No Arabic abstract
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 positive or sign changing), in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. Our approach is motivated by Flemings tangent cone analysis technique for minimal surfaces and Federers dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.
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 [partial_{x_i} u_alpha(x)=f^alpha_i(x,u(x)),quad iin I_alphasubseteq{1,dots,n}.] For a given point $bar xin mathbb{R}^n$ it is assumed that the values of the unknown $u_alpha$ are given locally near $bar x$ along ${x,|, x_i=bar x_i , text{for each}, iin I_alpha}$. The more general of the theorems, Theor`eme III, was proved by Darboux only for the cases $n=2$ and $3$. In this work we formulate and prove a generalization of Darbouxs Theor`eme III which applies to systems of the form [{mathbf r}_i(u_alpha)big|_x = f_i^alpha (x, u(x)), quad iin I_alphasubseteq{1,dots,n}] where $mathcal R={{mathbf r}_i}_{i=1}^n$ is a fixed local frame of vector fields near $bar x$. The data for $u_alpha$ are prescribed along a manifold $Xi_alpha$ containing $bar x$ and transverse to the vector fields ${{mathbf r}_i,|, iin I_alpha}$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $mathcal R$ and on the manifolds $Xi_alpha$; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $C^1$-solution via Picard iteration for any number of independent variables $n$.
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 $$C_1 Q_t e^{-frac{d^2}{4t}} le p_t le C_2 Q_t e^{-frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Caratheodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $| abla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups.