Elliptic problems involving the 1–Laplacian and a singular lower order term

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0 0$ a.e., the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit 1--dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of $L^\infty$--divergence--measure vector fields must be extended to deal with this equation.

[1]  OsherStanley,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[2]  V. Caselles,et al.  Parabolic Quasilinear Equations Min-imizing Linear Growth Functionals , 2004 .

[3]  Charles Alexander Stuart,et al.  Existence and approximation of solutions of non-linear elliptic equations , 1976 .

[4]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[5]  L. Boccardo,et al.  Critical points for functionals with quasilinear singular Euler–Lagrange equations , 2013 .

[6]  F. Murat,et al.  Advances in the Study of Singular Semilinear Elliptic Problems , 2016 .

[7]  J. M. Mazón,et al.  The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow , 2013 .

[8]  Yoshikazu Nakai,et al.  Some remarks on strongly invariant rings , 1975 .

[9]  Zhijun Zhang,et al.  On a Dirichlet Problem with a Singular Nonlinearity , 1995 .

[10]  P. Takáč,et al.  Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation , 2007 .

[11]  L. Boccardo,et al.  Semilinear elliptic equations with singular nonlinearities , 2010 .

[12]  M. Novaga.  The Total Variation Flow , 2003 .

[13]  C. Ballester,et al.  The Dirichlet Problem for the Total Variation Flow , 2001 .

[14]  L. Orsina,et al.  A Lazer-Mckenna type problem with measures , 2015, Differential and Integral Equations.

[15]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[16]  Cristina Trombetti,et al.  Asymptotic behaviour of solutions to p‐Laplacian equation , 2003 .

[17]  W. Ziemer,et al.  Existence, uniqueness, and regularity for functions of least gradient. , 1992 .

[18]  Hermano Frid,et al.  Divergence‐Measure Fields and Hyperbolic Conservation Laws , 1999 .

[19]  BEHAVIOUR OF p¡LAPLACIAN PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS WHEN p GOES TO 1 , 2012 .

[20]  Linda Maria De Cave.  Nonlinear elliptic equations with singular nonlinearities , 2013, Asymptot. Anal..

[21]  G. Anzellotti,et al.  Pairings between measures and bounded functions and compensated compactness , 1983 .

[22]  Bernhard Kawohl,et al.  On a familiy of torsional creep problems. , 1990 .

[23]  K. Schmitt,et al.  Boundary Value Problems for Singular Elliptic Equations , 2011 .

[24]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[25]  W. Fulks,et al.  A singular non-linear equation , 1960 .

[26]  S. D. León,et al.  On the solutions to 1-Laplacian equation with L1 data , 2009 .

[27]  M. M. Coclite,et al.  On a Dirichlet problem in bounded domains with singular nonlinearity , 2013 .

[28]  Zvisinei Sandi.  DEFINITION , 1961, A Philosopher Looks at Sport.

[29]  A. Trombetta,et al.  Existence and uniqueness for p-Laplace equations involving singular nonlinearities , 2016 .

[30]  Andy M. Yip,et al.  Total Variation Image Restoration: Overview and Recent Developments , 2006, Handbook of Mathematical Models in Computer Vision.

[31]  L. Mi.  Existence and boundary behavior of solutions to p-Laplacian elliptic equations , 2016 .

[32]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[33]  K. Perera,et al.  On singular $p$-Laplacian problems , 2007, Differential and Integral Equations.

[34]  D. Arcoya,et al.  Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity , 2014 .

[35]  D. Giachetti,et al.  Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0 , 2016, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE.

[36]  Francesco Petitta,et al.  Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness , 2016, 1611.08283.

[37]  W. Ziemer.  Weakly differentiable functions , 1989 .

[38]  D. Giachetti,et al.  A semilinear elliptic equation with a mild singularity at $u=0$: existence and homogenization , 2015, 1502.06234.

[39]  Fuensanta Andreu,et al.  Bounded solutions to the 1-Laplacian equation with a critical gradient term , 2012, Asymptot. Anal..

[41]  Marco Degiovanni,et al.  A variational approach to a class of singular semilinear elliptic equations , 2004 .

[42]  G. Huisken,et al.  The inverse mean curvature flow and the Riemannian Penrose Inequality , 2001 .

[43]  D. Giachetti,et al.  The Dirichlet problem for singular elliptic equations with general nonlinearities , 2018, Calculus of Variations and Partial Differential Equations.

[44]  Alan C. Lazer,et al.  On a singular nonlinear elliptic boundary-value problem , 1991 .

[45]  Ahmed Mohammed.  Positive solutions of the p-Laplace equation with singular nonlinearity , 2009 .

[46]  D. Giachetti,et al.  DIRICHLET PROBLEMS FOR SINGULAR ELLIPTIC EQUATIONS WITH GENERAL NONLINEARITIES , 2017 .

[47]  D. Giachetti,et al.  Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at $u=0$ in a domain with many small holes , 2017, 1705.09527.

[48]  V. Caselles,et al.  On the entropy conditions for some flux limited diffusion equations , 2011 .

[49]  S. D. León,et al.  On the behaviour of the solutions to p-Laplacian equations as p goes to 1 , 2008 .

[50]  Petri Juutinen,et al.  P-harmonic approximation of functions of least gradient , 2005 .

[51]  Françoise Demengel.  On some nonlinear equation involving the 1-Laplacian and trace map inequalities , 2002 .