## Illinois Journal of Mathematics

### The best constant and extremals of the Sobolev embeddings in domains with holes: The $L^\infty$ case

#### Abstract

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. We study the best constant of the Sobolev trace embedding $W^{1,\infty}(\Omega) \hookrightarrow L^\infty (\partial\Omega)$ for functions that vanish in a subset $A\subset \Omega$, which we call the hole. That is we deal with the minimization problem $S_A^T =\inf\|u\|_{W^{1,\infty}(\Omega)}/\|u\|_{L^\infty(\partial\Omega)}$ for functions that verify $u|_A = 0$. We find that there exists an optimal hole that minimizes the best constant $S_A^T$ among subsets of $\Omega$ of prescribed volume and we give a geometrical characterization of this optimal hole. In fact, minimizers associated to these holes are cones centered at some points $x_0^*$ on $\partial\Omega$ with respect to the arc-length metric in $\Omega$ and the best holes are of the form $A^*=\Omega\setminus B_d(x_0^*,r^*)$ where the ball is taken again with respect of the arc-length metric.

A similar analysis can be performed for the best constant of the embedding $W^{1,\infty}(\Omega) \hookrightarrow L^\infty( \Omega)$ with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points $x_0^*$ on $\partial\Omega$ and the best holes are of the form $A^*=\Omega\setminus B_d(x_0^*,r^*)$.

#### Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1111-1121.

Dates
First available in Project Euclid: 18 November 2009

https://projecteuclid.org/euclid.ijm/1258554352

Digital Object Identifier
doi:10.1215/ijm/1258554352

Mathematical Reviews number (MathSciNet)
MR2595757

Zentralblatt MATH identifier
1181.49006

#### Citation

Fernández Bonder, Julián; Rossi, Julio D.; Schönlieb, Carola-Bibiane. The best constant and extremals of the Sobolev embeddings in domains with holes: The $L^\infty$ case. Illinois J. Math. 52 (2008), no. 4, 1111--1121. doi:10.1215/ijm/1258554352. https://projecteuclid.org/euclid.ijm/1258554352

#### References

• G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41 (2004), 439–505.
• T. Aubin, Équations différentielles non linéaires et le problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269–296.
• E. N. Barron and R. Jensen, Minimizing the $L^\infty$ norm of the gradient with an energy constraint, Comm. Partial Differential Equations 30 (2005), 1741–1772.
• R. J. Biezuner, Best constants in Sobolev trace inequalities, Nonlinear Anal. 54 (2003), 575–589.
• A. Cherkaev and E. Cherkaeva, Optimal design for uncertain loading condition, Homogenization, Ser. Adv. Math. Appl. Sci., vol. 50, World Sci. Publishing, River Edge, NJ, 1999, pp. 193–213.
• O. Druet and E. Hebey, The $AB$ program in geometric analysis: Sharp Sobolev inequalities and related problems, Mem. Amer. Math. Soc. 160 (2002), 761.
• M. del Pino and C. Flores, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations 26 (2001), 2189–2210.
• J. F. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Math. J. 37 (1988), 687–698.
• J. Fernández Bonder, E. Lami Dozo and J. D. Rossi, Symmetry properties for the extremals of the Sobolev trace embedding, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 795–805.
• J. Fernandez Bonder, R. Ferreira and J. D. Rossi, Uniform bounds for the best Sobolev trace constant, Adv. Nonlinear Studies 3 (2003), 181–192.
• J. Fernández Bonder, P. Groisman and J. D. Rossi, Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach, Ann. Mat. Pura Appl. 186 (2007), 341–358.
• J. Fernández Bonder and J. D. Rossi, Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains, Comm. Pure Appl. Anal. 1 (2002), 359–378.
• J. Fernández Bonder, J. D. Rossi and N. Wolanski, Behavior of the best Sobolev trace constant and extremals in domains with holes, Bull. Sci. Math. 130 (2006), 565–579.
• J. Fernández Bonder, J. D. Rossi and N. Wolanski, Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant, SIAM J. Control Optim. 44 (2005), 1614–1635.
• J. Garcia-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, Steklov eigenvalues for the $\infty$-Laplacian, Rend. Lincei Mat. Appl. 17 (2006), 199–210.
• A. Henrot, Minimization problems for eigenvalues of the Laplacian, J. Evol. Equ. 3 (2003), 443–461.
• P. Juutinen, P. Lindqvist and J. J. Manfredi, The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), 89–105.
• E. Lami Dozo and O. Torne, Symmetry and symmetry breaking for minimizers in the trace inequality, Commun. Contemp. Math. 7 (2005), 727–756.
• A. Le, On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian, Electron. J. Differential Equations 2006 (2006), 1–9.
• Y. Li and M. Zhu, Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries, Comm. Pure Appl. Math. 50 (1997), 449–487.
• S. Martinez and J. D. Rossi, Isolation and simplicity for the first eigenvalue of the p-laplacian with a nonlinear boundary condition, Abst. Appl. Anal. 7 (2002), 287–293.