January  2013, 12(1): 253-267. doi: 10.3934/cpaa.2013.12.253

Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1

1. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli "Federico II", via Cintia, I-80126 Napoli

2. 

Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, 03080 Alicante

3. 

Departament d'Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain

4. 

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli

Received  June 2011 Revised  August 2011 Published  September 2012

We consider the solution $u_p$ to the Neumann problem for the $p$--Laplacian equation with the normal component of the flux across the boundary given by $g\in L^\infty(\partial\Omega)$. We study the behaviour of $u_p$ as $p$ goes to $1$ showing that they converge to a measurable function $u$ and the gradients $|\nabla u_p|^{p-2}\nabla u_p$ converge to a vector field $z$. We prove that $z$ is bounded and that the properties of $u$ depend on the size of $g$ measured in a suitable norm: if $g$ is small enough, then $u$ is a function of bounded variation (it vanishes on the whole domain, when $g$ is very small) while if $g$ is large enough, then $u$ takes the value $\infty$ on a set of positive measure. We also prove that in the first case, $u$ is a solution to a limit problem that involves the $1-$Laplacian. Finally, explicit examples are shown.
Citation: Anna Mercaldo, Julio D. Rossi, Sergio Segura de León, Cristina Trombetti. Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1. Communications on Pure & Applied Analysis, 2013, 12 (1) : 253-267. doi: 10.3934/cpaa.2013.12.253
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[2]

F. Andreu, C. Ballester, V, Caselles and J. M. Mazón, Minimizing total variation flow,, Differential Integral Equations, 14 (2001), 321.   Google Scholar

[3]

F. Andreu, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,", Progress in Mathematics, (2004).   Google Scholar

[4]

F. Andreu, J. M. Mazón and J. S. Moll, The total variation flow with nonlinear boundary conditions,, Asymptot. Anal., 43 (2005), 9.   Google Scholar

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G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, Ann. Mat. Pura Appl., 135 (1983), 293.   Google Scholar

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M. Cicalese and C. Trombetti, Asymptotic behaviour of solutions to $p$-Laplacian equation,, Asymptot. Anal., 35 (2003), 27.   Google Scholar

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

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J. M. Mazón and S. Segura de León, The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow,, to appear in Adv. Calc. Var., ().   Google Scholar

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A. Mercaldo, S. Segura de León and C. Trombetti, On the Behaviour of the solutions to $p$-Laplacian equations as $p$ goes to 1,, Publ. Mat., 52 (2008), 377.   Google Scholar

[10]

A. Mercaldo, S. Segura de León and C. Trombetti, On the solutions to $1$-Laplacian equation with $L^1$ data,, J. Func. Anal., 256 (2009), 2387.   Google Scholar

[11]

J. M. Rakotoson, Generalized solutions in a new type of sets for problems with measure as data,, Differential Integral Equations, 6 (1993), 27.   Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

[2]

F. Andreu, C. Ballester, V, Caselles and J. M. Mazón, Minimizing total variation flow,, Differential Integral Equations, 14 (2001), 321.   Google Scholar

[3]

F. Andreu, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,", Progress in Mathematics, (2004).   Google Scholar

[4]

F. Andreu, J. M. Mazón and J. S. Moll, The total variation flow with nonlinear boundary conditions,, Asymptot. Anal., 43 (2005), 9.   Google Scholar

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, Ann. Mat. Pura Appl., 135 (1983), 293.   Google Scholar

[6]

M. Cicalese and C. Trombetti, Asymptotic behaviour of solutions to $p$-Laplacian equation,, Asymptot. Anal., 35 (2003), 27.   Google Scholar

[7]

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

[8]

J. M. Mazón and S. Segura de León, The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow,, to appear in Adv. Calc. Var., ().   Google Scholar

[9]

A. Mercaldo, S. Segura de León and C. Trombetti, On the Behaviour of the solutions to $p$-Laplacian equations as $p$ goes to 1,, Publ. Mat., 52 (2008), 377.   Google Scholar

[10]

A. Mercaldo, S. Segura de León and C. Trombetti, On the solutions to $1$-Laplacian equation with $L^1$ data,, J. Func. Anal., 256 (2009), 2387.   Google Scholar

[11]

J. M. Rakotoson, Generalized solutions in a new type of sets for problems with measure as data,, Differential Integral Equations, 6 (1993), 27.   Google Scholar

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