Study shows solutions to certain equations form smooth manifolds.
problem Understanding moduli spaces of solutions to non-linear elliptic equations.
method Analyzes moduli spaces as derived log smooth manifolds.
result Moduli spaces of solutions are derived log smooth manifolds.
The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.
problem Boundary estimates for solutions to fully non-linear elliptic equations on Hermitian manifolds.
method Unified approach using quantitative boundary estimates, gradient estimates, and existence results.
result Established gradient estimates and unified approach to Dirichlet problem solutions.
Solves Dirichlet problem for elliptic equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds.
method Establishing a quantitative boundary estimate under a subsolution assumption.
result Derives solvability and regularity of the Dirichlet problem.
Derives estimates for geometric elliptic equations on complex manifolds.
problem Estimating solutions of geometric elliptic equations on complex manifolds.
method Derives a priori real Hessian estimates independent of the right-hand side.
result Establishes optimal C1,1 regularity of geometric envelopes. Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
problem Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
method Comparison with auxiliary complex Monge-Ampère equations, Hölder-Young inequality, and De Giorgi iteration lemma.
result Improved L∞ estimates for fully non-linear elliptic equations on Kähler and Hermitian manifolds.
Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.
problem Solving elliptic equations on Hermitian manifolds with boundary conditions.
method Derives quantitative boundary estimates and proves existence of solutions.
result Proves existence of solutions under almost optimal structural conditions.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
Study fully nonlinear elliptic equations on complex manifolds.
problem Solving fully nonlinear elliptic equations on complex manifolds.
method Derive C2,α-estimate and prove existence theorems for solutions and Dirichlet problems. result Existence theorems for solutions on closed Hermitian manifolds with unbounded conditions.
Paper establishes estimates for solutions on compact manifolds.
problem Solving fully non-linear equations on compact almost Hermitian manifolds.
method Establishes a priori estimates for solutions.
result Solves complex Hessian and Monge-Ampère equations.
Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
problem Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
method Fully non-linear elliptic equation
result Solves if datas are strictly convex
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent …
Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.
problem Solving elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting.
result Prove a priori estimates for solutions to elliptic equations.
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
Paper solves complex equations on noncompact manifolds.
problem Establishing estimates and existence for fully non-linear equations.
method General class of fully non-linear equations on Kähler and Hermitian manifolds.
result Constructs complete Kähler metrics with prescribed volume forms.
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…
We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces wit…
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of He…
Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.
problem Solving fully nonlinear elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting, proving a priori estimates.
result Proves solvability of quaternionic Hessian and Monge-Ampère equations on compact flat hyperkähler manifolds.
A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.
New Fueter sections solve monopole equations for 3/2-spinors.
problem Existence of solutions to monopole equations for 3/2-spinors.
method Introduced 3/2-Fueter sections as solutions to an overdetermined non-linear elliptic differential equation.
result Non-compactness of moduli space of solutions is equivalent to existence of 3/2-Fueter sections.
Study on slow convergence in geometric variational problems.
problem Slow convergence of solutions in geometric variational problems.
method Identifying necessary conditions for slowly converging solutions and characterizing their convergence rate and direction.
result Characterization of the rate and direction of convergence for slowly converging solutions.
A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and i…
The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.
problem Existence and uniqueness of radial solutions to a Weingarten equation in a disk.
method Analyzes the linear Weingarten equation in a disk of small radius, considering elliptic, hyperbolic, and parabolic cases.
result Proves existence and uniqueness of radial solutions in the elliptic case, and no solutions in the hyperbolic case.
`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing tec…
The paper constructs special Lagrangian n-folds in arbitrary dimensions.
problem Developing a construction for special Lagrangian n-folds in arbitrary dimensions.
method Reduction of special Lagrangian condition to a quasilinear elliptic system of 2D non-linear Cauchy-Riemann equations.
result The structure and multiplicity of singularities are governed by an associated polynomial.
We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over th…
The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.
problem Analyzing Bernstein property for minimal spacelike surfaces in 4D Minkowski space.
method Study of an extension of the Bernstein Theorem for minimal spacelike surfaces in R^4_1.
result The Bernstein property does not hold in general for graphic spacelike surfaces in R^4_1.
Calculates spectral flow bounds for reducible solutions to Vafa-Witten equations.
problem Bounding spectral flow between diverging reducible solutions.
method Localization and excision techniques to calculate spectral flow.
result Bounds on spectral flow are given for reducible solutions.
In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conforma…
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
PEA improves PCA and k-means for non-linear data and complex clusters.
problem Non-linear dimensionality reduction and clustering challenges.
method Principal Elliptical Analysis (PEA) for efficient non-linear approximation.
result PEA outperforms k-means in complex data clustering.
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
New method for analyzing elliptic and parabolic equations.
problem Analyzing elliptic and parabolic equations.
method Level set version of partial uniform ellipticity.
result Effective approach to investigate equations.
The paper proves stability of certain cosmological models with negative spatial curvature.
problem Stability of Friedmann-Lemaître-Robertson-Walker cosmological models with negative spatial curvature.
method Linear stability analysis using Hodge decomposition and energy estimates.
result Uniform boundedness and decay of solutions to the linearized Einstein-Euler system.
The Ryu-Takayanagi conjecture connects the entanglement entropy in the boundary CFT to the area of open co-dimension two minimal surfaces in the bulk. Especially in AdS(4), the latter are two-dimensional surfaces, and, thus, solutions of a Euclidean non-linear sigma model on a symmetric target space that can be reduced…
We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.
Let (Mn,g), n≥3 be a noncompact complete Riemannian manifold with compact boundary and f a smooth function on ∂M. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of g that is complete, has zero scalar curvature on M and has mean curv…
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
The paper examines ellipticity of specific equations on vector bundles.
problem Investigating ellipticity of vector bundle versions of Monge-Ampère equations.
method Analyzing continuity paths and preserving ellipticity of equations.
result Not all equations preserve ellipticity along continuity paths, but σ2 does. Note on advancements in nonlinear elliptic equations' regularity theory.
problem Nonlinear elliptic equations and their regularity.
method De Giorgi-Nash-Moser theory, Krylov-Safonov theory, Evans-Safonov theory.
result Contributions to Hilbert's 19th problem and fully nonlinear equations.
Given an elliptic operator P on a non-compact manifold (with proper asymptotic conditions), there is a discrete set of numbers called indicial roots. It's known that P is Fredholm between weighted Sobolev spaces if and only if the weight is not indicial. We show that an elliptic theory exists even when the weight i…
Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.
problem Proving a Liouville theorem for a generalized elliptic equation on H-type groups.
method Proof based on an a priori integral estimate and a generalized differential identity.
result Obtained a Liouville type theorem for the semilinear subcritical elliptic equation on H-type groups.
New solutions found for elliptic sinh-Gordon and sine-Gordon equations.
problem Elliptic sinh-Gordon and sine-Gordon equations on the real plane.
method Backlund transformation connecting the equations.
result New families of solutions introduced.
Local solubility of Bao--Ratiu equations proven for surfaces with specific curvature conditions.
problem Existence of asymptotic directions for volume-preserving diffeomorphisms on surfaces.
method Analysis of degenerate Monge--Ampère equation following Han's work.
result Asymptotic directions always exist locally about a point on surfaces with specific curvature conditions.
Uniform bounds derived for fully non-linear equations.
problem Bounding fully non-linear equations uniformly in background metrics.
method Auxiliary Monge-Ampère equations and entropy-like quantities.
result Uniform L∞ bounds for systems coupling fully non-linear equations to their linearizations. Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.