Remarks on Surjectivity of Gradient Operators

Raffaele Chiappinelli,David E Edmunds

doi:10.3390/math8091538

Raffaele Chiappinelli, David E Edmunds

Open Access

https://doi.org/10.3390/math8091538

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Journal: Mathematics	Publication Date: Sep 8, 2020
Citations: 6	License type: CC BY 4.0

Affiliation: University of Siena, University of Sussex

Abstract

Let X be a real Banach space with dual X∗ and suppose that F:X→X∗. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent result of ours, we prove that any gradient map that has this property and is additionally bounded, coercive and continuous is surjective. As before, the main tool for the proof is the Ekeland Variational Principle. Comparison with known surjectivity results is made; finally, as an application, we discuss a Dirichlet boundary-value problem for the p-Laplacian (1<p<∞), completing our previous result which was limited to the case p≥2.

Highlights

This is partly a research paper and partly a review paper, whose main purpose is to discuss a variant of the surjectivity result that we have proved in [1]
Be a real Banach space with dual X ∗, let h x, yi denote the pairing with x ∈ X ∗ and y ∈ X and let
Suppose that F is strongly coercive in the sense that for some c > 0 and some p > 1, h F ( x ), x i ≥ ckxkp for all x ∈ X, and suppose that F is locally proper, by which we mean that given any closed bounded set M of X, the set M ∩ F −1 (K ) is compact whenever K ⊂ X ∗ is compact

Summary

Introduction

This is partly a research paper and partly a review paper, whose main purpose is to discuss a variant of the surjectivity result that we have proved in [1]. For u, v in the Sobolev space X = W 1p (Ω), is not strongly monotone but merely strictly monotone, this last meaning that for all x, y ∈ X with x 6= y, h T ( x ) − T (y), x − yi > 0 This difficulty made us reconsider the local properness property itself, and seek different but equivalent forms of expressing it. Brézis) and that have their prototype in the famous Minty–Browder Theorem (see, e.g., Theorem 2 of [6]) stating that if X is a reflexive Banach space and if T : X → X ∗ is monotone, continuous and coercive, T ( X ) = X ∗ We prove that (6) has a solution u ∈ W 1p (Ω) for any h ∈ L p0 (Ω) provided that f satisfies, besides the standard regularity and growth assumptions, a coercivity condition of the form s f ( x, s) ≥ m |s| p , for some m > 0 and all ( x, s) ∈ Ω × R

A Variant of a Surjectivity Theorem

An Application to the p-Laplacian

Conclusions