相关论文: Convex functions with unbounded gradient

In this paper, we provide some characterizations of strong pseudoconvexity by the boundary behavior of intrinsic invariants for smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2$. As a consequence, if such domain is…

We prove that the derivative of a non-linear entire function is unbounded on the preimage of an unbounded set.

We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.

We prove that if $\Omega$ is a simply connected quadrature domain for a distribution with compact support and the infinity point belongs the boundary, then the boundary has an asymptotic curve that is either a straight line or a parabola or…

We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We…

Given a pseudoconvex domain D in C^N, N>1, we prove that there is a holomorphic function f on D such that the lengths of paths p: [0,1]--> D along which Re f is bounded above, with p(0) fixed, grow arbitrarily fast as p(1)--> bD. A…

We study the geometry of $m$-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every $m$-hyperconvex domain…

We give a short proof of Wolff-Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff-Denjoy theorem for weakly convex domains, again without any smoothness assumption…

We identity the optimal non-infinitesimal direction of descent for a convex function. An algorithm is developed that can theoretically minimize a subset of (non-convex) functions.

We give a precise description of Bergman complete bounded pseudoconvex Reinhardt domains.

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean…

We prove some sharp extremal distance results for functions in weighted Bergman spaces on the upper halfplane.We also prove such results in the context of bounded strictly pseudoconvex domains with smooth boundary

We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space $\R^N$, $N\geq4$. These domains arise as perturbations of complements of straight cylinders in $\R^N$, and by definition they support a positive…

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as…

We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a…

We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up…

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the…

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…

We study the problem of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman…

All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.