Supersolutions and superharmonic functions for nonlocal operators with Orlicz growth

We study supersolutions and superharmonic functions related to problems involving nonlocal operators with Orlicz growth, which are crucial tools for the development of nonlocal nonlinear potential theory. We provide several fine properties of supersolutions and superharmonic functions, and reveal the relation between them. Along the way we prove some results for nonlocal obstacle problems such as the well-posedness and (both interior and boundary) regularity estimates, which are of independent interest.Comment: 42 page

Estimation of the Available Rooftop Area for Installing the Rooftop Solar Photovoltaic (PV) System by Analyzing the Building Shadow Using Hillshade Analysis

AbstractFor continuous promotion of the solar PV system in buildings, it is crucial to analyze the rooftop solar PV potential. However, the rooftop solar PV potential in urban areas highly varies depending on the available rooftop area due to the building shadow. In order to estimate the available rooftop area accurately by considering the building shadow, this study proposed an estimation method of the available rooftop area for installing the rooftop solar PV system by analyzing the building shadow using Hillshade Analysis. A case study of Gangnam district in Seoul, South Korea was shown by applying the proposed estimation method

The Wiener criterion for nonlocal Dirichlet problems

We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.Comment: 39 page

Robust Near-Diagonal Green Function Estimates

Kaßmann M, Kim M, Lee K-A. Robust Near-Diagonal Green Function Estimates. International Mathematics Research Notices. 2023: rnad106.We prove sharp near-diagonal pointwise bounds for the Green function G(O)(x,y) for nonlocal operators of fractional order a ? (0, 2). The novelty of our results is two-fold: the estimates are robust as a? 2- and we prove the bounds without making use of the Dirichlet heat kernel p(O)(t; x, y). In this way, we can cover cases, in which the Green function satisfies isotropic bounds but the heat kernel does not

Harnack inequality for nonlocal operators on manifolds with nonnegative curvature

Kim J, Ki M, Lee K-A. Harnack inequality for nonlocal operators on manifolds with nonnegative curvature. Calculus of Variations and Partial Differential Equations. 2022;61(1): 22.We establish the Krylov-Safonov Harnack inequalities and Holder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity estimates for these operators which recover the classical estimates for second order local operators as limits

The Wiener Criterion for Nonlocal Dirichlet Problems

Ki M, Lee K-A, Lee S-C. The Wiener Criterion for Nonlocal Dirichlet Problems. Communications in Mathematical Physics . 2023.We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability s is an element of (0, 1) and summability p > 1. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory

PsyNet: Self-Supervised Approach to Object Localization Using Point Symmetric Transformation

Existing co-localization techniques significantly lose performance over weakly or fully supervised methods in accuracy and inference time. In this paper, we overcome common drawbacks of co-localization techniques by utilizing self-supervised learning approach. The major technical contributions of the proposed method are two-fold. 1) We devise a new geometric transformation, namely point symmetric transformation and utilize its parameters as an artificial label for self-supervised learning. This new transformation can also play the role of region-drop based regularization. 2) We suggest a heat map extraction method for computing the heat map from the network trained by self-supervision, namely class-agnostic activation mapping. It is done by computing the spatial attention map. Based on extensive evaluations, we observe that the proposed method records new state-of-the-art performance in three fine-grained datasets for unsupervised object localization. Moreover, we show that the idea of the proposed method can be adopted in a modified manner to solve the weakly supervised object localization task. As a result, we outperform the current state-of-the-art technique in weakly supervised object localization by a significant gap