Published in Potential Analysis
We prove in this paper that there exists no immersed closed curve in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb R}^2$\end{document} with curvature increasi...
Published in Potential Analysis
We consider the mixed boundary value problem, or Zaremba’s problem, for the Laplacian in a bounded Lipschitz domain Ω in Rn, n ≥ 2. We decompose the boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...
Published in Potential Analysis
We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.
Published in Potential Analysis
The upper escape rate for a Markov process is a natural partial generalization of the celebrated Khintchine’s law of the iterated logarithm. In this article, we present a new proof of the sharp volume growth criterion for upper escape rate of continuous time random walks on weighted graphs in Huang and Shiozawa (Stoch. Process Appl. 124(1), 317–347...
Published in Potential Analysis
In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov equations are n-times continuously Fréchet differentiable, then so are the generalized solutio...
Published in Potential Analysis
The lowest eigenvalue of the Schrödinger operator −Δ+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-{\Delta }+\mathcal {V}$\end{document} on a compact Riemannian mani...
Published in Potential Analysis
We consider the Kirchhoff equation −1+λ∫|∇u|2Δu+V(x)u=f(u)inℝN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Big(1+\lambda \int |\nabla u|^{2}\Big){\Delta} u+V(x)u=...
Published in Potential Analysis
We suggest a new approach to stochastic integration in infinite-dimensional spaces that is based on representing random variables on Banach spaces as real-valued processes on an interval. We prove stochastic integrability of operator-valued processes on general separable Banach spaces under the conditions that do not depend on the norm of the space...
Published in Potential Analysis
We compute the best constant in the embedding of WN,1(ℝN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^{N,1}(\mathbb {R} ^{N})$\end{document} into L∞(ℝN)\documentcla...
Published in Potential Analysis
In this article we study nonnegative solutions of quasilinear equation model of which is −△pu+V(x)f(u)=h(x)|∇u|p−1+g(x),p>1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen...
