arXiv subjects
Publications and source records attributed to Wentao Huo.
This paper focus on a class of singular or degenerate fully nonlinear elliptic equations with Hamiltonian terms under oblique boundary conditions on $C^{1}$ domains. Under quite general conditions on the singularity/degeneracy of the model, we establish the sharp $C^{1,\alpha}$ regularity up to the boundary within a unified framework.
We develop the regularity theory of viscosity solutions to degenerate fully nonlinear free transmission problems with Hamiltonian terms. By framing the equation in the context of viscosity inequalities, we establish local H\"{o}lder regularity of the gradient. In addition, based on a new improved oscillation-type estimate combined with a localized analysis, we obtain sharp pointwise $C^{1,\alpha}$ regularity.
This paper focuses on the regularity of viscosity solutions to normalized $p$-Laplacian equations with variable-exponent double phase type degeneracy/singularity and Hamiltonian terms. Based on a new improved oscillation-type estimate combined with a localized analysis, we establish sharp interior $C^{1,\alpha}$ regularity estimates in a unified way. In addition, in the degenerate case, we obtain improved gradient H\"{o}lder regularity results at points where the Hamiltonian coefficient and source term vanish, and establish a Schauder-type estimate at local extrema. Notably, our results are still novel even restricted to single power-type singularity or degeneracy law.
This paper focuses on a class of fully nonlinear elliptic equations with general double phase degeneracy/singularity law and Hamiltonian terms of the form $$Φ(|Du|,x)F(D^2 u, x)+H(Du,x) =f(x) \quad \text{in} \quad B_{1},$$ where $Φ$ takes one of two typical forms: $$Φ(|Du|,x)=σ_{1}(|Du|)+a(x)σ_{2}(|Du|)\quad {\rm or}\quad Φ(|Du|,x)=\frac{σ_{1}(|Du|)}{|Du|}+a(x)\frac{σ_{2}(|Du|)}{|Du|}.$$ Under suitable assumptions on the operator $F$, Hamiltonian term $H$, source term $f$ and modulating coefficient $a$, we establish $C^{1}$ regularity for viscosity solutions, provided that $σ_{1},σ_{2}$ are moduli of continuity and their inverses are Dini continuous. Our argument is based on a tangential analysis via approximating hyperplanes combined with a new recursive renormalization algorithm adapted to the present framework. It is noteworthy that our results are new even for the case $a(x)\equiv 0$.
This paper focuses on a class of fully nonlinear elliptic equations with variable double phase type degeneracy law and Hamiltonian terms. We obtain improved gradient Hölder regularity results at points where the Hamiltonian coefficients and source terms vanish. Furthermore, we establish a Schauder-type estimate at local extrema, which is sharp with respect to the vanishing rate of the Hamiltonian coefficient and source term. Our approach adapts compactness and dichotomy arguments to capture the interplay between the degeneracy rate and the growth of the Hamiltonian term.
We investigate the regularity of the viscosity solutions to a class of degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms. To overcome the difficulty caused by the simultaneous presence of the general degenerate/singular gradient terms and Hamiltonian terms, we analyze the coupled interplay between the degeneracy/singularity law and the growth of Hamiltonian terms and establish lower regularity results. Finally, we obtain sharp interior $C^{1,α}$ regularity estimates via a geometric tangential method.
We establish the homogenization results for a class of nonlocal operators of convolution type with integrable jumping kernel $p$ multiplied by rapidly oscillating periodic or locally periodic coefficients. The associated measure $p(z)dz$ is assumed to belong to the domain of attraction of a symmetric $α$-stable law. We also assume that $p$ satisfies a pointwise Lévy type lower bound and an averaged annular upper bound for points bounded away from the origin, and that the local $L^1$ oscillation of $p$ decays faster at infinity than its local $L^1$-norm. Under these assumptions, we prove the resolvent convergence of the nonlocal operators and explicitly determine the corresponding homogenized nonlocal operator, which is shown to be comparable to the fractional Laplacian. The proof relies on compactness arguments and a refined analysis based on the annular integral upper bound and an $\varepsilon$-cube decomposition.
This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional Laplacian and some operators compared to it. Based on the $Γ$-convergence method and compactness arguments, we prove the homogenization theorems for these nonlocal operators with product-type and symmetric coefficient-structured kernels respectively. Furthermore, these results are extended to general nonlinear nonlocal equations.