Monotonicity with respect to \(p\) of the best constants associated with Sobolev immersions of type \(W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega)\) when \(q\in\{1,p,\infty\}\)

Mihai Mihăilescu, Denisa Stancu-Dumitru

Abstract


The goal of this paper is to collect some known results on the monotonicity with respect to \(p\) of the best constants associated with Sobolev immersions of type \(W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega)\) when \(q\in\{1,p,\infty\}\). More precisely, letting \(\lambda(p,q;\Omega):=\inf_{u\in W_0^{1,p}(\Omega)\setminus\{0\}}{\|\;|\nabla u|_D\;\|_{L^p(\Omega)}}{\|u\|_{L^q(\Omega)}^{-1}}\) we recall some monotonicity results related with the following functions
\begin{eqnarray*}
(1,\infty)\ni p&\mapsto &|\Omega|^{p-1}\lambda(p,1;\Omega)^p\,,\\
(1,\infty)\ni p&\mapsto &\lambda(p,p;\Omega)^p\,,\\
(D,\infty)\ni p&\mapsto &\lambda(p,\infty;\Omega)^p\,,
\end{eqnarray*}
when \(\Omega\subset \mathbb{R}^{D}\) is a given open, bounded and convex set with smooth boundary.

 


Keywords


\(p\)-Laplacian; \(p\)-torsional rigidity; distance function to the boundary

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References


Benedikt, J., Drabek, P., Asymptotics for the principal eigenvalue of the p-Laplacian on the ball as p approaches 1, Nonlinear Anal., 93(2013), 23-29.

Bhattacharya, T., DiBenedetto, E., Manfredi, J.J., Limits as $ptoinfty$ of $Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1991), 15-68.

Bobkov, V., Tanaka, M., On positive solutions for (p,q)-Laplace equations with two parameters, Calc. Var. Partial Differential Equations, 54(2015), 3277-3301.

Bocea, M., Mihailescu, M., Minimization problems for inhomogeneous Rayleigh quotients, Communications in Contemporary Mathematics, 20(2018), 1750074, 13 pp.

Bocea, M., Mihailescu, M., On the monotonicity of the principal frequency of the p-Laplacian, Adv. Calc. Var., 14(2021), 147-152.

Bocea, M., Mihailescu, M., Stancu-Dumitru, D., The monotonicity of the principal frequency of the anisotropic p-Laplacian, Comptes Rendus Mathematique, 360(2022), 993-1000.

Brasco, L., On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique, ESAIM: Control, Optimisation and Calculus of Variations, 20(2014), 315-338.

Brasco, L., On principal frequencies and inradius of convex sets, Bruno Pini Mathematical Analysis Seminar, 9(2018), 78-101.

Brezis, H., Functional Analysis, Sobolev Spaces and Partial Di erential Equations, Universitext, Springer, New York, 2011, xiv+599 pp.

Briani, L., Buttazzo, G., Prinari, F., Inequalities between torsional rigidity and principal eigenvalue of the p-Laplacian, Calc. Var. Partial Differential Equations, 61(2022), no. 2, Paper No. 78, 25 pp.

Bueno, H., Ercole, G., Solutions of the Cheeger problem via torsion functions, J. Math. Anal. Appl., 381(2011), 263-279.

Bueno, H., Ercole, G., Macedo, S.S., Asymptotic behavior of the p-torsion functions as p goes to 1, Arch. Math., 107(2016), 63-72.

Della Pietra, F., Gavitone, N., Guarino Lo Bianco, S., On functionals involving the torsional rigidity related to some classes of nonlinear operators, J. Differential Equations, 265(2018), 6424-6442.

Enache, C., Mihailescu, M., A Monotonicity Property of the p-Torsional Rigidity, Non-linear Analysis, 208(2021), Article 112326.

Enache, C., Mihailescu, M., Stancu-Dumitru, D., The monotonicity of the p-torsional rigidity in convex domains, Mathematische Zeitschrift, 302(2022), 419-431.

Ercole, G., Pereira, G.A., Asymptotics for the best Sobolev constants and their extremal functions, Math. Nachr., 289(2016), 1433-1449.

F arcaseanu, M., Mihailescu, M., On the monotonicity of the best constant of Morrey's inequality in convex domains, Proceedings of the American Mathematical Society, 150(2022), 651-660.

Fukagai, N., Ito, M., Narukawa, K., Limit as $ptoinfty$ of p-Laplace eigenvalue problems and $L^infty$-inequality of Poincare type, Differential Integral Equations, 12(1999), 183-206.

Grecu, A., Mihailescu, M., Monotonicity of the principal eigenvalue of the p-Laplacian on an annulus, Math. Reports, 23(2021), 149-155.

Hynd, R., Lindgren, E., Extremal functions for Morrey's inequality in convex domains, Math. Ann., 375(2019), 1721-1743.

Juutinen, P., Lindqvist, P., Manfredi, J.J., The $infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148(1999), 89-105.

Kajikiya, R., A priori estimate for the first eigenvalue of the p-Laplacian, Differential Integral Equations, 28(2015), 1011-1028.

Kajikiya, R., Tanaka M., Tanaka S., Bifurcation of positive solutions for the one-dimensional (p,q)-Laplace equation, Electron. J. Differential Equations, 107(2017), 1-37.

Kawohl, B., On a family of torsional creep problems, J. Reine Angew. Math., 410(1990), 1-22.

Le, A., Eigenvalue problems for the p-Laplacian, Nonlinear Analysis, 64(2006), 1057-1099.

Lindqvist, P., On the equation $div(|nabla u|^{p-2}nabla u)+lambda|u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109(1990), 157-164.

Lindqvist, P., On non-linear Rayleigh quotients, Potential Anal., 2(1993), 199-218.

Lindqvist, P., Note on a nonlinear eigenvalue problem, Rocky Mountain J. Math., 23(1993), 281-288.

Lindqvist, P., A nonlinear eigenvalue problem, Ciatti, Paolo (ed.) et al., Topics in Mathematical Analysis, 175-203, Hackensack, NJ: World Scientific (ISBN 978-981-281-105-

/hbk). Series on Analysis Applications and Computation 3, 2008).

Mihailescu, M., Monotonicity properties for the variational Dirichlet eigenvalues of the p-Laplace operator, Journal of Differential Equations, 335(2022), 103-119.

Mihailescu, M., Rossi, J.D., Monotonicity with respect to p of the first nontrivial eigenvalue of the p-Laplacian with homogeneous Neumann boundary conditions, Comm. Pure Appl. Anal., 19(2020), 4363-4371.

Payne, L.E., Philippin, G.A., Some applications of the maximum principle in the problem of torsional creep, SIAM J. Appl. Math., 33(1977), 446-455.




DOI: http://dx.doi.org/10.24193/subbmath.2023.1.08

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