Abstract
Abstract We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.
Highlights
This paper deals with quasilinear elliptic systems in divergence form
When N =, that is in the case of one single equation, the celebrated De Giorgi-Nash-Moser theorem ensures that weak solutions u ∈ W, (Ω) are locally bounded and even Hölder continuous, see section 2.1 in [27]
In the vectorial case N ≥, the aforementioned result is no longer true due to the De Giorgi’s counterexample, see [6], section 3 in [27] and the recent paper [29]; see [32] and [20]. It arises the question of nding additional structural restrictions on the coe cients a α,β i,j that keep away De Giorgi’s counterexample and allow for local boundedness of weak solutions u, see Section 3.9 in
Summary
Where u : Ω ⊂ Rn → RN and a : Ω × RN → RN n is matrix valued with components aαi,,jβ(x, y) where i, j ∈ { , ..., n} and α, β ∈ { , ..., N}. In the vectorial case N ≥ , the aforementioned result is no longer true due to the De Giorgi’s counterexample, see [6], section 3 in [27] and the recent paper [29]; see [32] and [20]. It arises the question of nding additional structural restrictions on the coe cients a α,β i,j that keep away De Giorgi’s counterexample and allow for local boundedness of weak solutions u, see Section 3.9 in [28]. (see Figure 1 and note that the support has the shape of a butter y in the plane yβ − yα)
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