A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Researchers find higher symmetries in symplectic Dirac operator.
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work…
New method constructs solution operators for PDEs with prescribed support properties.
Solves a 60-year-old compatibility problem on manifolds with boundary.
BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor conn…
The aim of this work is to develop a systematic manner to close overdetermined systems arising from conformal Killing tensors (CKT). The research performs this action for 1-tensor and 2-tensors. This research makes it possible to develop a new general method for any rank of CKT. This method can also be applied to other…
Mathematicians study surfaces with constant nonlocal mean curvature.
The paper optimizes domains for the first nontrivial eigenvalue in Riemannian manifolds.
These notes are based on my lectures at IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations." Here I try to explain basic ideas of the ambient metric construction by studying the Szego kernel of the sphere.
Solves overdetermined boundary problems for semilinear equations.
Study proves radial symmetry of solutions to certain nonlinear equations in space forms.
Study solves overdetermined problem on compact surfaces.
The purpose of this present paper is to investigate the geometric structure of regular overdetermined systems of second order with two independent and one dependent variables from the point of view of rank 2 prolongations. Utilizing this notion of prolongations, we characterize the type of these overdetermined systems.…
Study proves radial symmetry in convex cones using subharmonic functions.
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as detour complexes. This elaborates on a lecture given at the IMA Summer Programme ``S…
Rigidity theorem for spherical sectors in Riemannian manifolds.
New domains found in hyperbolic space solve a specific elliptic problem.
The paper characterizes gauge balls in the Heisenberg group and solves overdetermined problems.
We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqu…
Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function.
Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.
Paper solves overdetermined -Hessian equation in exterior domains.
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.
This article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux.
Method of moving planes used for proving symmetry in PDEs and geometric analysis.
We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces.
Researchers create compatibility complexes for Einstein metrics.
The paper resolves compactness and non-compactness for fourth- and sixth-order Q-curvature problems.
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
Paper solves a mixed boundary value problem in space forms with umbilical boundaries.
In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal th…
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a Lie algebra of symmetries has invariant solutions with respect to this Lie alge…
New Fueter sections solve monopole equations for 3/2-spinors.
Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.
Non-trivial conservation law found for a specific system.
In this paper we prove a version of Lie-Bäcklund theorem for overdetermined systems of scalar PDEs, whose general solution depends on 1 function of 1 variable. This generalizes the case of involutive system of the second order on the plane treated by E.Cartan in 1910. Many examples are provided.
Solves division problem for L. Hörmander's systems.
A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which cover all elliptic, overdetermined elliptic, subelliptic and parabolic…
The symmetry operators for the Laplacian in flat space were recently described and here we consider the same question for the square of the Laplacian. Again, there is a close connection with conformal geometry. There are three main steps in our construction. The first is to show that the symbol of a symmetry is constra…
Study proves a sharp upper bound for the zero set area of a static manifold's potential.
This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation …
Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.
Paper constructs solutions for a class of overdetermined systems.
We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators can be glued in a chosen region in order to obtain a new smooth soluti…
Unified treatment of stability problems in geometry and analysis.
We study necessary conditions on the geometry and the topology of domains in that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Be…
A regular normal parabolic geometry of type on a manifold gives rise to sequences of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\na^\om$ on the corresponding tractor bundle where $\om$ is…