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On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary

Díaz Díaz, Jesús Ildefonso and Rakotoson, Jean Michel Theresien (2009) On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary. Journal of Functional Analysis , 257 (3). pp. 807-831. ISSN 0022-1236

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Abstract

We study the differentiability of very weak solutions v is an element of L(1) (Omega) of (v, L* phi)(0) = (f, phi)(0) for all phi is an element of C(2)((Omega) over bar) vanishing at the boundary whenever f is in L(1) (Omega, delta), with delta = dist(x, partial derivative Omega), and L* is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function (f) over tilde of f.

Item Type:Article
Uncontrolled Keywords:Very weak solutions; Distance to the boundary; Regularity; Linear PDE; Monotone rearrangement; Lorentz space
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:15112
References:

C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, London, 1983.

I. Birindelli, F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators,

Ann. Fac. Sci. Toulouse Math. (6) 13 (2) (2004) 261–287.

L. Boccardo, T. Gallouët, Non-linear elliptic and parabolic equations involving measure as data, J. Funct. Anal. 87

(1989) 149–169.

H. Brezis, Personal communication to J.I. Díaz: Une équation semi-linéaire avec conditions aux limites dans L1,

unpublished.

H. Brezis, T. Cazenave, Y. Martel, A. Ramiandrisoa, Blow up for ut − _u = g(u) revisited, Adv. Differential

Equations 1 (1996) 73–90.

X. Cabré, Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. 156

(1998) 30–56.

J.I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Res. Notes Math., vol. 106, Pitman, London,

1985.

J.I. Díaz, J.M. Rakotoson, On very weak solutions of semilinear elliptic equations with right hand side data integrable

with respect to the distance to the boundary, in preparation.

D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.

H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,

Math. Nachr. 307 (1997) 89–102.

L. Korkut, M. Pasic, D. Zubrini´c, A singular ODE related to quasilinear elliptic equations, Electron. J. Differential

Equations 12 (2000) 1–37.

P. Quittner, P. Souplet, Superlinear Parabolic Problems, Birkhäuser, Basel, 2007.

J.M. Rakotoson, Quasilinear elliptic problems with measure as data, Differential Integral Equations 4 (3) (1991)

449–457.

J.M. Rakotoson, Réarrangement Relatif: un instrument d’estimation dans les problèmes aux limites, Springer-

Verlag, Berlin, 2008.

J.E. Rakotoson, J.M. Rakotoson, Analyse fonctionnelle appliquée aux équations aux dérivées partielles, P.U.F.,

Paris, 1999.

C. Simader, On Dirichlet’s Boundary Value Problem and Lp Theory Based on Generalization of Gårding’s Inequality,

Lecture Notes in Math., vol. 268, Springer-Verlag, New York, 1972.

L. Veron, Singularities of Solutions of Second Order Quasilinear Equations, Longman, Edinburgh Gate, Harlow,

1995, pp. 176–180.

Deposited On:07 May 2012 08:54
Last Modified:06 Feb 2014 10:16

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