We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the ...distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of
L
q
-properties of the gradient and of the self-adjointness of Schrödinger-type operators.
The $$ L^1 $$-Liouville Property on Graphs Adriani, Andrea; Setti, Alberto G.
The Journal of fourier analysis and applications,
08/2023, Letnik:
29, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Abstract
In this paper we investigate the
$$ L^1 $$
L
1
-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a ...characterization of the
$$ L^1 $$
L
1
-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the
$$ L^1 $$
L
1
-Liouville property and prove some comparison theorems for general graphs based on inner–outer curvatures. We also introduce the Dirichlet
$$L^1$$
L
1
-Liouville property of subgraphs and prove that if a graph has a Dirichlet
$$L^1$$
L
1
-Liouville subgraph, then it is
$$L^1$$
L
1
-Liouville itself. As a consequence, we obtain that the
$$ L^1$$
L
1
-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is
$$ L^1$$
L
1
-Liouville provided that at least one of its ends is Dirichlet
$$ L^1$$
L
1
-Liouville.
The L1-Liouville Property on Graphs Adriani, Andrea; Setti, Alberto G.
The Journal of fourier analysis and applications,
2023/8, Letnik:
29, Številka:
4
Journal Article
Recenzirano
Odprti dostop
In this paper we investigate the
L
1
-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the
L
1
...-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the
L
1
-Liouville property and prove some comparison theorems for general graphs based on inner–outer curvatures. We also introduce the Dirichlet
L
1
-Liouville property of subgraphs and prove that if a graph has a Dirichlet
L
1
-Liouville subgraph, then it is
L
1
-Liouville itself. As a consequence, we obtain that the
L
1
-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is
L
1
-Liouville provided that at least one of its ends is Dirichlet
L
1
-Liouville.
We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. ...For changing sign integrable data, we show existence and uniqueness under extra assumptions such as local finiteness or a uniform lower bound on the node measure.
The aim of this paper is to introduce the reader to various forms of the maximum principle, starting from its classical formulation up to generalizations of the Omori-Yau maximum principle at ...infinity recently obtained by the authors. Applications are given to a number of geometrical problems in the setting of complete Riemannian manifolds, under assumptions either on the curvature or on the volume growth of geodesic balls.
Height Estimates for Killing Graphs Impera, Debora; de Lira, Jorge H.; Pigola, Stefano ...
The Journal of geometric analysis,
07/2018, Letnik:
28, Številka:
3
Journal Article
Recenzirano
Odprti dostop
The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with non-empty boundary. To this end, we first point out how the geometric analysis on a Killing ...graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height function in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled.
This book presents very recent results involving an extensive use of analytical tools in the study of geometrical and topological properties of complete Riemannian manifolds. It analyzes in detail an ...extension of the Bochner technique to the non compact setting, yielding conditions which ensure that solutions of geometrically significant differential equations either are trivial (vanishing results) or give rise to finite dimensional vector spaces (finiteness results). The book develops a range of methods from spectral theory and qualitative properties of solutions of PDEs to comparison theorems in Riemannian geometry and potential theory. All needed tools are described in detail, often with an original approach. Some of the applications presented concern the topology at infinity of submanifolds, Lp cohomology, metric rigidity of manifolds with positive spectrum, and structure theorems for Kähler manifolds. The book is essentially self-contained and supplies in an original presentation the necessary background material not easily available in book form.
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and
L
p
-Liouville type ...results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under
L
p
conditions on the relevant quantities.
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