A Priori Estimates and Comparison Principle for Some Nonlinear Elliptic Equations
We present a priori estimates and comparison principle for second order quasilinear elliptic operators in divergence form with a first order term. We deduce existence and uniqueness results for weak solutions or “solution obtained as limit of approximatio
- PDF / 268,892 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 95 Downloads / 158 Views
Abstract We present a priori estimates and comparison principle for second order quasilinear elliptic operators in divergence form with a first order term. We deduce existence and uniqueness results for weak solutions or “solution obtained as limit of approximations” to Dirichlet problems related to these types of operators when data belong to suitable Lorentz spaces. Moreover it is also shown how the summability of these solutions increases when the summability of the datum increases. Keywords A priori estimates · Existence · Comparison principle · Uniqueness · Nonlinear elliptic operators
1 Introduction Let us consider the homogeneous Dirichlet problem −div a(x, u, ∇u) = H (x, ∇u) + f u=0
in Ω on ∂Ω,
(1)
where Ω is a bounded open subset of RN , N ≥ 2. We assume that a : Ω × R × RN → RN and H : Ω × RN → R are Carathéodory functions which satisfy the ellipticity condition a(x, s, ξ ) · ξ ≥ α|ξ |p ,
α > 0,
the monotonicity condition a(x, s, ξ ) − a(x, s, η) · (ξ − η) > 0,
(2) ξ = η,
(3)
A. Mercaldo (B) Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Complesso di Monte S. Angelo, 80126 Napoli, Italy e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_14, © Springer-Verlag Italia 2013
223
224
and the growth conditions a(x, s, ξ ) ≤ a0 |ξ |p−1 + a1 |s|p−1 + a2 , a0 , a1 , a2 > 0, H (x, ξ ) ≤ h|ξ |q , h > 0,
A. Mercaldo
(4) (5)
with 1 < p < +∞, p − 1 < q < p for almost every x ∈ RN , for every s ∈ R, for every ξ, η ∈ RN . Finally we assume that f belongs to suitable Lorentz spaces. The purpose of the present note is to announce some recent results dealing with existence, uniqueness and regularity for solutions to problem (1). The notion of solution to which we refer depends on the summability of the datum. We use the classical weak solution when the datum f is an element of W −1,p (Ω), the dual 1,p space of W0 (Ω), while, if f is not in such a space, a different notion of solution has to be adopted. Various equivalent notion of solutions are available in literature, i.e. renormalized solution [24, 27], entropy solution [6] or “solution obtained as limit of approximations” ([15], see also [16]). We will refer to “solution obtained as limit of approximations”, whose definition is based on a procedure of passage to the limit which we recall in Sect. 2. Existence results for solution to problem (1) are well-known in literature (see e.g. [18, 20, 21]). Their proofs are based on a priori estimates for solutions and its gradients which are the first step of a standard approach, whose main idea consists in a passage to the limit in a sequence of approximated problems having regular data. By using the a priori estimates, one can prove that weak solutions to such problems converge in a suitable sense to a function which is the solution to problem (1). A priori estimates for weak solutions or renormalized solutions to (1) are proved in [
Data Loading...
