Harnack inequality for nonlocal operators on manifolds with nonnegative curvature

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators [Formula: see text]. Our focus is on non-Euclidean situations.

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Local versus nonlocal elliptic equations: short-long range field interactions

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0166 ◽

2020 ◽

Vol 10 (1) ◽

pp. 895-921

Author(s):

Daniele Cassani ◽

Luca Vilasi ◽

Youjun Wang

Keyword(s):

Asymptotic Behavior ◽

Maximum Principle ◽

Weak Solutions ◽

Elliptic Equations ◽

Spectral Properties ◽

Fractional Laplacian ◽

Coupling Parameter ◽

Nonlocal Operators ◽

The Maximum Principle ◽

Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

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Harnack’s inequality for doubly nonlinear equations of slow diffusion type

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02044-z ◽

2021 ◽

Vol 60 (6) ◽

Author(s):

Verena Bögelein ◽

Andreas Heran ◽

Leah Schätzler ◽

Thomas Singer

Keyword(s):

Vector Field ◽

Harnack Inequality ◽

Parabolic Equations ◽

Full Range ◽

Nonlinear Parabolic Equations ◽

Growth Conditions ◽

Diffusion Type ◽

Slow Diffusion ◽

Nonlinear Parabolic ◽

Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

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A path integral formulation for particle detectors: the Unruh-DeWitt model as a line defect

Journal of High Energy Physics ◽

10.1007/jhep03(2021)076 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Ivan M. Burbano ◽

T. Rick Perche ◽

Bruno de S. L. Torres

Keyword(s):

Path Integral ◽

Degrees Of Freedom ◽

Wilson Line ◽

Transition Probability ◽

Relativistic Quantum ◽

Quantum Fields ◽

Line Defect ◽

Particle Detectors ◽

Nonlocal Operators ◽

Detector Model

Abstract Particle detectors are an ubiquitous tool for probing quantum fields in the context of relativistic quantum information (RQI). We formulate the Unruh-DeWitt (UDW) particle detector model in terms of the path integral formalism. The formulation is able to recover the results of the model in general globally hyperbolic spacetimes and for arbitrary detector trajectories. Integrating out the detector’s degrees of freedom yields a line defect that allows one to express the transition probability in terms of Feynman diagrams. Inspired by the light-matter interaction, we propose a gauge invariant detector model whose associated line defect is related to the derivative of a Wilson line. This is another instance where nonlocal operators in gauge theories can be interpreted as physical probes for quantum fields.

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