Recent zbMATH articles in MSC 31B35https://zbmath.org/atom/cc/31B352024-03-13T18:33:02.981707ZWerkzeugCharacterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional \(p\)-Laplace and the fractional \(p\)-convexityhttps://zbmath.org/1528.310022024-03-13T18:33:02.981707Z"Shi, Shaoguang"https://zbmath.org/authors/?q=ai:shi.shaoguang"Zhai, Zhichun"https://zbmath.org/authors/?q=ai:zhai.zhichun"Zhang, Lei"https://zbmath.org/authors/?q=ai:zhang.lei.15Summary: In this paper, when studying the connection between the fractional convexity and the fractional \(p\)-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that \(u(x)\) is the viscosity sub-solution of the equation if and only if \(u(x)\) is so-called \((\alpha,p)\)-convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an \((\alpha,p)\)-convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.On single-layer potentials, pseudo-gradients and a jump theorem for an isotropic \(\alpha\)-stable stochastic processhttps://zbmath.org/1528.600422024-03-13T18:33:02.981707Z"Mamalyha, Khrystyna"https://zbmath.org/authors/?q=ai:mamalyha.khrystyna"Osypchuk, Mykhailo"https://zbmath.org/authors/?q=ai:osypchuk.m-mSummary: The aim of this paper is a behavior investigation of pseudo-gradients with respect to the spatial variable of a single-layer potential if the spatial point tends to some point in the carrier surface of the potential. The potentials connect to the generator of an isotropic \(\alpha\)-stable stochastic process with the power \(\alpha \in (1,2]\). This generator is the fractional Laplacian of the order \(\alpha\). Pseudo-gradient or fractional gradient is the pseudo-differential operator of the gradient type. Its order \(\beta\) is a positive number less than \(\alpha\). The jump theorem is known in the case of \(\beta =\alpha -1\). We present here the corresponding results in the cases of \(\beta <\alpha -1\) and \(\beta >\alpha -1\). In the first case there are no jumps, and in the second case there are no finite limits of the fractional gradients of the single-layer potential, when the spatial argument tends to some point placed on the potential carrier.