arXiv research

Study solves inverse problems for equations with fractional nonlinearities.

Note on advancements in nonlinear elliptic equations' regularity theory.

Solves nonlinear problems on metric structures through eigenvalue counting.

Solves a specific Dirichlet problem on Riemannian manifolds.

Study finds lower bounds for solutions on Riemannian orbifolds.

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

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…

Study fully nonlinear elliptic equations on complex manifolds.

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

Solves open problems for fully nonlinear elliptic equations on manifolds.

Study improves understanding of solutions to complex equations in geometry.

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and opt…

This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

Interdisciplinary study linking potential theory and elliptic PDEs.

Proves existence and compactness of solutions to $σ_2$-Nirenberg problem on sphere.

The paper concerns singular solutions of nonlinear elliptic equations.

In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

Study solves equations on tori for Calabi-Yau problems.

Study moduli spaces of elliptic PDEs using derived $C^{\infty}$-geometry.

We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the bound…

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived.

In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on $R^n$. We show that the problem has infinite positive solutions in $C^τ(R^n)\bigcap H^α_{loc}(R^n)$. Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …

In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of ba…

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac…

Solves curvature problems on manifolds with negative curvature.

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.

Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.

Study on a nonlinear elliptic equation on compact Hermitian manifolds.

New method for analyzing elliptic and parabolic equations.

In this short note, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $Δu+cu^α=0,$ where $c, α$ are two real constants and $c\neq 0$.

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…

We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.

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.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.

Estimates for complex equations on manifolds derived from a conjecture.

The paper studies gradient estimates for solutions of a nonlinear elliptic equation on Riemannian manifolds.

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.