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Spectral stability of the Neumann Laplacian
 J. Diff. Eq
"... We prove the equivalence of Hardy and Sobolevtype inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of th ..."
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We prove the equivalence of Hardy and Sobolevtype inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.
A Remark On A Weighted Landau Inequality Of Kwong And Zettl
"... In this note we extend a Theorem of Kwong and Zettl concerning the inequality Z 1 0 t fi ju 0 j p K `Z 1 0 t fl juj p ' 1=2 `Z 1 0 t ff ju 00 j p ' 1=2 to all ff; fi; fl such that fi = (ff + fl)=2 except for the triple: ff = p \Gamma 1, fi = \Gamma1, fl = \Gamma1 \Gamma p. In this case the inequalit ..."
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In this note we extend a Theorem of Kwong and Zettl concerning the inequality Z 1 0 t fi ju 0 j p K `Z 1 0 t fl juj p ' 1=2 `Z 1 0 t ff ju 00 j p ' 1=2 to all ff; fi; fl such that fi = (ff + fl)=2 except for the triple: ff = p \Gamma 1, fi = \Gamma1, fl = \Gamma1 \Gamma p. In this case the inequality is false; however u satisfies the inequality Z 1 0 t fi ju 0 j p K 1 ( `Z 1 0 t fl juj p ' 1=2 `Z 1 0 t ff ju 00 j p ' 1=2 + Z 1 0 t fl juj p ).