A Talenti-type comparison theorem for RCD(K, N) spaces and applications
We prove pointwise and L p -gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an RCD(K, N) metric measure space, with K > 0 and N ∈ (1, ∞)). The obtained Talenti-type c...
| मुख्य लेखकों: | , |
|---|---|
| स्वरूप: | Journal article |
| भाषा: | English |
| प्रकाशित: |
Springer Nature
2021
|
| _version_ | 1826277126669598720 |
|---|---|
| author | Mondino, A Vedovato, M |
| author_facet | Mondino, A Vedovato, M |
| author_sort | Mondino, A |
| collection | OXFORD |
| description | We prove pointwise and L
p
-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more
precisely of an RCD(K, N) metric measure space, with K > 0 and N ∈ (1, ∞)). The obtained
Talenti-type comparison is sharp, rigid and stable with respect to L
2/measured-Gromov-Hausdorff
topology; moreover it seems new even for smooth Riemannian manifolds. As applications of such
Talenti-type comparison, we prove a series of improved Sobolev-type inequalities, and an RCD
version of the St. Venant-Pólya torsional rigidity comparison theorem (with associated rigidity and
stability statements). Finally, we give a probabilistic interpretation (in the setting of smooth Riemannian manifolds) of the aforementioned comparison results, in terms of exit time from an open
subset for the Brownian motion. |
| first_indexed | 2024-03-06T23:24:12Z |
| format | Journal article |
| id | oxford-uuid:69d015db-d727-4547-9dbf-3739187dba92 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:24:12Z |
| publishDate | 2021 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | oxford-uuid:69d015db-d727-4547-9dbf-3739187dba922022-03-26T18:53:26ZA Talenti-type comparison theorem for RCD(K, N) spaces and applicationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:69d015db-d727-4547-9dbf-3739187dba92EnglishSymplectic ElementsSpringer Nature2021Mondino, AVedovato, MWe prove pointwise and L p -gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an RCD(K, N) metric measure space, with K > 0 and N ∈ (1, ∞)). The obtained Talenti-type comparison is sharp, rigid and stable with respect to L 2/measured-Gromov-Hausdorff topology; moreover it seems new even for smooth Riemannian manifolds. As applications of such Talenti-type comparison, we prove a series of improved Sobolev-type inequalities, and an RCD version of the St. Venant-Pólya torsional rigidity comparison theorem (with associated rigidity and stability statements). Finally, we give a probabilistic interpretation (in the setting of smooth Riemannian manifolds) of the aforementioned comparison results, in terms of exit time from an open subset for the Brownian motion. |
| spellingShingle | Mondino, A Vedovato, M A Talenti-type comparison theorem for RCD(K, N) spaces and applications |
| title | A Talenti-type comparison theorem for RCD(K, N) spaces and applications |
| title_full | A Talenti-type comparison theorem for RCD(K, N) spaces and applications |
| title_fullStr | A Talenti-type comparison theorem for RCD(K, N) spaces and applications |
| title_full_unstemmed | A Talenti-type comparison theorem for RCD(K, N) spaces and applications |
| title_short | A Talenti-type comparison theorem for RCD(K, N) spaces and applications |
| title_sort | talenti type comparison theorem for rcd k n spaces and applications |
| work_keys_str_mv | AT mondinoa atalentitypecomparisontheoremforrcdknspacesandapplications AT vedovatom atalentitypecomparisontheoremforrcdknspacesandapplications AT mondinoa talentitypecomparisontheoremforrcdknspacesandapplications AT vedovatom talentitypecomparisontheoremforrcdknspacesandapplications |