On the regularity of very weak solutions for linear elliptic equations in divergence form
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Nonlinear Differential Equations and Applications NoDEA
On the regularity of very weak solutions for linear elliptic equations in divergence form Domenico Angelo La Manna, Chiara Leone Roberta Schiattarella
and
Abstract. In this paper we consider a linear elliptic equation in divergence form Dj (aij (x)Di u) = 0 in Ω. (0.1) i,j
Assuming the coefficients aij in W 1,n (Ω) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very 1,2 weak solution u ∈ Ln loc (Ω) of (0.1) is actually a weak solution in Wloc (Ω). Mathematics Subject Classification. Primary 35-02, Secondary 35B65. Keywords. Elliptic equations, Very weak solutions.
1. Introduction Let n ≥ 2 and Ω ⊂ Rn be a bounded open set. In this paper we study regularity properties of very weak solutions to the linear elliptic equation Dj (aij (x)Di u) = 0 in Ω, (1.1) i,j
where the matrix-field A : Ω → Rn×n , A(x) = (aij (x))i,j , is elliptic and belongs to W 1,n (Ω, Rn×n ) ∩ L∞ (Ω, Rn×n ), i.e. sup i,j=1,...,n
and λ|ξ|2 ≤
aij (x)ξi ξj ≤ Λ|ξ|2
aij W 1,n (Ω) ≤ M
∀ξ = (ξ1 , . . . , ξn ) ∈ Rn , a.e. in Ω,
(1.2)
(1.3)
i,j
for some positive constants λ, Λ, and M . Moreover, the matrix A is symmetric, that is aij = aji a.e. in Ω for all i, j ∈ {1, ..., n}. 0123456789().: V,-vol
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Finally we assume that the coefficients (aij (x))i,j are double-Dini continuous in Ω, i.e. aij ∈ C 0 (Ω) and A¯Ω (r) :=
i,j
satisfies
ˆ
sup |aij (x) − aij (y)|,
r > 0,
x,y∈Ω |x−y|≤r
ˆ 1 t A¯Ω (s) ds dt < ∞. (1.4) t 0 s 0 A common type of double-Dini continuous functions are, of course, ω(r) = rα , 0 < α ≤ 1, thus an example of a matrix-field A satisfying (1.2) and (1.4) is A ∈ W 1,p (Ω, Rn×n ), with p > n. On the other hand, condition (1.4) occurs not only for ω(r) = rα , but more generally for ω(r) = logβ 1r , β < −2. Given a measurable matrix A(x) = (aij (x))i,j satisfying (1.3), a function 1,2 u ∈ Wloc (Ω) is called a weak solution of (1.1) if ˆ aij (x)Di uDj ϕ dx = 0, ∀ϕ ∈ Cc∞ (Ω). diam(Ω)
i,j
The celebrated result by De Giorgi in [5] states that if u is a weak solution of (1.1) then u is locally H¨ older continuous. Subsequently, J. Serrin produced in [14] a famous example, constructing an equation of the form (1.1) which has a solution u ∈ W 1,p (Ω), with 1 < p < 2, and u ∈ / L∞ loc (Ω). Serrin conjectured that if the coefficients aij are locally H¨ older continuous, then any solution (in the sense of distributions) 1,1 1,2 (Ω) of (1.1) must be a (usual) weak solution, i.e. u ∈ Wloc (Ω). u ∈ Wloc Serrin’s conjecture was established by Hager and Ross [9], and then in full 1,1 (Ω), generality by Brezis [2] (see also [1] for a full proof) starting with u ∈ Wloc 1 or even with u ∈ BVloc (Ω), i.e., u ∈ Lloc (Ω) and its derivatives (in the sense of distributions) being Radon measures. Let us remark that in Brezis’s result the coefficients aij , satisfying (1.3), are Dini continuous functions in Ω. The Dini continuity of the coefficients is optimal in some sense: for the unit ball B1
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