scholarly journals The complement value problem for non-local operators

2020 ◽  
Vol 20 (03) ◽  
pp. 2050043
Author(s):  
Wei Sun
Keyword(s):  
Weak Solution ◽  
Lipschitz Domain ◽  
Dirichlet Forms ◽  
Probabilistic Representation ◽  
Kernel Estimates ◽  
Mild Conditions ◽  
Heat Kernel Estimates ◽  
Bounded Lipschitz Domain ◽  
Local Operators ◽  
Non Local

Let [Formula: see text] be a bounded Lipschitz domain of [Formula: see text]. We consider the complement value problem [Formula: see text] Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in our approach.

scholarly journals Maximum principles for time-fractional Cauchy problems with spatially non-local components

2018 ◽  
Vol 21 (5) ◽  
pp. 1335-1359 ◽  
Author(s):  
Anup Biswas ◽  
József Lőrinczi
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Strong Maximum Principle ◽  
Maximum Principles ◽  
Inverse Source Problem ◽  
Cauchy Problems ◽  
Probabilistic Representation ◽  
Local Operators ◽  
Non Local

Abstract We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate for the weak solutions of a Cauchy problem featuring Caputo time-derivatives and non-local operators in space variables given in terms of Bernstein functions of the Laplacian. To achieve this, first we propose a suitable meaning of a weak solution, show their existence and uniqueness, and establish a probabilistic representation in terms of time-changed Brownian motion. As an application, we also discuss an inverse source problem.

scholarly journals Stability of heat kernel estimates for symmetric non-local Dirichlet forms

2021 ◽  
Vol 271 (1330) ◽  
Author(s):  
Zhen-Qing Chen ◽  
Takashi Kumagai ◽  
Jian Wang
Keyword(s):  
Heat Kernel ◽  
Jump Processes ◽  
Metric Measure Spaces ◽  
Kernel Estimates ◽  
Open Problems ◽  
Content Type ◽  
Heat Kernel Estimates ◽  
Measure Spaces ◽  
Non Local ◽  
Volume Doubling

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for α \alpha -stable-like processes even with α ≥ 2 \alpha \ge 2 when the underlying spaces have walk dimensions larger than 2 2 , which has been one of the major open problems in this area.

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