Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains
By Stathis Filippas, Luisa Moschini, A. Tertikas. 2006
66 citations recorded by the source corpus.
On a smooth bounded domain $$\Omega \subset {\bf {\rm R}}^N$$ we consider the Schrödinger operators − Δ − V, with V being either the critical borderline potential V(x) = (N − 2)2/4 |x|−2 or V(x) = (1/4) dist(x, ∂Ω)−2, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/179f7a319fad1299746c98b679000577dda4ed14
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In Applied Mathematics
- Tomasz Jakubowski
- T. Grzywny
- Krzysztof Bogdan
- Junfang Li
- Dominika Pilarczyk
- R. Lewis
- Yanyan Li
- M. Ryznar
In Mathematical Physics
- T. Grzywny
- Krzysztof Bogdan
- Tomasz Jakubowski
- Junfang Li
- R. Lewis
- Yanyan Li
- M. Ryznar
- Luis Silvestre
Citations in the corpus
- A geometric characterization of a sharp Hardy inequality (2011, background): “Filippas, Maz’ya, and Tertikas [15] proved that in a small enough tubular neighborhood of the boundary of a bounded domain, a Hardy-Sobolev inequality holds.”
- Fractional Laplacian with Hardy potential (2017, background): “[11] obtained the critical case in bounded domains.”