Unified Poincaré and Hardy inequalities with sharp constants for convex domains. (English) Zbl 1145.26005
The authors prove five attractive theorems. The main result reads as follows:
Let \(\Omega\) be an open and convex set in \({\mathbb R}^n\). If the inradius \(\delta_0:=\delta_0(\Omega)\) is finite, then
\[ \int_\Omega| \nabla f| ^2\,dx\geq \frac 14 \int_\Omega\frac{| f| ^2}{\delta^2}\,dx+ \frac{\lambda_0^2}{\delta^2_0} \int_\Omega| f| ^2\,dx,\quad\forall\,f\in H_0^1(\Omega),\tag {\(\ast\)} \]
where \(\lambda_0\) is the Lamb constant. The inequality \((*)\) is sharp for all dimensions \(n\geq 1.\) The factor \(\lambda_0^2\) in \((*)\) is actually attained for any finite interval \((a,b)\subset{\mathbb R}\) and for all domain of the form \((a,b)\times\mathbb R^{n-1}\subset\mathbb R^n\) \(n\geq 2\), and their linear transformations.
Let \(\Omega\) be an open and convex set in \({\mathbb R}^n\). If the inradius \(\delta_0:=\delta_0(\Omega)\) is finite, then
\[ \int_\Omega| \nabla f| ^2\,dx\geq \frac 14 \int_\Omega\frac{| f| ^2}{\delta^2}\,dx+ \frac{\lambda_0^2}{\delta^2_0} \int_\Omega| f| ^2\,dx,\quad\forall\,f\in H_0^1(\Omega),\tag {\(\ast\)} \]
where \(\lambda_0\) is the Lamb constant. The inequality \((*)\) is sharp for all dimensions \(n\geq 1.\) The factor \(\lambda_0^2\) in \((*)\) is actually attained for any finite interval \((a,b)\subset{\mathbb R}\) and for all domain of the form \((a,b)\times\mathbb R^{n-1}\subset\mathbb R^n\) \(n\geq 2\), and their linear transformations.
Reviewer: László Leindler (Szeged)
MSC:
| 26D15 | Inequalities for sums, series and integrals |
References:
| [1] | and , (eds.) Handbook of Mathematical Functions, (Dover Publ., New York, 1968). |
| [2] | Avkhadiev, Lobachevskii J. Math. 21 pp 3– (2006) |
| [3] | Avkhadiev, Proc. Steklov Inst. Math 255255 pp 1– (2006) |
| [4] | Isoperimetric Inequalities and Applications (Pitman Adv. Publ. Program, Boston-London-Melbourne, 1980). · Zbl 0436.35063 |
| [5] | and , Table of inequalities in elliptic boundary value problems, in: Recent Progress in Inequalities, edited by V. Milovanovic (Kluwer Academic Publ., Dordrecht, 1998), pp. 97–125. · Zbl 0914.35042 |
| [6] | Brezis, Ann. Scuola Sup. Pisa Cl. Sci. 25(4) pp 217– |
| [7] | Davies, Quart. J. Math. Oxford 46 pp 2– (1995) |
| [8] | Davies, Oper. Theory Adv. Appl. 110 pp 55– (1999) |
| [9] | Fillippas, Calc. Var. Partial Differential Equations 25(4) pp 491– (2005) |
| [10] | Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957). · Zbl 0078.35703 |
| [11] | , and , Inequalities (Cambridge Univ. Press, Cambridge, 1934). |
| [12] | Hersch, J. Math. Phys. Appl. 11 pp 387– (1960) |
| [13] | Hoffmann-Ostenhof, J. Func. Anal. 189 pp 539– (2002) |
| [14] | Differentialgleichungen. Lösungsmethoden und Lösungen (B. G. Teubner, Stuttgart, 1977). |
| [15] | Lamb, Proc. Lond. Math. Soc. pp 270– (1884) |
| [16] | Marcus, Trans. Amer. Math. Soc. 350 pp 3237– (1998) |
| [17] | Matskewich, Nonlinear Anal. 28 pp 1601– (1997) |
| [18] | Payne, Applicable Anal. 3 pp 295– (1973) |
| [19] | and , Isoperimetric Inequalities in Mathematical Physics (Princeton University Press, Princeton, 1951). · Zbl 0044.38301 |
| [20] | Theory of the Bessel Functions, Second edition (Cambridge Univ. Press, Cambridge, 1962). |
| [21] | Committee for the calculation of math. tables, British Assoc. Adv. Sci. Mathematical Tables, Vol. VI Bessel Functions, Part I, Functions of orders zero and unity (Cambridge Univ. Press, Cambridge, 1937). |
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