On n-superharmonic functions and some geometric applications

In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.

Extremals in nonlinear potential theory

We consider the PDE - Δ p ⁢ u = ρ -\Delta_{p}u=\rho , where 𝜌 is a signed Borel measure on R n \mathbb{R}^{n} . For each p > n p>n , we characterize solutions as extremals of a generalized Morrey inequality determined by 𝜌.

Equivalence between distributional and viscosity solutions for the double-phase equation

We investigate the different notions of solutions to the double-phase equation - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = 0 , -{\operatorname{div}(\lvert Du\rvert^{p-2}Du+a(x)\lvert Du\rvert^{q-2}Du)}=0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the A H ⁢ ( ⋅ ) \mathcal{A}_{H(\,{\cdot}\,)} -harmonic functions of nonlinear potential theory and then show that A H ⁢ ( ⋅ ) \mathcal{A}_{H(\,{\cdot}\,)} -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.

Gradient continuity estimates for the normalized p-Poisson equation

In this paper, we obtain gradient continuity estimates for viscosity solutions of [Formula: see text] in terms of the scaling critical [Formula: see text]-norm of [Formula: see text], where [Formula: see text] is the normalized [Formula: see text]-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential [Formula: see text]. Moreover, for [Formula: see text] with [Formula: see text], we also obtain [Formula: see text] estimates. This improves one of the regularity results in [A. Attouchi, M. Parviainen and E. Ruosteenoja, [Formula: see text] regularity for the normalized [Formula: see text]-Poisson problem, J. Math. Pures Appl. (9) 108(4) (2017) 553–591], where a [Formula: see text] estimate was established depending on the [Formula: see text]-norm of [Formula: see text] under the additional restriction that [Formula: see text] and [Formula: see text]. We also mention that differently from the approach in the above paper, which uses methods from divergence form theory and nonlinear potential theory, the method in this paper is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107(2) (1994) 341–350]. Moreover, for [Formula: see text] continuous, our approach also gives a somewhat different proof of the [Formula: see text] regularity result.

Numerical Modelling of Wave Resonance in a Narrow Gap Between Two Floating Bodies in Close Proximity Using a Hybrid Model

This paper presents a numerical investigation on the wave resonance in a narrow gap between two floating bodies in close proximity using a hybrid model, qaleFOAM, which combines a two-phase Navier-Stokes model (NS) and the fully nonlinear potential theory (FNPT) using a spatially hierarchical approach. The former governs the computational domain near the floating bodies and the gap, where the viscous effects are significant, and is solved by using OpenFOAM/InterDyMFoam. The latter covers the rest of the domain and solved by using the Quasi Lagrangian Eulerian Finite Element Method (QALE-FEM). The model is validated by comparing its numerical predictions with experimental data in the cases with linear incident waves. Systematic investigations using incident waves with different steepness are then followed to explore the nonlinear effects on the wave resonance.

Self-Adaptive Wave Absorbing Technique for Nonlinear Shallow Water Waves

A key challenge in long-duration modelling of ocean waves or wave-structure interactions in numerical wave tanks (NWT) is how to effectively absorb undesirable waves on the boundaries of the wave tanks. The self-adaptive wavemaker theory is one technique developed for this purpose. However, it was derived based on the linear wavemaker theory, in which the free surface elevation and the motion of the wavemaker are assumed to be approximately zero. Numerical investigations using the fully nonlinear potential theory based Quasi Arbitrary Lagrangian Eulerian Finite Element Method (QALE-FEM) suggested that its efficiency is relatively lower when dealing with nonlinear waves, especially for shallow water waves due to three typical issues associated with the wave nonlinearity including (1) significant wavemaker motion for extreme waves; (2) the mean wave elevation (i.e. the component corresponding to zero frequency), leading to a constant velocity component, thus a significant slow shift of the wavemaker; (3) the nonlinear components, especially high-order harmonics, may significantly influence the wavemaker transfer functions. The paper presents a new approach to numerically implement the existing self-adaptive wavemaker theory and focuses on its application on the open boundary, where all incident waves are expected to be fully absorbed. The approach is implemented by the NWT based on the QALE-FEM method. A systematic numerical investigation on uni-directional waves is carried out, following the corresponding validation through comparing the numerical prediction with experimental data for highly nonlinear shallow water waves.