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ELLIPTIC EQUATIONS WITH COMPACTLY SUPPORTED SOLUTIONS

  • Arena, Orazio (Dipartimento di Costruzioni e Restauro Universita di Firenze) ;
  • Giannotti, Cristina (Scuola di Scienze e Tecnologie Universita di Camerino)
  • Received : 2012.05.27
  • Published : 2013.03.01

Abstract

For any $p{\in}(1,2)$ and arbitrary $f{\in}L^p(\mathbb{R}^2)$ with compact support, it is proved that there exists a pair (L, $u$), with L second order uniformly elliptic operator and $u{\in}W_0^{2,p}(\mathbb{R}^2)$ such that $Lu=f$ a.e. in $\mathbb{R}^2$.

Keywords

second order elliptic equations;compactly supported solutions

References

  1. C. Giannotti, A compactly supported solution to a three-dimensional uniformly elliptic equation without zero order term, J. Differential Equations 201 (2004), no. 2, 234-249. https://doi.org/10.1016/j.jde.2003.12.003
  2. C. Giannotti and P. Manselli, Expansions with Poisson kernels and related topics, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 1, 153-173. https://doi.org/10.1017/S0013091507001150
  3. C. Giannotti and P. Manselli, On elliptic extensions in the disk, Potential Anal. 33 (2010), no. 3, 249-262. https://doi.org/10.1007/s11118-009-9168-y
  4. T. H. Wolff, Some constructions with solutions of variable coefficient elliptic equations, J. Geom. Anal. 3 (1993), no. 5, 423-511. https://doi.org/10.1007/BF02921289
  5. L. Bers, F. John, and M. Schechter, Partial Differential Equations, Interscience, 1964.
  6. P. Buonocore and P. Manselli, Solutions to two dimensional, uniformly elliptic equations, that lie in Sobolev spaces and have compact support, Rend. Circ. Mat. Palermo (2) 51 (2002), no. 3, 476-484. https://doi.org/10.1007/BF02871855
  7. K. Astala, T. Iwaniec, and G. Martin, Pucci's conjecture and the Alexandrov inequality for elliptic PDEs in the plane, J. Reine Angew. Math. 591 (2006), 49-74.