On the strong maximum principle (Q2758977)

This paper is concerned with the variational problem: NEWLINE\[NEWLINE\text{minimize }\int_\Omega f(|\nabla u|) \,dx \quad\text{on }u^0+ W_0^{1,1}(\Omega), \tag{1}NEWLINE\]NEWLINE where \(u^0\) is a given function in \(W^{1,1}(\Omega)\) and \(f\) is a nonnegative, extended valued, lower semicontinuous, and convex function on \(\mathbb R\) such that \(f(0)= 0\), and where \(\Omega\) is a domain in \(\mathbb R^N\). \(f\) is said to have the strong maximum principle property, if for every continuous nonnegative solution \(u\) of (1), \(u(x^0)= 0\) for some \(x^0\in \Omega\) implies \(u(x)\equiv 0\) on \(\Omega\). The main theorem is that \(f\) has the strong maximum principle property if and only if both conditions: (i) \(\partial f(0)= \{0\}\) and (ii) \((\partial f)^{-1}(0)= \{0\}\) hold true. Here \(\partial f\) denotes the subdifferential of \(f\), that is, \(\partial f(s)= \{a\in \mathbb R: f(t)\geq f(s)+ a(t-s)\) for all \(t\in \mathbb R\}\).