A regularity result for a class of elliptic equations with lower order terms

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A regularity result for a class of elliptic equations with lower order terms Claudia Capone1 · Teresa Radice2  Received: 20 June 2020 / Accepted: 20 July 2020 © The Author(s) 2020

Abstract In this paper we establish the higher differentiability of solutions to the Dirichlet problem { div(A(x, Du)) + b(x)u(x) = f in Ω u=0 on 𝜕Ω under a Sobolev assumption on the partial map x → A(x, 𝜉) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions. Keywords  A priori estimate · Boundedness of solution · Regularizing effect · Approximation Mathematics Subject Classification  35B65 · 35J60 · 49N60 · 49N99

1 Introduction This paper concerns the higher differentiability and the higher integrability of the gradient of local weak solution of the Dirichlet problem { div(A(x, Du)) + b(x)u(x) = f in Ω (1.1) u=0 on 𝜕Ω

* Teresa Radice [email protected] Claudia Capone [email protected] 1

Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino 111, 80131 Napoli, Italy


Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Sudi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy



C. Capone, T. Radice

The operator A ∶ Ω × ℝn → ℝn is a Carathéodory mapping, satisfying for positive constants 𝛼, 𝛽 > 0 the following assumptions

𝛼�𝜉 − 𝜂�2 ⩽ ⟨A(x, 𝜉) − A(x, 𝜂), 𝜉 − 𝜂⟩,


|A(x, 𝜉) − A(x, 𝜂)| ⩽ 𝛽 |𝜉 − 𝜂|.


|A(x, 𝜉) − A(y, 𝜉)| ⩽ (k(x) + k(y)) |x − y| (1 + |𝜉|)


|f (x)| ⩽ Qb(x)


Concerning the dependence on x-variable, it is clear that no extra differentiability can be obtained for solutions even if the data f(x) and b(x) are smooth, unless some differentiability assumption is made on the operator A(x, 𝜉). To this aim, we shall assume that there exists a non negative function n k(x) ∈ Lloc (Ω) , such that for every 𝜉 ∈ ℝn and a.e. x, y ∈ Ω . Finally, we shall assume

for a.e. x ∈ Ω. Conditions (1.2) and (1.3) express the uniform ellipticity and lipschitz continuity of the operator A(x, 𝜉) with respect to the variable 𝜉. Condition (1.4), in view of the pointwise characterization of the Sobolev spaces [12], means that the partial map x → A(x, 𝜉) belongs to the Sobolev class 1,n Wloc (Ω). Finally, condition (1.5), introduced in Ref. [1] (see also Ref. [2]), relates the coefficient of the lower order term with the right hand side. This interplay yields a regularizing effect on the solution of the Dirichlet problem (1.1). More precisely, it is sufficient to assume (1.5) to obtain

‖u‖L∞ (Ω) ⩽ Q .

The key tool to deal with equations with lower order terms, assuming a low integrability for b(x) and f(x) as in (1.1), is the result in Ref. [1] (see also Ref. [2]). Recall that the boundedness of the solution of equation (1.1) is well known if n b(x), f (x) ∈ Ls (Ω) for some s > [10], and usually it is the first step in the analysis 2 of the regularity of the solutions and open the way to the investigation of s