In a bounded open set Ω ⊂ Rn, n > 3, we consider the nonlinear fourth-order partial differential equation ∑|α|=1,2(−1)^|α|D^α A_α(x,u,Du,D^2 u)+B(x,u,Du,D^2 u) = 0. It is assumed that the principal coefficients {A_α}|α|=1,2 satisfy the growth and coercivity conditions suitablefor the energy space W˚_(1,q)^(2,p)(Ω) = W˚_(1,q)(Ω)∩W˚^(2,p)(Ω), 1 < p < n/2, 2p < q < n. The lower-order term B(x,u,Du,D^2 u) behaves as b(u)(|Du|^q +|D^2 u|^p) +g(x) where g ∈ L^τ(Ω), τ > n/q. Weestablish the Holder continuity up to the boundary of any solution u ∈W˚_(1,q)^(2,p)(Ω)∩L^∞(Ω) by using the measure density condition on ∂Ω, an interior local result and a modified Moser method with special test function.

HOLDER CONTINUITY UP TO THE BOUNDARY OF ¨SOLUTIONS TO NONLINEAR FOURTH–ORDER ELLIPTIC EQUATIONS WITH NATURAL GROWTH TERMS

BONAFEDE, Salvatore;
2019

Abstract

In a bounded open set Ω ⊂ Rn, n > 3, we consider the nonlinear fourth-order partial differential equation ∑|α|=1,2(−1)^|α|D^α A_α(x,u,Du,D^2 u)+B(x,u,Du,D^2 u) = 0. It is assumed that the principal coefficients {A_α}|α|=1,2 satisfy the growth and coercivity conditions suitablefor the energy space W˚_(1,q)^(2,p)(Ω) = W˚_(1,q)(Ω)∩W˚^(2,p)(Ω), 1 < p < n/2, 2p < q < n. The lower-order term B(x,u,Du,D^2 u) behaves as b(u)(|Du|^q +|D^2 u|^p) +g(x) where g ∈ L^τ(Ω), τ > n/q. Weestablish the Holder continuity up to the boundary of any solution u ∈W˚_(1,q)^(2,p)(Ω)∩L^∞(Ω) by using the measure density condition on ∂Ω, an interior local result and a modified Moser method with special test function.
File in questo prodotto:
File Dimensione Formato  
Bonafede_2019_dea_ HOLDER-editor.pdf

accesso aperto

Tipologia: Versione Editoriale (PDF)
Licenza: Creative commons
Dimensione 249.59 kB
Formato Adobe PDF
249.59 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12318/4605
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
social impact