Nonlocal Harnack inequalities in the Heisenberg group
We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group H n , whose prototype is the Dirichlet problem for the p -fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase trans...
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| Veröffentlicht in: | Calculus of variations and partial differential equations 2022-10, Vol.61 (5), Article 185 |
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| container_title | Calculus of variations and partial differential equations |
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| creator | Palatucci, Giampiero Piccinini, Mirco |
| description | We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group
H
n
, whose prototype is the Dirichlet problem for the
p
-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is
p
=
2
, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent
s
goes to 1. |
| doi_str_mv | 10.1007/s00526-022-02301-9 |
| format | Article |
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H
n
, whose prototype is the Dirichlet problem for the
p
-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is
p
=
2
, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent
s
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H
n
, whose prototype is the Dirichlet problem for the
p
-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is
p
=
2
, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent
s
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H
n
, whose prototype is the Dirichlet problem for the
p
-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is
p
=
2
, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent
s
goes to 1.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00526-022-02301-9</doi><orcidid>https://orcid.org/0000-0002-3706-9349</orcidid><oa>free_for_read</oa></addata></record> |
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| language | eng |
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| source | SpringerNature Journals |
| subjects | Analysis Asymptotic properties Calculus of Variations and Optimal Control Optimization Control Dirichlet problem Euclidean geometry Ferromagnetism Inequalities Mathematical and Computational Physics Mathematics Mathematics and Statistics Operators (mathematics) Phase transitions Quantum mechanics Systems Theory Theoretical |
| title | Nonlocal Harnack inequalities in the Heisenberg group |
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