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Feida Jiang

Publications and source records attributed to Feida Jiang.

At least 19 recordsLinked to original sources
arXiv2026

Degeneracy Set Comparison Principle and Free Boundary Estimates for H\'enon-type Infinity Laplace equations

In this work, we study nonnegative viscosity solutions to H\'enon-type equations driven by the infinity-Laplacian with a degenerate weight, strong absorption and an additional source term. We focus on free boundary points lying in the degeneracy set of the weight. We prove the degeneracy set comparison principle, which yields uniqueness within each slice determined by the value on the degeneracy set of the weight, although global uniqueness is not available in general. We establish sharp improved regularity estimates near free boundary points, with the intrinsic growth rate $r^{\frac{4+\alpha}{3-m}}$. Finally, we obtain a matching non-degeneracy estimate at free boundary points; the property holds in the inhomogeneous case with suitable condition. Consequently, solutions detach from their zero phase at this rate.

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arXiv2026

Regularity theory for a class of degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms and applications

In this paper, we investigate regularity properties for viscosity solutions to a general class of degenerate or singular fully nonlinear elliptic equations with Hamiltonian terms, \[ \left\{ \begin{alignedat}{2} \Phi(|Du|,x)F(D^{2}u,x)+H(|Du|,x) &=f(x) \quad && \text{in } \Omega,\\ u&=g \quad && \text{on } \partial\Omega. \end{alignedat} \right. \] Here, $F$ is uniformly elliptic, while $\Phi$ and $H$ satisfy suitable structural and growth conditions allowing for both degenerate and singular regimes. Our first result establishes sharp global $C^{1,\alpha}$ regularity for this general class of equations, thereby providing a unified regularity framework for both degenerate and singular regimes. We next develop an oscillation-based approach for fully nonlinear equations with unbalanced variable degeneracy and Hamiltonian terms. Under a H\"older-type decay assumption on $ f $, we derive sharp boundary $C^{1,\beta'}$ estimates with an explicit exponent, and establish a quantitative non-degeneracy estimate in the singular region. Finally, as applications of the general theory, under a suitable viscosity curvature condition on the level sets of the solution, we establish global $C^{1,\frac{1}{3}}$ regularity for the infinity-Poisson and global $C^{1,\frac{1}{p-1}}$ regularity for the $p$-Poisson with $p>2$. These results may provide a new perspective on these two long-standing open problems.

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arXiv2026

Global $ C^{1}$ regularity for degenerate fully nonlinear elliptic equations

In this paper, we prove global $ C^{1}$ regularity for Dirichlet problem of a class of degenerate fully nonlinear elliptic equations on $ C^{2}$ domain. This corresponds to the boundary counterpart of the interior $ C^{1}$ regularity results by \cite{APPT22} and \cite{AN25}. By an example, we show that $ C^{1, \alpha}$ regularity of boundary datum is sharp within the scale of H\"{o}lder spaces. Furthermore, we also provide boundary $ C^{1, \beta}$ regularity for a class of singular fully nonlinear elliptic equations.

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arXiv2026

Near Field Refraction Problem With Loss of Energy in Negative Refractive Index Material

This paper studies the near field refraction problem with loss of energy in negative refractive index material. Based on the relative refractive index $\kappa$, the analysis is categorized into two cases, namely $\kappa < -1$ and $-1 < \kappa < 0$. For each case, we give the definition of the refractor and discuss some crucial properties of it. The properties of Fresnel coefficients are also discussed. Based on these properties, the existence of the weak solution when the target measure is either discrete or a finite Radon measure are proved. Besides, the critical case $\kappa = -1$ is also discussed briefly at the end of this paper.

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arXiv2026

Far field refraction problem with loss of energy in negative refractive index material

This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index $\kappa$, that is, $\kappa<-1$ and $-1<\kappa<0$. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Amp\`ere type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.

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arXiv2026

On a class of Cauchy problems with applications in nonlinear partial differential equations

In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of nonlinear partial differential equations in n-dimensional Euclidean space. As applications of the general framework, we present such results for two series of nonlinear equations: a series of k-Hessian type equations and a new series of P-k-Hessian type equations. These results are also proved for the more general p-Hessian matrices, with p > 1, and degenerate non homogeneous terms.

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arXiv2025

Regularity for fully nonlinear degenerate parabolic equations with strong absorption

In this paper, we investigate dead-core problems for fully nonlinear degenerate parabolic equations with strong absorption, \begin{equation*} |Du|^{p} F(D^{2}u) - u_{t} = \lambda_{0}(x,t)\, u^{\mu}\, \chi_{\{u>0\}}(x,t) \qquad \text{in } \quad Q_{T} := Q \times (0,T), \end{equation*} where $0 \leq p < \infty$ and $0 < \mu < 1$. We establish a sharp and improved parabolic $C^{\alpha}$-regularity estimate along the free boundary $\partial \{ u > 0 \}$, where \[ \alpha := \frac{2+p}{1+p-\mu} > 1 + \frac{1}{1+p}. \] Moreover, we establish weak geometric properties of solutions, such as non-degeneracy and uniform positive density. As an application, we obtain a Liouville-type theorem for entire solutions and gradient bounds. Finally, as a byproduct of our approach, we derive a novel $L^{\delta}$-average estimate for fully nonlinear singular elliptic equations and present a new formulation of the gradient decay property. It is worth noting that the results presented here extend those in da Silva {\it et al.} ({\it Pacific J. Math}., \textbf{300} (2019), 179--213) and ({\it J. Differential Equations}., \textbf{264} (2018), 7270--7293) to the degenerate setting, and can be viewed as a parabolic analogue of da Silva {\it et al.} ({\it Math. Nachr}., \textbf{294} (2021), 38--55) and Teixeira ({\it Math. Ann}., \textbf{364} (2016), 1121--1134). Additionally, of independent mathematical interest, we emphasize that our manuscript establishes a comparison principle result and the compactness of viscosity solutions to fully nonlinear degenerate parabolic models with continuous and bounded forcing terms. These compactness and comparison properties serve as key ingredients in deriving enhanced regularity estimates along free boundary points for our model problem with strong absorption.

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arXiv2025

Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and H\'{e}non-type equations

In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary, non-degeneracy, a measure estimate of the free boundary, Liouville-type results, and the behavior of blow-up solution. We also derive sharp regularity estimates for viscosity solutions to H\'{e}non-type equations with a degenerate weight and strong absorption, governed by a degenerate fully nonlinear operator. Our results are new even for the model equations involving degenerate Laplacian operators.

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arXiv2024

Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations

We study a series of regularity results for solutions to a degenerate or singular fully nonlinear integro-differential equation of the form $$- \big( \sigma_{1}(|Du|) + a(x) \sigma_{2}(|Du|) \big) \mathcal{I}_{\tau}(u,x) = f(x).$$ In the degenerate case, we establish borderline regularity, provided the inverse of the degeneracy law $ \sigma_{2}$ is Dini-continuous. In addition, we show Schauder-type higher regularity at local extremum points for a specific non-local degenerate equation. In the singular case, we establish H\"{o}lder continuity of the gradient for solutions to a general non-local equation. It is noteworthy that these results are new even in the case $ a(x) \equiv 0 $. Finally, as a byproduct of the borderline regularity analysis, we demonstrate how our methods can be applied to study of the corresponding regularity for a class of degenerate non-local normalized $ p$-Laplacian equations.

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arXiv2024

Uniqueness of the critical points of solutions to two kinds of semilinear elliptic equations in higher dimensional domains

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $\Omega\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the symmetry of positive stable solutions for two kinds of semilinear elliptic equations. To do this, when $f(\cdot,s)$ is convex with respect to $s$, we show that the positivity of the first eigenvalue of the corresponding linearized operator in somehow symmetric domains is a sufficient condition for the symmetry of $u$. Moreover, we prove the uniqueness of critical points of a positive stable solution to semilinear elliptic equation $-\triangle u=f(\cdot,u)$ with zero Dirichlet boundary condition for simple rotationally symmetric domains in $\mathbb{R}^n$ by continuity method and a variety of maximum principles.

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arXiv2023

Regularity of solutions to degenerate normalized $p$-Laplacian equation with general variable exponents

In this paper, we consider a kind of degenerate normalized $p$-Laplacian equation with general variable exponents. We establish local $C^{1,\alpha'}$ regularity of viscosity solutions by making use of the compactness argument, scaling techniques and the localized oscillating method. In addition, we also obtain almost optimal pointwise $C^{1,\tau} $ regularity for degenerate free transmission problem related to normalized $ p$-Laplacian. Our argument is based on a new improved oscillation-type estimate combined with a localized analysis.

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arXiv2022

Entire subsolutions of a kind of k-Hessian type equations with gradient terms

In this paper, we consider a kind of $k$-Hessian type equations $S_k^{\frac{1}{k}}(D^2u+\mu|D u|I)= f(u)$ in $\mathbb{R}^n$, and provide a necessary and sufficient condition of $f$ on the existence and nonexistence of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $\mu$ respectively, which embrace the standard Hessian equation case ($\mu=0$) by Ji and Bao (Proc Amer Math Soc 138: 175--188, 2010) as a typical example. The difference between the semilinear case ($k=1$) and the fully nonlinear case ($k\ge 2$) is also concerned.

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arXiv2021

On the solutions to weakly coupled system of $\boldsymbol{k_i}$-Hessian equations

In this paper, the existence and multiplicity of nontrivial radial convex solutions to general coupled system of $k_i$-Hessian equations in a unit ball are studied via a fixed-point theorem. In particular, we obtain the uniqueness of nontrivial radial convex solution and nonexistence of nontrivial radial $\boldsymbol{k}$-admissible solution to a power-type system coupled by $k_i$-Hessian equations in a unit ball. Moreover, using a generalized Krein-Rutman theorem, the existence of $\boldsymbol{k}$-admissible solutions to an eigenvalue problem in a general strictly $(k-1)$-convex domain is also obtained.

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arXiv2019

On the Dirichlet problem for general augmented Hessian equations

In this paper we apply various first and second derivative estimates and barrier constructions from our treatment of oblique boundary value problems for augmented Hessian equations, to the case of Dirichlet boundary conditions. As a result we extend our previous results on the Monge-Ampere and k-Hessian cases to general classes of augmented Hessian equations in Euclidean space

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arXiv2018

Oblique boundary value problems for augmented Hessian equations III

In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for degenerate, (and non-degenerate), augmented Hessian equations, with strictly regular associated matrix functions. By establishing local second derivative estimates at the boundary and proving viscosity comparison principles, we show that the solution is correspondingly smooth near boundary points where the appropriate uniform convexity is satisfied.

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arXiv2018

Interior C^{1,1} regularity of solutions to degenerate Monge-Ampère type equations

In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Ampère type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in C^{1,1}(\barΩ\times\mathbb{R}\times \mathbb{R}^n). We prove that u\in C^{1,1}(Ω) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently.

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arXiv2018

On the second boundary value problem for Monge-Ampere type equations and geometric optics

In this paper, we prove the existence of classical solutions to second boundary value prob- lems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family, thereby enabling application of the degree theory for nonlinear oblique boundary value problems.

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