Solves overdetermined boundary problems for semilinear equations.
problem Overdetermined boundary problems for semilinear equations with position-dependent nonlinearities.
method Analyzes Euclidean and compact Riemannian manifolds, proving existence of nontrivial solutions.
result Nontrivial solutions exist for a wide range of problems.
Study proves radial symmetry of solutions to certain nonlinear equations in space forms.
problem Proving radial symmetry of solutions to nonlinear equations in space forms.
method Establishing Rellich-Pohožaev type identities for Hessian quotient and k-Hessian equations.
result Radial symmetry of solutions for Hessian quotient and k-Hessian equations in space forms.
Study proves radial symmetry in convex cones using subharmonic functions.
problem Proving radial symmetry in convex cones with boundary conditions.
method Using maximum principle and integral identities for subharmonic functions.
result Proves radial symmetry and Serrin-type results for partially overdetermined problems.
Study solves overdetermined problem on compact surfaces.
problem Overdetermined problem on compact Riemannian surfaces.
method Analyzes Steklov eigenvalues and geometric properties.
result Characterizes domains for which the problem is solvable.
New domains found in hyperbolic space solve a specific elliptic problem.
problem Solving an overdetermined elliptic problem in nontrivial exterior domains of hyperbolic space.
method Constructing nontrivial domains and solving the elliptic equation.
result Positive bounded solutions found in $C^{2,α}\left(Ω
ight) \cap H^1\left(Ω
ight)$.
Rigidity theorem for spherical sectors in Riemannian manifolds.
problem Rigidity of spherical sectors in Riemannian manifolds under overdetermined conditions.
method Analyzing solutions to the inhomogeneous Helmholtz equation with constant Dirichlet and Neumann boundary conditions.
result Spherical sectors are the only solutions under given conditions.
The paper characterizes gauge balls in the Heisenberg group and solves overdetermined problems.
problem Characterizing gauge balls in the Heisenberg group and solving overdetermined problems.
method Discussing a one-parameter family of overdetermined problems related to the geometry of the Heisenberg group.
result Uniqueness results for domains with partial symmetries of cylindrical type in the Heisenberg group.
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.
problem Characterizing umbilical hypersurfaces in space forms.
method Using a Serrin-type partially overdetermined problem with inhomogeneous Robin boundary condition.
result Any contact angle θ ∈ (0, π) can be achieved, generalizing previous results.
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.
Method of moving planes used for proving symmetry in PDEs and geometric analysis.
problem Proving symmetry properties in PDEs and geometric analysis.
method Method of the moving planes for quantitative studies.
result Quantitative approximate symmetry results obtained.
Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.
problem Overdetermined problem for p-Laplacian equation on compact Riemannian manifolds.
method Introduced a new P-function related to the first nonzero eigenvalue for p-Laplacian, derived integral identities, and applied them to achieve inequalities and the Soap Bubble Theorem.
result Achieved the Heintze-Karcher type inequality and the Soap Bubble Theorem.
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.
Paper solves overdetermined k-Hessian equation in exterior domains.
problem Overdetermined problem for k-Hessian equation in exterior domains. method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for k-admissible solutions. 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 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.
problem Solving a partially overdetermined mixed boundary value problem in space forms.
method Generalizing previous results to domains with partial umbilical boundaries.
result A partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface.
Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.
problem Rigidity in Serrin's overdetermined problems in Riemannian manifolds.
method Prove a Pohozoaev-type identity, use conformal vector field, and apply P-function approach.
result Show Serrin's type rigidity result in Riemannian manifolds.
Solves division problem for L. Hörmander's systems.
problem Division problem for L. Hörmander's overdetermined systems.
method Formulates and proves divisibility criterion, coherence theorem.
result Establishes effective divisibility criterion and extends coherence theorem.
Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.
problem Solving overdetermined problems for rotationally invariant Poisson equations in model manifolds.
method Analyzes specific cases of overdetermined problems and uses geometric properties of model manifolds to deduce radial solutions.
result Conditions on f, φ and κ imply that the solution u is radial and the domain Ω is a geodesic ball centered at O. The paper optimizes domains for the first nontrivial eigenvalue in Riemannian manifolds.
problem Optimizing domains for the first nontrivial eigenvalue in Riemannian manifolds.
method Analyzing locally extremal domains and using an extended shape derivative formula.
result Critical domains for the first nontrivial eigenvalue are identified.
Solves a 60-year-old compatibility problem on manifolds with boundary.
problem Finding a compatibility operator for Lie derivatives of the metric tensor on compact Riemannian manifolds.
method Develops a framework for elliptic pre-complexes and pseudodifferential operators to correct and yield Hodge-like decompositions.
result Explicit integrability conditions for overdetermined boundary-value problems are derived, resolving the Saint-Venant problem.
We study necessary conditions on the geometry and the topology of domains in R2 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…
The paper proves rigidity for capillary graphs with specific curvature properties.
problem Understanding capillary graphs with height-dependent mean curvature.
method Analyzing the capillary overdetermined problem with a general function f.
result Boundedness of solutions and rigidity of domains under specific conditions.
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 symmetry of bounded domains in Riemannian manifolds.
problem Symmetry of bounded domains in Riemannian manifolds.
method Integral identities and P-function method. result Equality implies the domain is isometric to a Euclidean ball.
For all N≥9, we find smooth entire epigraphs in RN, namely smooth domains of the form Ω:={x∈RN / xN>F(x1,…,xN−1)}, which are not half-spaces and in which a problem of the form Δu+f(u)=0 in Ω has a positive, bounded solution with 0 Dirichlet boundary data and constant Neum…
Develops Hodge theory for boundary-value problems on general geometric structures.
problem Solvability and uniqueness conditions for linearized overdetermined boundary-value problems.
method Introduces elliptic pre-complex and order-reduction property to generalize Hodge theory.
result Provides tools to study cohomology explicitly for general geometric structures.
Mathematicians study surfaces with constant nonlocal mean curvature.
problem Overdetermined boundary value problems.
method Properties of Nonlocal Mean Curvature and related differential operators.
result Surfaces with constant NMC have a close connection to overdetermined problems.
Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.
problem Proving mass-capacity inequalities for critical area-normalized capacitors.
method Analyzes asymptotically flat manifolds with boundary capacity potential satisfying an overdetermined problem.
result Improves Schwarzschild metric uniqueness and results for spin asymptotically flat spacetimes.
In this paper we provide a new method for establishing the rotational symmetry of the solutions to a couple of very classical overdetermined problems arising in potential theory, in both the exterior and the interior punctured domain. Thanks to a conformal reformulation of the problems, we obtain Riemannian manifolds w…
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Extends Serrin's symmetry result to model manifolds.
problem Proving symmetry in solutions to a specific PDE on manifolds.
method Uses an extension of Weinberger's argument to prove symmetry.
result Euclidean symmetry result under compatibility assumption.
Study on p-Laplace equation in convex cones, proving rigidity under specific conditions.
problem Overdetermined problem for p-Laplace equation in convex cones. method Established properties of capacitary potential, used P-function, isoperimetric inequality, and Heintze-Karcher inequality. result Rigidity result under orthogonal intersection assumption.
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.
Analyzes constant heat flow properties on Riemannian manifolds.
problem Classify manifolds with constant heat flow property.
method Analyzes overdetermined parabolic problems and uses isoparametric tubes.
result Analytic manifolds with constant flow property are isoparametric tubes.
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…
Non-trivial conservation law found for a specific system.
problem Conservation law for a specific system with a vanishing characteristic.
method Analyzing overdetermined system with given characteristics.
result Non-trivial conservation law despite vanishing characteristic.
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 paper concerns rigidity results to Serrin's overdetermined problem in an epigraph {Δu+f(u)=0, in Ω={(x′,xn):xn>φ(x′)},u>0, in Ω,u=0, on ∂Ω,∣∇u∣=const.on∂Ω.. We prove that up to isometry the ep…
Study proves a sharp upper bound for the zero set area of a static manifold's potential.
problem Proving a sharp upper bound for the zero set area of a static manifold's potential.
method Proved a rigidity theorem for the Euclidean closed unit ball in R^3.
result Sharp upper bound for the area of the zero set of the 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…
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 study the geometry and the topology of unbounded domains in the Hyperbolic Space Hn supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show tha…
The paper connects minimal surfaces to cones in a ball.
problem Understanding minimal surfaces with free boundaries.
method Establishing a connection between minimal surfaces and cones.
result Doubly connected minimal surfaces in a ball are catenoids.
Researchers find higher symmetries in symplectic Dirac operator.
problem Understanding symmetries in symplectic Dirac operator.
method Constructing higher symmetry algebra of symplectic Dirac operator Daise0.22ex/s. result Higher symmetry algebra structure corresponds to a completely prime primitive ideal.
The paper classifies domains with constant boundary temperature in various 3D spaces.
problem Domains with constant boundary temperature in 3D space-forms.
method Analyzes heat diffusion problems and uses minimal surface theory.
result Minimal surfaces bound domains with constant boundary temperature in various 3D spaces.