In a bounded open subset Ω⊂Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rnwith values into RN, defined by −divA(x,u(x),Du(x))=μ in Ω,u=0 on ∂Ω, where Aiα(x,y,ξ)=∑β=1N∑j=1nai,jα,β(x,y)ξjβwith α∈1,…,N the equation index. We prove the existence of a (distributional) solution u:Ω→RN, obtained as the limit of approximations, by assuming: (i) that coefficients ai,jα,βare bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients ai,jα,α; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients ai,jα,β(i.e. α≠β) verifying a r-staircase support condition with r>0. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) ai,jα,β=−aj,iβ,α(skew-symmetry); (b) |aα,βi,j| is small; (c) ai,jα,βmay be decomposed into two parts, the first enjoying skew-symmetry and the second being small in absolute value. We give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works. A Brezis's type nonexistence result is also given for general (smooth) elliptic-hyperbolic systems.

Smallness and cancellation in some elliptic systems with measure data

Leonetti, Francesco;
2018-01-01

Abstract

In a bounded open subset Ω⊂Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rnwith values into RN, defined by −divA(x,u(x),Du(x))=μ in Ω,u=0 on ∂Ω, where Aiα(x,y,ξ)=∑β=1N∑j=1nai,jα,β(x,y)ξjβwith α∈1,…,N the equation index. We prove the existence of a (distributional) solution u:Ω→RN, obtained as the limit of approximations, by assuming: (i) that coefficients ai,jα,βare bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients ai,jα,α; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients ai,jα,β(i.e. α≠β) verifying a r-staircase support condition with r>0. Such a smallness condition is satisfied, for instance, in each one of these cases: (a) ai,jα,β=−aj,iβ,α(skew-symmetry); (b) |aα,βi,j| is small; (c) ai,jα,βmay be decomposed into two parts, the first enjoying skew-symmetry and the second being small in absolute value. We give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works. A Brezis's type nonexistence result is also given for general (smooth) elliptic-hyperbolic systems.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

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: https://hdl.handle.net/11697/127768
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 8
social impact