Regularity Theorems and Maximum Principles
This chapter provides a comprehensive presentation of regularity theorems and maximum principles that are essential for the subsequent study of nonlinear elliptic boundary value problems. In addition to the presentation of fundamental results, the chapter
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Regularity Theorems and Maximum Principles
Abstract This chapter provides a comprehensive presentation of regularity theorems and maximum principles that are essential for the subsequent study of nonlinear elliptic boundary value problems. In addition to the presentation of fundamental results, the chapter offers, to a large extent, a novel approach with clarification of tedious arguments and simplification of proofs. The first section of this chapter treats two major topics related to weak solutions of nonlinear elliptic problems: boundedness and regularity. The second section has as its objective to report on maximum and comparison principles. It comprises two parts: local results and strong maximum principles. Comments and related references are given in a remarks section.
8.1 Regularity of Solutions In this section, we prove regularity results for the weak solutions of certain nonlinear elliptic problems, which include as a particular case problems driven by the pLaplace differential operator. Usually, the regularity of a weak solution of a nonlinear elliptic problem is obtained by arguing in two steps: first, one shows that the weak solution is bounded, and second, relying on this boundedness property, one establishes the regularity of the weak solution up to the boundary of its domain of definition. Accordingly, this section is organized in two parts: in the first part, we provide a criterion that guarantees the boundedness of weak solutions, and in the second part, we present results that establish the regularity of bounded weak solutions.
Boundedness of Weak Solutions of Nonlinear Elliptic Problems Let Ω be a bounded domain in RN (N ≥ 1) with a Lipschitz boundary ∂ Ω , and let p ∈ (1, +∞). We consider a general operator a : Ω × RN → RN satisfying the following hypotheses: D. Motreanu et al., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, DOI 10.1007/978-1-4614-9323-5__8, © Springer Science+Business Media, LLC 2014
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8 Regularity Theorems and Maximum Principles
H(a)1 (i) a : Ω × RN → RN is continuous; (ii) There is a constant c1 > 0 such that |a(x, ξ )| ≤ c1 (1 + |ξ | p−1 ) for all x ∈ Ω , all ξ ∈ RN ; (iii) There are constants c0 > 0 and R, σ ≥ 0 such that (a(x, ξ ), ξ )RN ≥ c0 (R + |ξ |) p−σ |ξ |σ for all x ∈ Ω , all ξ ∈ RN .
Hypothesis H(a)1 (ii) implies that we have a(·, ∇u(·)) ∈ L p (Ω , RN ) whenever p u ∈ W 1,p (Ω ), where p = p−1 . In particular, the divergence div a(x, ∇u) (in the distributional sense) is well defined. Example 8.1. Many interesting operators fit the setting of hypotheses H(a)1 : (a) a(x, ξ ) = |ξ | p−2 ξ , so that div a(x, ∇u) is the p-Laplacian in this case. If a1 satisfies H(a)1 and a2 satisfies H(a)1 (i), (ii), and (a2 (x, ξ ), ξ )RN ≥ 0, then the sum a1 + a2 also satisfies H(a)1 . Thus, we can derive other examples from (a): (b) a(x, ξ ) = |ξ | p−2 ξ + ln(1 + |ξ | p−2 )ξ ; (c) a(x, ξ ) =
τ = 2;
|ξ | p−2 ξ + |ξ |q−2 ξ |ξ | p−2 ξ
+
q−2 τ −2
|ξ |τ −2 ξ
if |ξ | ≤ 1 −
q−τ τ −2 ξ
if |ξ | > 1,
with 1 < τ ≤ p ≤ q
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