arXiv research

Proves a principle for one-phase Bernoulli problem minimizers.

In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral …

Study on maximum principles for nonlinear equations on Riemannian manifolds.

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein …

We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…

Study compares nodal sets of solutions to the Allen-Cahn equation.

We extend Bony's propagation of support argument \cite{Bony} to $C^1$ solutions of the non-homogeneous sub-elliptic $p-$Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that general…

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splittin…

Study solves a mathematical problem related to elliptic Schroedinger-to-Neumann maps.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

This paper is about the influence of Geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, in particular by the study of entire graphs with prescribed mean curvature, we consider classes of coercive d…

We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used …

In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \geq 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove $(i)$ the Paneitz operator satisfies a strong maximum principle; $(ii)$ the Paneitz operator is a positive operator; and $(iii)$ its Gree…

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

The so called "globalization" process (i.e. the inexorable integration of markets, currencies, nation-states, technologies and the intensification of consciousness of the world as a whole) has a behavior exactly equivalent to a system that is tending to a maximum entropy state. This globalization process obeys a collec…

The article explores constructing biharmonic and conformal biharmonic maps to spheres.

The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.

Study stabilizes translating solitons in hyperbolic space for MCF.

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic ave…

In this paper, we firstly establish an Interpolating curvature invariance between the well known nonnegative and 2-non-negative curvature invariant along the Ricci flow. Then a related strong maximum principle for the $(λ_1, λ_2)$-nonnegativity is also derived along Ricci flow. Based on these, finally we obtain a rigid…

Establishes a boundary maximum principle for varifolds with fixed contact angle.

Note establishes a local maximum principle for Ricci flow under curvature conditions.

We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …

Study proves Maximum Principles for unbounded Riemannian domains.

In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…

Study on 4D Ricci solitons with specific curvature properties.

In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …

We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and …

In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity v…

This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operato…

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

The study proves theorems about minimal and H-surfaces in 3D space.

New principle for harmonic maps helps study higher-dimensional submanifolds.

Study shows strong min-max principle for phase transitions.

Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature est…

Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…

In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over …

We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many a…

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum-entro…

Pontryagin's Maximum Principle is an outstanding result for solving optimal control problems by means of optimizing a specific function on some particular variables, the so called controls. However, this is not always enough for solving all these problems. A high order maximum principle (Krener, 1977) must be used in o…

We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.

MEP-Net uses MEP to generate solutions from limited data.

Paper develops MRCs for supervised classification using generalized maximum entropy.

Unified approach to Merton's portfolio problem using Pontryagin's principles.

We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an orientable, complete immersion with finite strong total curvature, then $X$ is proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite number of points $q_1, \dots q_k$. Adding some extra hypothesis, including $H_r=0,$ wh…