## Topological Methods in Nonlinear Analysis

### On certain variant of strongly nonlinear multidimensional interpolation inequality

#### Abstract

We obtain the inequality $\int_{\Omega}|\nabla u(x)|^ph(u(x))\,dx \leq C(n,p)\int_{\Omega} \Big( \sqrt{ |\nabla^{(2)} u(x)| |{\mathcal T}_{h,C}(u(x))|}\Big)^{p}h(u(x))\,dx,$ where $\Omega\subset \mathbb R^n$ and $n\ge 2$, $u\colon\Omega\rightarrow \mathbb R$ is in certain subset in second order Sobolev space $W^{2,1}_{\rm loc}(\Omega)$, $\nabla^{(2)} u$ is the Hessian matrix of $u$, ${\mathcal T}_{h,C}(u)$ is a certain transformation of the continuous function $h(\,\cdot\,)$. Such inequality is the generalization of a similar inequality holding in one dimension, obtained earlier by second author and Peszek.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 1 (2018), 49-67.

Dates
First available in Project Euclid: 24 April 2018

https://projecteuclid.org/euclid.tmna/1524535226

Digital Object Identifier
doi:10.12775/TMNA.2017.050

Mathematical Reviews number (MathSciNet)
MR3867979

Zentralblatt MATH identifier
07029861

#### Citation

Choczewski, Tomasz; Kałamajska, Agnieszka. On certain variant of strongly nonlinear multidimensional interpolation inequality. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 49--67. doi:10.12775/TMNA.2017.050. https://projecteuclid.org/euclid.tmna/1524535226

#### References

• S. Bloom, First and second order Opial inequalities, Studia Math. 126 (1997), no. 1, 27–50.
• R. Brown, V. Burenkov, S. Clark and D. Hinton, Second order Opial inequalities in ${\Cal L}^p$ spaces and applications, Analytic and Geometric Inequalities and Applications (Rassias, Themistocles et al., eds.), Mathematics and its Applications, vol. 478, pp. 37–52, Kluwer, Dordrecht, 1999.
• C. Capogne, A. Fiorenza and A. Kałamajska, Strongly nonlinear Gagliardo–Nirenberg inequality in Orlicz spaces and Boyd indices, Rend. Lincei Mat. Appl. 28 (2017), 119–141.
• P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, New York, Providence, 2010.
• H. Federer, Geometric Measure Theory, Springer, New York, 1969.
• E. Fermi, Un methodo statistico par la determinazione di alcune properitá dell' atoma, Rend. Accad. Naz. del Lincei Cl. Sci. Fis. Mat. e Nat. 6 (1927), 602–607.
• E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 8 (1959), 24–51.
• A. Kałamajska and K. Mazowiecka, Some regularity results to the generalized Emden–Fowler equation with irregular data, Math. Methods Appl. Sci. 38 (2015), no. 12, 2479–2495.
• A. Kałamajska and J. Peszek, On some nonlinear extensions of the Gagliardo–Nirenberg inequality with applications to nonlinear eigenvalue problems, Asymptot. Anal. 77 (2012), no. 3–4, 169–196.
• A. Kałamajska and J. Peszek, On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities, J. Fixed Point Theory Appl. 13 (2013), no. 1, 271–290.
• A. Kałamajska and K. Pietruska-Pałuba, Gagliardo–Nirenberg inequalities in Orlicz spaces, Indiana Univ. Math. J. 55 (2006), no. 6, 1767–1789.
• A. Kufner, O. John and S. Fučík, Function Spaces, Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff, Leyden; Academia, Prague, 1977.
• V.G. Mazy'a, Sobolev Spaces, Springer, Berlin, 1985.
• L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. di Pisa 13 (1959), 115–162.
• Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29–32.
• J. Peszek, Discrete Cucker–Smale flocking model with a weakly singular weight, SIAM J. Math. Anal. 47 (2015), no. 5, 3671–3686.
• L.H. Thomas, The calculation of atomic fields, Proc. Camb. Phil. Soc. 23 (1927), 1473–1484.