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We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
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We present a constructive proof of Alexandrovs theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be equal to the Hessian of the volume of the dual generalized polyhedron. We prove the non-degeneracy by generalizing the Alexandrov-Fenchel inequality. Our construction of a convex polytope from a given metric is implemented in a computer program.
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 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.
Let $Omegasubseteq M$ be a bounded domain with smooth boundary $partialOmega$, where $(M,J,g)$ is a compact almost Hermitian manifold. Our main result of this paper is to consider the Dirichlet problem for complex Monge-Amp`{e}re equation on $Omega$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolution which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds. %This work was already done by Plis when we assume there is a strictly $J$-psh defining function for $Omega$.
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.
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