A representation formula for the distributional normal derivative
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A representation formula for the distributional normal derivative Augusto C. Ponce1
· Nicolas Wilmet1
Received: 23 March 2020 / Accepted: 27 August 2020 © Universidad Complutense de Madrid 2020
Abstract We prove an integral representation formula for the distributional normal derivative of solutions of −u + V u = μ in , u=0
on ∂,
where V ∈ L 1loc () is a nonnegative function and μ is a finite Borel measure on . As an application, we show that the Hopf lemma holds almost everywhere on ∂ when V is a nonnegative Hopf potential. Keywords Schrödinger operator · Distributional normal derivative · Hopf lemma · Measure data Mathematics Subject Classification 35J10 · 31B10 · 35B50
1 Introduction and main results In [4], Brezis and the first author introduced a notion of distributional normal derivative that applies in particular to solutions of the Dirichlet problem
−u + V u = μ in , u=0
B
on ∂,
(1.1)
Nicolas Wilmet [email protected] Augusto C. Ponce [email protected]
1
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgium
123
A. C. Ponce, N. Wilmet
where ⊂ R N is a smooth connected bounded open set, V ∈ L 1loc () is a nonnegative function and μ is a finite Borel measure on . By a solution of (1.1), we mean a function u ∈ W01,1 () ∩ L 1 (; V dx) such that −u + V u = μ in the sense of distributions in . For example, when μ = f dx with f ∈ L 2 () the solution always exists and can be obtained as the minimizer of the functional E(z) =
1 2
(|∇z|2 + V z 2 )dx −
f zdx
in W01,2 () ∩ L 2 (; V dx) ; see [5, Theorem 2.4]. The assumption V ∈ L 1loc () ensures that Cc∞ () ⊂ L 2 (; V dx) and thus smooth functions with compact support are admissible in the Euler–Lagrange equation associated to E. The distributional normal derivative of u is an element of L 1 (∂), which we denote by ∂u/∂n and coincides with the classical normal derivative when u ∈ C 2 (), that is characterized by the identity
∇u · ∇φdx = −
φu −
∂
∂u φ dσ for every φ ∈ C ∞ (), ∂n
(1.2)
where σ = H N −1 ∂ denotes the surface measure on ∂ ; see [4, Theorem 1.2] or [11, Proposition 7.3]. We adopt the convention that n is the inward unit normal vector on ∂, which explains the minus sign in front of the second integral in the right-hand side of (1.2). Our goal in this paper is to prove an integral representation formula for ∂u/∂n that involves μ and is valid for all solutions of (1.1). To obtain the kernel of such a formula, we rely upon a counterpart of the notion of duality solution by A. Malusa and L. Orsina [7], inspired from the seminal paper [6]. We adapt this formalism to the boundary value problem
−v + V v = 0 v = δa
in , on ∂,
(1.3)
involving the Dirac measure δa on ∂. Definition 1.1 Given a ∈ ∂, a function Pa ∈ L 1 () is a duality solution of (1.3) whenever f ∂ζ (a) = Pa f dx for every f ∈ L ∞ (), ∂n where ζ f denotes the solution of (1.1) with datum
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