A sharp pointwise bound for functions with $L^2$-Laplacians on arbitrary domains and its applications
Abstract
For all functions on an arbitrary open set $\Omega\subset\R^3$ with zero boundary values, we prove the optimal bound \[ \sup_{\Omega}|u| \leq (2\pi)^{-1/2} \left(\int_{\Omega}|\nabla u|^2 \,dx\, \int_{\Omega}|\Delta u|^2 \,dx\right)^{1/4}. \] The method of proof is elementary and admits generalizations. The inequality is applied to establish an existence theorem for the Burgers equation.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- March 1992
- DOI:
- 10.48550/arXiv.math/9204239
- arXiv:
- arXiv:math/9204239
- Bibcode:
- 1992math......4239X
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 5 pages