arXiv research

Method of moving planes used for proving symmetry in PDEs and geometric analysis.

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…

Solves second-order PDEs using quotients and differential invariants.

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…

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.

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 …

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…

New method constructs solution operators for PDEs with prescribed support properties.

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…

Extends Serrin's symmetry result to model manifolds.

Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying approach will be discussed. It is rather close to the classical differential co…

The paper proves two theorems about solutions to certain PDE systems.

Study shows no hidden symmetries in specific spacetime metrics.

In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Darboux integrable syst…

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 $P$ can be glued in a chosen region in order to obtain a new smooth soluti…

Study several classes of Riemannian manifolds defined by Ricci tensor conditions.

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $\varphi(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut v…

BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of …

Stability results for geometric equations in warped product spaces.

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.…

Solves numerical computation of Killing and conformal Killing vector fields on compact Riemannian manifolds.

Study proves radial symmetry in convex cones using subharmonic functions.

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.

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 $k$-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.

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.

The study examines special domains in S^2 supporting specific solutions to a PDE.

Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web…

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.

Studies projective geometry and partial differential equations prolongation.

Paper solves a mixed boundary value problem in space forms with umbilical boundaries.

Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.

Non-trivial conservation law found for a specific system.

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic ob…

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined…

Solves division problem for L. Hörmander's systems.

Study proves a sharp upper bound for the zero set area of a static manifold's potential.

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…

Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.