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http://hdl.handle.net/10773/23634
Title: | Smallness and cancellation in some elliptic systems with measure data |
Author: | Leonetti, Francesco Rocha, Eugenio Staicu, Vasile |
Keywords: | Elliptic System Existence Measure Solution |
Issue Date: | 15-Sep-2018 |
Publisher: | Elsevier |
Abstract: | In a bounded open subset Ω ⊂ Rn, we study Dirichlet problems with elliptic systems, involving a finite Radon measure μ on Rn with values into RN , defined by { −div A(x, u(x), Du(x)) = μ in Ω, u = 0 on ∂Ω, where Aα i (x, y, ξ) = N∑ β=1 n∑ j=1 aα,β i,j (x, y) ξβ j with 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 aα,β i,j are bounded Carathéodory functions; (ii) ellipticity of the diagonal coefficients aα,α i,j ; and (iii) smallness of the quadratic form associated to the off-diagonal coefficients aα,β i,j (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) aα,β i,j = −aβ,α j,i (skew-symmetry); (b) |aα,β i,j | is small; (c) aα,β i,j may 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. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/23634 |
DOI: | 10.1016/j.jmaa.2018.05.047 |
ISSN: | 0022-247X |
Appears in Collections: | CIDMA - Artigos FAAG - Artigos |
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Smallness and cancellation in some elliptic systems with measure data.pdf | 416.14 kB | Adobe PDF | View/Open |
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