Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/existence-and-comparison-results-for-an-elliptic-equation-involving-EukfjqBd78
References (35)
-
J. Mazón, S. León (2013)
The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow, 6
-
L. Ambrosio, D. Pallara, N. Fusco (2000)
Functions of Bounded Variation and Free Discontinuity Problems -
R. Moser (2007)
The inverse mean curvature flow and p-harmonic functionsJournal of the European Mathematical Society, 9
-
W. Ziemer (1989)
Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation -
Sergio Segura de León E-mail: sergio.segura@uv.es -
L Boccardo, T Gallouët, L Orsina (1996)
Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure dataAnn. Inst. H. Poincaré (Non–Lineaire), 13
-
H. Brezis (2010)
Functional Analysis, Sobolev Spaces and Partial Differential Equations -
G. Huisken, T. Ilmanen (2001)
The inverse mean curvature flow and the Riemannian Penrose InequalityJournal of Differential Geometry, 59
-
L. Boccardo, T. Gallouët, L. Orsina (1996)
Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure dataAnnales De L Institut Henri Poincare-analyse Non Lineaire, 13
-
A Mercaldo, S Segura de León, C Trombetti (2009)
On the solutions to 1-Laplacian equation with $$L^1$$ L 1 dataJ. Funct. Anal., 256
-
B. Kawohl (1990)
On a familiy of torsional creep problems.Journal für die reine und angewandte Mathematik (Crelles Journal), 1990
-
G. Maso, F. Murat, L. Orsina, A. Prignet (1997)
Definition and existence of renormalized solutions of elliptic equations with general measure dataComptes Rendus De L Academie Des Sciences Serie I-mathematique, 325
-
J. Mazón, S. León (2015)
A non-homogeneous elliptic problem dealing with the level set formulation of the inverse mean curvature flowJournal of Differential Equations, 259
-
S. Superiore, P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. Vázquez, Bénilan (2018)
An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations -
F. Andreu, C. Ballester, V. Caselles, J. Mazón (2001)
The Dirichlet Problem for the Total Variation FlowJournal of Functional Analysis, 180
-
A. Mercaldo, S. León, C. Trombetti (2009)
On the solutions to 1-Laplacian equation with L1 dataJournal of Functional Analysis, 256
-
M. Cicalese, C. Trombetti (2003)
Asymptotic behaviour of solutions to p‐Laplacian equationAsymptotic Analysis, 35
-
G. Anzellotti (1983)
Pairings between measures and bounded functions and compensated compactnessAnnali di Matematica Pura ed Applicata, 135
-
M Cicalese, C Trombetti (2003)
Asymptotic behaviour of solutions to $$p$$ p -Laplacian equationAsymptot. Anal., 35
-
A. Mercaldo, S. León, C. Trombetti (2008)
On the behaviour of the solutions to p-Laplacian equations as p goes to 1Publicacions Matematiques, 52
-
G. Huisken, T. Ilmanen (2008)
Higher regularity of the inverse mean curvature flowJournal of Differential Geometry, 80
-
M. Latorre, S. Le'on (2016)
Elliptic equations involving the 1--Laplacian and a total variation term with $L^{N,\infty}$--dataarXiv: Analysis of PDEs
-
V. Caselles (2011)
On the entropy conditions for some flux limited diffusion equationsJournal of Differential Equations, 250
-
E-mail address: marta.latorre@uv.es S. Segura de León: Departament d'Anàlisi Matemàtica -
G Dal Maso, F Murat, L Orsina, A Prignet (1999)
Renormalized solutions of elliptic equations with general measure dataAnn. Sc. Norm. Sup. Pisa Cl. Sci. (4), 28
-
Charlie Harper (2005)
Partial Differential EquationsMultivariable Calculus with Mathematica
-
L. Evans (1992)
Measure theory and fine properties of functions -
O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, Frank Lenzen (2009)
Weakly Differentiable Functions -
Gui-Qiang Chen, H. Frid (1999)
Divergence‐Measure Fields and Hyperbolic Conservation LawsArchive for Rational Mechanics and Analysis, 147
-
Ronald Gariepy (2001)
FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs)Bulletin of The London Mathematical Society, 33
-
Gerald Teschl (2004)
Functional Analysis -
A Mercaldo, S Segura de León, C Trombetti (2008)
On the behaviour of the solutions to $$p$$ p -Laplacian equations as $$p$$ p goes to $$1$$ 1Publ. Mat., 52
-
P Bénilan, L Boccardo, T Gallouët, R Gariepy, M Pierre, JL Vázquez (1995)
An $$L^1$$ L 1 theory of existence and uniqueness of solutions of nonlinear elliptic equationsAnn. Scuola Norm. Sup. Pisa Cl. Sci., 22
-
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. Vázquez (1995)
An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equationsAnnali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 22
-
R Moser (2007)
The inverse mean curvature flow and $$p$$ p -harmonic functionsJ. Eur. Math. Soc. (JEMS), 9
- Publisher
- Springer Journals
- Copyright
- Copyright © 2017 by Springer International Publishing
- Subject
- Mathematics; Analysis
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-017-0388-0
- Publisher site
- See Article on Publisher Site
Abstract
This paper is devoted to analyze the Dirichlet problem for a nonlinear elliptic equation involving the 1-Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the inverse mean curvature flow. We study this problem in an open bounded set with Lipschitz boundary. We prove an existence result and a comparison principle for nonnegative $$L^1$$ L 1 -data. Moreover, we search the summability that the solution reaches when more regular $$L^p$$ L p -data, with $$1<p<N$$ 1 < p < N , are considered and we give evidence that this summability is optimal. To prove these results, we apply the theory of $$L^\infty $$ L ∞ -divergence measure fields which goes back to Anzellotti (Ann Mat Pura Appl (4) 135:293–318, 1983). The main difficulties of the proofs come from the absence of a definition for the pairing of a general $$L^\infty $$ L ∞ -divergence measure field and the gradient of an unbounded BV-function.
Journal
Journal of Evolution Equations – Springer Journals
Published: Apr 5, 2017