A class of weighted Hardy inequalities and applications to evolution problems. (English) Zbl 1445.35015
Summary: We state the following weighted Hardy inequality: \[ c_{o, \mu }\int_{\mathbb{R}^N}\frac{\varphi^2 }{|x|^2}\, \text{d}\mu \leq \int_{\mathbb{R}^N} |\nabla \varphi |^2 \, \text{d}\mu + K \int_{\mathbb{R}^N}\varphi^2 \, \text{d}\mu \quad \forall \, \varphi \in H_\mu^1, \] in the context of the study of the Kolmogorov operators: \[ Lu=\Delta u+\frac{\nabla \mu }{\mu }\cdot \nabla u, \] perturbed by inverse square potentials and of the related evolution problems. The function \(\mu\) in the drift term is a probability density on \(\mathbb{R}^N\). We prove the optimality of the constant \(c_{o, \mu}\) and state existence and nonexistence results following the Cabré-Martel’s approach [X. Cabré and Y. Martel, C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 11, 973–978 (1999; Zbl 0940.35105)] extended to Kolmogorov operators.
MSC:
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 34G10 | Linear differential equations in abstract spaces |
| 47D03 | Groups and semigroups of linear operators |
Citations:
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