Abstract
In this paper we proof a Harnack inequality and a regularity theorem for local-minima of scalar intagral functionals with general growth conditions.
Highlights
In this paper we proof a Harnack inequality for local-minima of scalar intagral functionals of the calculus of variation of that typeJ u, f x,u x, u x dx (1.1)where Ω is a bounded open subset of N, Φ:[0,+∞)→[0,+∞) is a N-function and Φ globally satisfies the Δ′-| z | f x,s,z L2 | z |for a. e. x∈Ω and for every (s,z)∈R ×R N
In [15] and [16] we have shown that the following hypothesis can be used in order to give a new estimation of the measure of the livel set A(k,R): H-1) Φ globally satisfies the Δ′-condition in [0,+∞); H-2) there exists a constant cH2 0
Theorem 1: If u W1LΦ(Ω) is a quasi-minima of the functional (1.1) and if Φ confirm the hypotheses H-1, H-2 and H-3; u is locally hölder continuous. In these pages we show that the hypotheses H-2 and H-3 are purely technical and they can be eliminated
Summary
Where Ω is a bounded open subset of N , Φ:[0,+∞)→[0,+∞) is a N-function and Φ globally satisfies the Δ′-. The only present novelty in the demonstrative technique is the use of an ɛ-Young inequality This simple trick allows to recover the results introduced in [15,16,17,24,26] in a simple way and without using the properties of the functions Δ2∩ 2 (see Lemma of [15,24] and [26]). Definition 2: Let p be a real valued function defined on [0,+∞) and having the following properties: p(0) = 0, p(t) > 0 if t > 0, p is nondecreasing and right continuous on (0,+∞). Using the previous results we obtain the following theorems: Theorem 5 (Weak Harnack inequality): Let Φ be a N-function. Elisa Albano who translated the article into English supporting and encouraging me so much
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
