On lower semicontinuity and relaxation (Q2731102)

The paper is addressed to the study of the lower semicontinuity properties of multiple integrals of the calculus of variations of the form NEWLINE\[NEWLINEF(\Omega,u)=\int_\Omega f(x,u(x),\nabla u(x))dx\quad u\in W^{1,1}_{\text{ loc}}(\Omega;{\mathbb{R}}^d),NEWLINE\]NEWLINE where \(\Omega\) is an open subset of \({\mathbb{R}}^N\).NEWLINENEWLINENEWLINEA first result deals with the case \(d=1\). It states that if \(f:\Omega\times{\mathbb{R}}\times{\mathbb{R}}^N\to[0,+\infty[\) is Borel, if \(f(x,u,\cdot)\) is convex, and if for every \((x_0,u_0)\in\Omega\times{\mathbb{R}}\) and \(\varepsilon>0\) there exists \(\delta>0\) such that NEWLINE\[NEWLINEf(x_0,u_0,z)-f(x,u,z)\leq\varepsilon(1+f(x,u,z)){\text{ for all }}(x,u)\in\Omega\times{\mathbb{R}}NEWLINE\]NEWLINE with NEWLINE\[NEWLINE|x-x_0|+|u-u_0|\leq\delta{\text{ and for all }}z\in{\mathbb{R}}^N,NEWLINE\]NEWLINE then for every \(u\in \text{BV}_{\text{ loc}}(\Omega;{\mathbb{R}})\) and for every \(\{u_n\}\subseteq W^{1,1}_{\text{ loc}}(\Omega;{\mathbb{R}})\) converging to \(u\) in \(L^1_{\text{ loc}}(\Omega;{\mathbb{R}})\) it results that NEWLINE\[NEWLINE\begin{multlined} \int_\Omega f(x,u(x),\nabla u(x))dx+\int_\Omega f^\infty(x,u(x),dC(u))\\ +\int_{S(u)\cap\Omega}\left(\int_{u^-(x)}^{u^+(x)}f^\infty(x,s,\nu_u)ds\right)d{\mathcal H}^{N-1}(x) \leq\liminf_{n\to\infty}\int_\Omega f(x,u_n(x),\nabla u_n(x))dx.\end{multlined}NEWLINE\]NEWLINE Here \(f^\infty\) is the recession function of \(f\), that is, \(f^\infty(x,u,z)=\limsup_{t\to\infty}f(x,u,tz)/t\), \(C(u)\) is the Cantor part of the measure \(Du\), \(u^+-u^-\) is the jump of \(u\) across the interface \(S(u)\), and \(\nu_u\) is the unit vector normal to \(S(u)\).NEWLINENEWLINENEWLINEIf, in addition to the previous assumptions, there exists \(C>0\) such that NEWLINE\[NEWLINEf(x,u,z)\leq C(1+|z|){\text{ for all }}(x,u,z)\in\Omega\times{\mathbb{R}}\times{\mathbb{R}}^N,NEWLINE\]NEWLINE it is also proved that the left hand side of the above inequality is the relaxed functional of \(F\) in the \(L^1_{\text{ loc}}(\Omega;{\mathbb{R}}^N)\)-topology on the whole \(\text{BV}(\Omega;{\mathbb{R}}^N)\).NEWLINENEWLINENEWLINESimilar statements are proved under different sets of assumptions of the type of those existing in the literature on the subject, extending the corresponding results.NEWLINENEWLINENEWLINELower semicontinuity and relaxation results of the same type are also proved when \(d>1\). NEWLINENEWLINENEWLINEThe main tool in the proofs of the paper is a blow-up method.