives insight into the principle that the maximum is in general attained through a single large jump.
We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_