The Orlicz Brunn-Minkowski inequality for the first eigenvalue of \(p\)-Laplacian. (English) Zbl 1526.35127
In this paper, the authors considered the following nonlinear eigenvalue problem
\[
\left\{ \begin{array}{ll} \Delta_p u =- \lambda(K)|u|^{p-2}u, \ \ \ \ & x\in \Omega,\\
u=0 , & x\in {\partial\Omega}, \end{array} \right. \tag{S1}
\]
where \(p>1\), \(K\in \mathcal{K}^n\), \(\Omega\) is the interior of \(K\), \(\mathcal{K}^n\) is the set of convex body in \(\mathbb{R}^n\) and \(\lambda(K)\) is the first eigenvalue of \(p-\)Laplacian operator. Similar to the Orlicz Hadamard variational formulas for volume and motivated by the works [A. Colesanti et al., Adv. Math. 285, 1511–1588 (2015; Zbl 1327.31024); W. Wang et al., J. Funct. Spaces 2020, Article ID 1670617, 8 p. (2020; Zbl 1454.52007)], the authors show the Orlicz Hadamard variational formulas for \(\lambda(K)\) and obtain the Orlicz Brunn-Minkowski inequality under some sooth assumptions on \(K,L\) and \(\phi\). They also prove the uniqueness of the Orlicz mixed first eigenvalue of \(p\)-Laplace with \(p>1\).
Reviewer: Qin Dongdong (Changsha)
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
the first eigenvalue of \(p\)-Laplacian; the Orlicz Hadamard variational formula; the Orlicz Brunn-Minkowski inequalityReferences:
| [1] | Caglar, U.; Ye, D., Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory, Adv. Appl. Math., 81, 78-114 (2016) · Zbl 1357.52001 |
| [2] | Colesanti, A.; Cuoghi, P.; Salani, P., Brunn-Minkowski inequalities for two functionals involving the p-Laplace operator, Appl. Anal., 85, 1-3, 45-66 (2006) · Zbl 1151.52307 |
| [3] | Colesanti, A.; Nyström, K.; Salani, P.; Xiao, J.; Yang, D.; Zhang, G., The Hadamard variational formula and the Minkowski problem for p-capacity, Adv. Math., 285, 1511-1588 (2015) · Zbl 1327.31024 |
| [4] | Firey, W., p-means of convex bodies, Math. Scand., 10, 17-24 (1962) · Zbl 0188.27303 |
| [5] | Gardner, R.; Hug, D.; Weil, W., The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities, J. Differ. Geom., 97, 3, 427-476 (2014) · Zbl 1303.52002 |
| [6] | Gardner, R.; Hug, D.; Weil, W.; Ye, D., The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl., 430, 2, 810-829 (2015) · Zbl 1320.52008 |
| [7] | Haberl, C.; Lutwak, E.; Yang, D.; Zhang, G., The even Orlicz Minkowski problem, Adv. Math., 224, 6, 2485-2510 (2010) · Zbl 1198.52003 |
| [8] | Huang, Q.; He, B., On the Orlicz Minkowski problem for polytopes, Discrete Comput. Geom., 48, 2, 281-297 (2012) · Zbl 1255.52006 |
| [9] | Huang, Y.; Lutwak, E.; Yang, D.; Zhang, G., Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216, 2, 325-388 (2016) · Zbl 1372.52007 |
| [10] | Huang, Y.; Song, C.; Xu, L., Hadamard variational formulas for p-torsion and p-eigenvalue with applications, Geom. Dedic., 197, 1, 61-76 (2018) · Zbl 1403.35117 |
| [11] | Hug, D.; Lutwak, E.; Yang, D.; Zhang, G., On the \(L_p\) Minkowski problem for polytopes, Discrete Comput. Geom., 33, 4, 699-715 (2005) · Zbl 1078.52008 |
| [12] | Ludwig, M., General affine surface areas, Adv. Math., 224, 6, 2346-2360 (2010) · Zbl 1198.52004 |
| [13] | Ludwig, M.; Reitzner, M., A classification of SL(n) invariant valuations, Ann. Math. (2), 172, 2, 1219-1267 (2010) · Zbl 1223.52007 |
| [14] | Lutwak, E., The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differ. Geom., 38, 1, 131-150 (1993) · Zbl 0788.52007 |
| [15] | Lutwak, E.; Yang, D.; Zhang, G., Orlicz centroid bodies, J. Differ. Geom., 84, 2, 365-387 (2010) · Zbl 1206.49050 |
| [16] | Lutwak, E.; Yang, D.; Zhang, G., Orlicz projection bodies, Adv. Math., 223, 1, 220-242 (2010) · Zbl 1437.52006 |
| [17] | Sakaguchi, S., Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 14, 3, 403-421 (1987) · Zbl 0665.35025 |
| [18] | Schneider, R., Convex Bodies: the Brunn-Minkowski Theory (2014), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1287.52001 |
| [19] | Wang, W.; Li, J.; He, R.; Liu, L., The functional Orlicz Brunn-Minkowski inequality for q-capacity, J. Funct. Spaces, Article 1670617 pp. (2020) · Zbl 1454.52007 |
| [20] | Xi, D.; Jin, H.; Leng, G., The Orlicz Brunn-Minkowski inequality, Adv. Math., 260, 350-374 (2014) · Zbl 1357.52004 |
| [21] | Zou, D.; Xiong, G., A unified treatment for \(L_p\) Brunn-Minkowski type inequalities, Commun. Anal. Geom., 26, 2, 435-460 (2018) · Zbl 1391.52011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
